Domination in Graphs: Core Concepts 3031094956, 9783031094958

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Springer Monographs in Mathematics

Teresa W. Haynes Stephen T. Hedetniemi Michael A. Henning

Domination in Graphs: Core Concepts

Springer Monographs in Mathematics Editor-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea, International Centre for Mathematical Sciences, Edinburgh, UK Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NJ, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Teresa W. Haynes • Stephen T. Hedetniemi Michael A. Henning

Domination in Graphs: Core Concepts

Teresa W. Haynes Department of Mathematics and Statistics East Tennessee State University Johnson City, TN, USA

Stephen T. Hedetniemi School of Computing Clemson University Clemson, SC, USA

Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg, South Africa

Michael A. Henning Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg, South Africa

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-031-09496-5 (eBook) ISBN 978-3-031-09495-8 https://doi.org/10.1007/978-3-031-09496-5 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedication The authors dedicate this book to their coauthor and long-time good friend Pete Slater, who coauthored the predecessor of this volume, Fundamentals of Domination in Graphs, with Teresa and Steve.

Peter J. Slater (1946–2016)

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Teresa pays a special tribute to Ulysses Grant (Lit) Haynes. I love you, Dad. Steve pays a special tribute to Gerd H. Fricke (1946–2016), an exceptional mathematician and graph theorist, who made special contributions to domination in graphs, and in particular to the understanding of the concept of irredundance in graphs. Steve also pays a special tribute to Sandee, his wife of 43 years, with whom he has coauthored 78 papers. Michael pays a special tribute to his brother Paul Henning, for the countless lives he has saved as an Emergency Medicine Physician over the years.

Preface While concepts related to domination in graphs can be traced back to the Roman Empire in the fourth century AD and to the mid-1800s in connection with various chessboard problems, the mathematical concept of domination in graphs was first suggested by Kőnig in 1936, and then defined as a graph theoretical parameter by Berge in 1958. Domination in graphs experienced rapid growth from its introduction, resulting in over 1200 papers published on domination in graphs by the late 1990s. Much of the interest in domination theory in graphs is due to its applications in many areas of study, such as genetics, chemistry, computer communication networks, facility location, social networking, surveying, transporting hazardous materials, monitoring electrical power networks, school bus routing, voting, and several areas of mathematics, to name a few. Noting the need for a comprehensive survey of the literature on domination in graphs, in 1998 Haynes, Hedetniemi, and Slater published the first two books on domination, writing Fundamentals of Domination in Graphs (ISBN: 9780429157769) and editing Domination in Graphs: Advanced Topics (ISBN: 9781315141428). The explosive growth of this field has continued since 1998, and today more than 5000 papers have been published on domination in graphs, and the material in these two books is now more than 20 years old. Thus, we thought it was time for an update on the developments in domination theory since 1998. We also wanted to give a comprehensive treatment of only the major topics in domination. This coverage of domination, including the major results and updates, is in the form of three books: this book and its two companion books, Topics in Domination in Graphs (ISBN: 9783030511173) and Structures of Domination in Graphs (ISBN: 9783030588915), which we will call Books I, II, and III, respectively. This book, Domination in Graphs: Core Concepts, is limited to, as the title suggests, the most core concepts of domination in graphs: domination, total domination, and independent domination. It contains major results on these three types of domination, including a wide variety of proofs, both short and long, of selected results that illustrate many of the proof techniques used in domination theory. For the companion books, Books II and III, we invited leading researchers in domination theory to contribute chapters. Book II focuses on the most-studied types of domination that are not covered in Book I. Although well over 70 types of domination have been defined, Book II focuses on those that have received the most attention in the literature, and contains chapters on paired domination, connected domination, restrained domination, multiple domination, distance domination, dominating functions, fractional domination, Roman domination, rainbow domination, locating-domination, eternal and secure domination, global domination, stratified domination, and power domination. Book III is divided into three parts. The first part covers several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings, irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination, and domination in spectral graph theory. The second part contains chapters on domination in hypergraphs, chessboards, and digraphs and tournaments. vii

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The third part focuses on the development of algorithms and complexity of signed, minus, and majority domination, power domination, and alliances in graphs. The third part also includes a chapter on self-stabilizing algorithms for domination. This book (Book I) is intended as a reference resource for researchers and is written to reach the following audiences: first, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; second, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and third, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for master’s and doctoral theses topics. We also believe that this book provides a good basis for use in a seminar on either domination theory or domination algorithms and complexity, including the new algorithm paradigm of self-stabilizing domination algorithms. This book is intended as an in-depth introduction to domination in graphs, limited to its most core concepts of domination, total domination, and independent domination. We have therefore intentionally focused more on depth than breadth in Book I, and supplied several in-depth proofs for the reader to acquaint themselves with a tool box of proof techniques and methods with which to attack open problems in the field. We have identified many unsolved problems and open conjectures, which can be used as a launching pad for future researchers in the field. With the enormous literature that exists on domination in graphs and the dynamic nature of the subject, we were faced with the challenge of determining which topics to include and perhaps even more importantly which topics to exclude, even for the core concepts of domination, total domination, and independent domination. We have therefore been selective in the material included in this core domination book and wish to apologize in advance for omitting many important results and proofs due to space limitations. We assume that the reader is acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. However, since graph theory terminology sometimes varies, we provide a glossary as a reference source for the reader regarding terminology and notation adopted in this book. Assuming that the reader has some familiarity with graph theory, this book is selfcontained as we include the terminology and definitions involving domination in the glossary in Appendix A. The material in this book has been organized into 18 chapters, an epilogue, and three appendices. It contains an extensive bibliography of more than 900 references, which we have cited throughout the book. A brief summary of the material covered in each chapter is presented below. Chapter 1 In the Beginning: Roots of Domination in Graphs discusses the many origins, both historical and mathematical, of domination in graphs, dating as far back as the Roman Empire in the fourth century AD under Emperor Constantine. Chapter 2 Fundamentals of Domination discusses how it is that the domination number, total domination number, and independent domination number can be defined

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in a variety of equivalent ways, each of which suggests natural generalizations of these three types of domination. Chapter 3 Complexity and Algorithms for Domination in Graphs provides an overview of the core results on NP-completeness and algorithms for domination, total domination, and independent domination in graphs. It presents NP-completeness proofs for each type of domination, when restricted to several subclasses of graphs, and provides linear algorithms for computing each type of domination on trees. Chapter 4 General Bounds presents some of the more basic lower and upper bounds on the domination, total domination, and independent domination numbers of graphs. Chapter 5 Domination in Trees presents a wide variety of domination results for the class of trees, including lower and upper bounds, bounds in terms of the number of leaves in a tree, the Slater lower bound for trees, vertices in all or no minimum dominating sets in a tree, trees in which every vertex is a member of some minimum dominating set, trees having unique minimum dominating sets, trees in which the domination number equals the independent domination number, and trees in which the domination number equals the total domination number. Chapter 6 Upper Bounds in Terms of Minimum Degree presents results which establish upper bounds on the core domination numbers in terms of the order of a graph and the minimum degree of a vertex in the graph, where for the domination number and total domination number the minimum degree ranges between one and six. Chapter 7 Probabilistic Bounds and Domination in Random Graphs presents probabilistic bounds for the core domination numbers of a graph in terms of its order and minimum degree, and also considers bounds for the domination numbers of random graphs. It covers the basic questions of the probability that a randomly chosen set 𝑆 of vertices in a graph 𝐺 is a dominating set of one of the three basic types, if each vertex in the graph is chosen to be in the set 𝑆 with a given probability. Chapter 8 Bounds in Terms of Size discusses how the number of edges of a graph affects the values of the core domination numbers. Chapter 9 Efficient Domination in Graphs considers classes of graphs that have dominating or total dominating sets 𝑆 in which specified vertices are adjacent to exactly one vertex in 𝑆. Included in these classes of graphs are certain circulants, Cayley graphs, grid graphs, cylindrical graphs, toroidal graphs, prisms, Möbius ladders, and lexicographic graphs. Also included is a section on NP-completeness results for graphs having an efficient dominating set. Chapter 10 Domination and Forbidden Subgraphs presents bounds on the three core domination numbers in classes of graphs which have certain subgraph restrictions, such as bipartite (no odd cycles), cubic (every vertex has degree three), and claw-free (no vertex has three neighbors, no two of which are adjacent). Chapter 11 Domination in Planar Graphs covers domination and total domination in planar triangulations, outerplanar graphs, and in planar graphs having small diameter. Results on independent domination in planar graphs are also presented, including bipartite, cubic, and minimum diameter planar graphs.

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Chapter 12 Domination Partitions covers vertex partitions 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of a graph 𝐺 such that each set 𝑉𝑖 is a dominating set. Partitions into total and independent dominating sets are also considered. The nine possible ways of partitioning the vertices of a graph into two sets, say 𝑉1 and 𝑉2 , such that 𝑉1 is one of the three types of domination and 𝑉2 is one of the three types of domination are also considered. Chapter 13 Domination Critical and Stable Graphs presents the classes of graphs whose types of domination numbers change upon the removal of any vertex, the removal of any edge, or the addition of any edge. It also considers the classes of graphs whose domination numbers do not change, regardless of which vertex or edge is removed or which new edge is added to the graph. Chapter 14 Upper Domination Parameters covers the upper domination number, the upper total domination number, and the independence number, that is, the maximum cardinalities of a minimal type of dominating set. Since the independence number, that is, the maximum cardinality of an independent set, is very well-studied in the literature, the focus of this chapter is mainly on the upper domination and upper total domination numbers, although several important results on the independence number are presented. Chapter 15 Relating the Core Parameters presents relationships, inequalities, and bounds that exist between the three types of domination numbers, for example bounds on the ratio of the independent domination number to the domination number and the total domination number to the domination number. Also considered are the classes of graphs in which two of these domination numbers are always equal, for example the classes of graphs in which the independence number equals the upper domination number. Chapter 16 Nordhaus-Gaddum Bounds discusses bounds on the sum and product of the domination numbers of a graph 𝐺 and its complement 𝐺. Bounds on the sum and product for total domination and independent domination numbers are also presented. Chapter 17 Domination in Grids and Hypercubes presents results on domination, total domination, and independent domination in grids, which are chessboard-like graphs. There is also a brief discussion of cylinders (chessboards with column wrap-arounds) and tori (chessboards with both column and row wrap-arounds). The chapter concludes with domination in hypercubes. Chapter 18 Domination and Vizing’s Conjecture provides an overview of the most well-known conjecture in domination theory, that the domination number of the Cartesian product of two graphs 𝐺 and 𝐻 is greater than or equal to the product of the domination number of 𝐺 and the domination number of 𝐻. Similar conjectures are also discussed for all six core domination numbers, including the lower and upper domination, total domination, and independent domination numbers. The authors would like to thank Elizabeth Loew, the Executive Editor, Mathematics at Springer, and Saveetha Balasundaram, the Project Coordinator (Books) for Springer Nature, for their continued support and encouragement, not only in producing this book but throughout the production of Books II and III. We are especially grateful to them for their patience in waiting for this manuscript from the date the contract was signed, and for their cooperation in all aspects of the production of this book.

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The authors thank Jason Hedetniemi for proofreading several chapters of this book. In addition, the authors thank the five Springer reviewers of an early version of this book for their many helpful suggestions, which lead to a much improved book. We extend a hearty and sincere thanks to Dr Werner Gründlingh for his tireless and outstanding efforts in typesetting the book and for producing superb graphics. We very much appreciate his expertise and skills, and the enormous time sacrifice he has made in assisting us during the typesetting process. Teresa Haynes would like to thank East Tennessee State University and the University of Johannesburg for their support during the writing of this book. In particular, she extends a special acknowledgement to the staff at Sherrod Library for their assistance. She also thanks Pamela Morgan for her friendship and helpful encouragement for this project. Stephen Hedetniemi would like to thank Clemson University and the Clemson University Library, the School of Computing, and the Emeritus College for their support in producing Books I, II, and III. He also thanks his wife, Sandee, and children Traci, Laura, Kevin, and Jason for their encouragement in writing this book. Michael Henning expresses his sincere thanks to the University of Johannesburg for their continued support and for creating a conducive research environment to enable him to work on the book. Special thanks are due to his wife Anne, son John, and daughter Alicen, for their much appreciated support and encouragement throughout the writing of the book. We have tried to eliminate errors, but surely several remain. We welcome any comments or corrections the reader may have. A list of typographical errors, corrections, and suggestions can be sent to any of our email addresses below. East Tennessee State University, USA and University of Johannesburg, South Africa Clemson University, USA University of Johannesburg, South Africa

Teresa W. Haynes e-mail: [email protected] Stephen T. Hedetniemi e-mail: [email protected] Michael A. Henning e-mail: [email protected]

Contents 1 In the Beginning: Roots of Domination in Graphs 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Defensive and Offensive Strategies of the Roman Empire 1.3.2 Chaturanga . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Eight Queens Problem . . . . . . . . . . . . . . . . . . 1.3.4 Five Queens Problem . . . . . . . . . . . . . . . . . . . 1.3.5 Queens Independent Domination Problem . . . . . . . . 1.3.6 Queens Total Domination Problem . . . . . . . . . . . . 1.3.7 Generalizations to Other Chess Pieces . . . . . . . . . . 1.4 Application Driven Origins . . . . . . . . . . . . . . . . . . . . 1.4.1 Radio Broadcasting . . . . . . . . . . . . . . . . . . . . 1.4.2 Computer Communication Networks . . . . . . . . . . 1.4.3 Sets of Representatives . . . . . . . . . . . . . . . . . . 1.4.4 School Bus Routing and Bus Stop Selection . . . . . . . 1.4.5 Electrical Power Domination . . . . . . . . . . . . . . . 1.4.6 Influence in Social Networks . . . . . . . . . . . . . . . 1.4.7 Topographic Maps . . . . . . . . . . . . . . . . . . . . 1.4.8 Transporting Hazardous Materials . . . . . . . . . . . . 1.5 Early Chronology of Domination in Graph Theory . . . . . . . . 2

Fundamentals of Domination 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Core Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Independent Sets . . . . . . . . . . . . . . . . . . . 2.2.2 Dominating Sets . . . . . . . . . . . . . . . . . . . 2.2.3 Irredundant Sets . . . . . . . . . . . . . . . . . . . 2.3 Parameters Suggested by the Definition of a Dominating Set 2.3.1 Total Dominating Sets . . . . . . . . . . . . . . . . 2.3.2 𝑘-Dominating Sets . . . . . . . . . . . . . . . . . . 2.3.3 𝐻-forming Dominating Sets . . . . . . . . . . . . . 2.3.4 Perfect and Efficient Dominating Sets . . . . . . . .

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2.3.5 Distance-𝑘 Dominating Sets . . . . . . 2.3.6 Fractional Domination . . . . . . . . . Equivalent Formulations of Domination . . . . 2.4.1 Pendant Edges in Spanning Forests . . 2.4.2 Enclaveless Sets . . . . . . . . . . . . 2.4.3 Spanning Star Partitions . . . . . . . . 2.4.4 Non-dominating Partitions . . . . . . . 2.4.5 Total Domination and Splitting Graphs 2.4.6 Dominating Sets and (1, 4 : 3)-Sets . . . Domination in Terms of Perfection in Graphs . Ore’s Lemmas and Their Implications . . . . .

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Complexity and Algorithms for Domination in Graphs 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Brief Review of NP-Completeness . . . . . . . . . . . . . . . . 3.3 NP-Completeness of Domination, Independent Domination, and Total Domination . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 NP-Completeness Results for Arbitrary Graphs . . . . . 3.3.2 NP-Completeness Results for Bipartite Graphs . . . . . 3.3.3 Summary of Complexity Results for Graph Families . . 3.4 A Representative Sample of Domination Algorithms for Trees . 3.4.1 Minimum Dominating Set . . . . . . . . . . . . . . . . 3.4.2 Minimum Independent Dominating Set . . . . . . . . . 3.4.3 Minimum Total Dominating Set . . . . . . . . . . . . . 3.5 Early Domination Algorithms and NP-Completeness Results . . 3.6 Other Sources for Domination Algorithms and Complexity . . .

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General Bounds 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Domination and Maximum Degree . . . . . . . . . . . . . . . . 4.2.1 Domination Number and Maximum Degree . . . . . . . 4.2.2 Total Domination Number and Maximum Degree . . . . 4.2.3 Independent Domination Number and Maximum Degree 4.3 Domination and Order . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Domination Number and Order . . . . . . . . . . . . . 4.3.2 Total Domination Number and Order . . . . . . . . . . 4.3.3 Independent Domination Number and Order . . . . . . . 4.4 Basic Relationships Among Core Parameters . . . . . . . . . . 4.5 Domination and Distance . . . . . . . . . . . . . . . . . . . . . 4.6 Domination and Packing . . . . . . . . . . . . . . . . . . . . . 4.7 Gallai Type Theorems . . . . . . . . . . . . . . . . . . . . . . . 4.8 Domination and Matching . . . . . . . . . . . . . . . . . . . .

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5 Domination in Trees 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Domination in Trees . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Domination Bounds in Trees . . . . . . . . . . . . . . . . . 5.2.2 Domination Lower Bounds Involving the Number of Leaves 5.2.3 Slater Lower Bound on the Domination Number . . . . . . 5.2.4 Vertices in All or No Minimum Dominating Sets . . . . . . 5.2.5 Domination and Packing in Trees . . . . . . . . . . . . . . 5.3 Total Domination in Trees . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Total Domination Bounds in Trees . . . . . . . . . . . . . . 5.3.2 Total Domination Bounds Involving the Number of Leaves . 5.3.3 Vertices in All or No Minimum Total Dominating Sets in Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Unique Minimum Total Dominating Sets in Trees . . . . . . 5.3.5 Total Domination and Open Packing in Trees . . . . . . . . 5.4 Independent Domination in Trees . . . . . . . . . . . . . . . . . . . 5.4.1 Independent Domination Bounds in Trees . . . . . . . . . . 5.4.2 Unique Minimum Independent Dominating Sets in Trees . . 5.5 Equality of Domination Parameters . . . . . . . . . . . . . . . . . . 5.5.1 (𝛾, 𝑖)-trees . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 (𝛾, 𝛾t )-trees . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Upper Bounds in Terms of Minimum Degree 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bounds on the Domination Number . . . . . . . . . . 6.2.1 Minimum Degree One . . . . . . . . . . . . . 6.2.2 Minimum Degree Two . . . . . . . . . . . . . 6.2.3 Minimum Degree Three . . . . . . . . . . . . 6.2.4 Minimum Degree Four . . . . . . . . . . . . . 6.2.5 Minimum Degree Five . . . . . . . . . . . . . 6.2.6 Minimum Degree Six . . . . . . . . . . . . . . 6.2.7 Minimum Degree Seven or Larger . . . . . . . 6.3 Bounds on the Total Domination Number . . . . . . . 6.3.1 Minimum Degree One . . . . . . . . . . . . . 6.3.2 Minimum Degree Two . . . . . . . . . . . . . 6.3.3 Minimum Degree Three . . . . . . . . . . . . 6.3.4 An Interplay with Transversals in Hypergraphs 6.3.5 Minimum Degree Four . . . . . . . . . . . . . 6.3.6 Minimum Degree Five . . . . . . . . . . . . . 6.3.7 Minimum Degree Six . . . . . . . . . . . . . . 6.3.8 A Heuristic Bound . . . . . . . . . . . . . . . 6.4 Bounds on the Independent Domination Number . . . 6.4.1 Minimum Degree One . . . . . . . . . . . . . 6.4.2 Arbitrary Minimum Degree . . . . . . . . . .

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6.4.3 Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . 200 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Probabilistic Bounds and Domination in Random Graphs 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Probabilistic Bounds . . . . . . . . . . . . . . . . . . 7.3 Domination in Random Graphs . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . .

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9 Efficient Domination in Graphs 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Efficient Dominating Sets . . . . . . . . . . . . . 9.1.2 Efficient Total Dominating Sets . . . . . . . . . . 9.1.3 Perfect Dominating Sets . . . . . . . . . . . . . . 9.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . 9.2 Efficient Domination . . . . . . . . . . . . . . . . . . . . 9.2.1 Efficient Graphs . . . . . . . . . . . . . . . . . . 9.2.2 Efficient Grid Graphs and Efficient Toroidal Graphs 9.2.3 Efficient Cube-connected Cycles . . . . . . . . . . 9.2.4 Efficient Vertex-transitive Graphs . . . . . . . . . 9.2.5 Efficient Cayley Graphs . . . . . . . . . . . . . . 9.2.6 Efficient Circulant Graphs . . . . . . . . . . . . . 9.2.7 Efficient Graphs with Efficient Complements . . . 9.3 Efficient Total Domination . . . . . . . . . . . . . . . . . 9.3.1 Total Efficient Trees . . . . . . . . . . . . . . . . 9.3.2 Total Efficient Grid Graphs . . . . . . . . . . . . . 9.3.3 Total Efficient Cylindrical Graphs . . . . . . . . . 9.3.4 Total Efficient Toroidal Graphs . . . . . . . . . . . 9.3.5 Total Efficient Product Graphs . . . . . . . . . . . 9.3.6 Total Efficient Circulant Graphs . . . . . . . . . . 9.4 Algorithms and Complexity of Efficient Domination . . . .

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259 259 259 260 261 261 262 263 264 266 267 268 270 272 272 273 274 275 276 280 282 282

10 Domination and Forbidden Subgraphs 10.1 Introduction . . . . . . . . . . . . . . . . 10.2 Domination and Forbidden Cycles . . . . 10.2.1 Domination Number . . . . . . . 10.2.2 Total Domination Number . . . . 10.2.3 Independent Domination Number 10.3 Domination in Claw-free Graphs . . . . .

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291 291 291 291 299 310 314

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Bounds in Terms of Size 8.1 Introduction . . . . . . . . . . . . 8.2 Domination and Size . . . . . . . 8.3 Total Domination and Size . . . . 8.4 Independent Domination and Size 8.5 Summary . . . . . . . . . . . . .

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10.3.1 Domination and Independent Domination Numbers . . . . . 314 10.3.2 Total Domination Number . . . . . . . . . . . . . . . . . . 318 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11 Domination in Planar Graphs 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Domination in Planar Graphs . . . . . . . . . . . . . . . . . . . 11.2.1 Domination in Planar Triangulations . . . . . . . . . . . 11.2.2 Domination in Outerplanar Graphs . . . . . . . . . . . . 11.2.3 Domination in Planar Graphs with Small Diameter . . . 11.3 Total Domination in Planar Graphs . . . . . . . . . . . . . . . . 11.3.1 Total Domination in Outerplanar Graphs . . . . . . . . 11.3.2 Total Domination in Planar Graphs with Small Diameter 11.4 Independent Domination in Planar Graphs . . . . . . . . . . . .

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325 325 326 326 333 337 341 341 344 347

12 Domination Partitions 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 12.2 Domatic Numbers . . . . . . . . . . . . . . . . . . . . 12.2.1 Domatically Full Graphs . . . . . . . . . . . . 12.2.2 Lower Bounds . . . . . . . . . . . . . . . . . 12.2.3 Generalizations of the Domatic Number . . . . 12.3 Idomatic Number . . . . . . . . . . . . . . . . . . . . 12.4 Total Domatic Number . . . . . . . . . . . . . . . . . 12.4.1 Total Domatic Number in Graph Families . . . 12.4.2 Total Domatic Number in Planar Graphs . . . . 12.5 Results of Zelinka on Domatic Numbers . . . . . . . . 12.6 Dominating Bipartitions of Graphs . . . . . . . . . . . 12.6.1 Dominating and Total Dominating Set Partition 12.6.2 Total and Independent Dominating Set Partition 12.6.3 Partitions into Two Total Dominating Sets . . .

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353 353 353 356 358 359 361 363 365 365 366 368 368 370 370

13 Domination Critical and Stable Graphs 13.1 Introduction . . . . . . . . . . . . . . . . . . 13.2 The Six Graph Families . . . . . . . . . . . . 13.2.1 CVR, CER, and CEA . . . . . . . . . 13.2.2 UVR, UER, and UEA . . . . . . . . 13.2.3 Relationships Among the Families . . 13.3 Domination Vertex-Critical Graphs (CVR) . . 13.3.1 Vertex-Critical Graphs . . . . . . . . 13.3.2 3-Vertex-Critical Graphs . . . . . . . 13.4 Domination Edge-Critical Graphs (CEA) . . 13.4.1 Properties of 𝑘-Edge-Critical Graphs 13.4.2 3-Edge-Critical Graphs . . . . . . . . 13.5 Total Domination Edge-Critical Graphs . . . 13.5.1 𝑘 t -Edge-Critical Graphs . . . . . . .

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381 381 383 384 385 386 393 393 395 397 397 398 401 402

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13.5.2 3t -Edge-Critical Graphs . . . . . . . . . . . . . . . . . . . 403

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411 411 412 413 414 419 426 428

15 Relating the Core Parameters 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Well-covered and Well-dominated Graphs . . . . . . . . . 15.2.1 Well-covered Graphs . . . . . . . . . . . . . . . . 15.2.2 Well-dominated Graphs . . . . . . . . . . . . . . 15.2.3 Well-total Dominated Graphs . . . . . . . . . . . 15.3 Domination Versus Independent Domination . . . . . . . . 15.3.1 (𝛾, 𝑖)-graphs . . . . . . . . . . . . . . . . . . . . 15.3.2 Domination Perfect Graphs . . . . . . . . . . . . . 15.3.3 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.4 Domination Versus Total Domination . . . . . . . . . . . 15.4.1 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.4.2 Claw-free Graphs . . . . . . . . . . . . . . . . . . 15.5 Upper Domination Versus Independence . . . . . . . . . . 15.6 Upper Domination Versus Upper Total Domination . . . . 15.6.1 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.7 Independence Versus Total Domination . . . . . . . . . . 15.7.1 Independent Domination Versus Total Domination 15.7.2 Independence Versus Total Domination . . . . . . 15.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

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435 435 436 437 438 440 440 441 442 445 451 451 454 454 458 461 463 463 465 465

16 Nordhaus-Gaddum Bounds 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Domination Number . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Minimum Degree One . . . . . . . . . . . . . . . . . 16.2.2 Minimum Degree Two . . . . . . . . . . . . . . . . . 16.2.3 Minimum Degree Three . . . . . . . . . . . . . . . . 16.2.4 Minimum Degree Four . . . . . . . . . . . . . . . . . 16.2.5 Minimum Degree Five . . . . . . . . . . . . . . . . . 16.2.6 Minimum Degree Six . . . . . . . . . . . . . . . . . . 16.2.7 Summary of Bounds with Specified Minimum Degree 16.2.8 Multiple Factors . . . . . . . . . . . . . . . . . . . . 16.2.9 Relative Complement . . . . . . . . . . . . . . . . . . 16.3 Total Domination Number . . . . . . . . . . . . . . . . . . .

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467 467 467 470 474 476 478 478 479 480 480 484 485

14 Upper Domination Parameters 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 14.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . 14.2.1 Upper Bounds in Terms of Minimum Degree 14.2.2 Upper Bounds in Regular Graphs . . . . . . 14.2.3 Upper Bounds in Claw-free Graphs . . . . . 14.3 Upper Domination Number . . . . . . . . . . . . . . 14.4 Independence Number . . . . . . . . . . . . . . . .

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16.4 Independent Domination Number . . . . . . . . . . . 16.5 Upper Domination Parameters . . . . . . . . . . . . . 16.5.1 Upper Domination and Independence Numbers 16.5.2 Upper Total Domination Number . . . . . . . 16.6 Domatic Numbers of 𝐺 and 𝐺 . . . . . . . . . . . . .

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505 505 505 506 510 512 520

18 Domination and Vizing’s Conjecture 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Vizing’s Conjecture for the Domination Number . . . . . 18.2.1 A Framework . . . . . . . . . . . . . . . . . . . 18.2.2 Key Preliminary Lemmas . . . . . . . . . . . . 18.2.3 Classical Results Related to Vizing’s Conjecture 18.3 Total Domination Number . . . . . . . . . . . . . . . . 18.4 Independent Domination Number . . . . . . . . . . . . 18.5 Independence Number . . . . . . . . . . . . . . . . . . 18.6 Upper Domination Number . . . . . . . . . . . . . . . . 18.7 Upper Total Domination Number . . . . . . . . . . . . .

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525 525 525 527 531 533 537 539 540 541 546

17 Domination in Grids and Hypercubes 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . 17.2 Domination in Grids . . . . . . . . . . . . . . . . 17.2.1 Domination Numbers of Grids . . . . . . . 17.2.2 Independent Domination Numbers of Grids 17.2.3 Total Domination Numbers of Grids . . . . 17.3 Domination in Hypercubes . . . . . . . . . . . . .

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Epilogue A Glossary A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Basic Graph Theory Definitions . . . . . . . . . . . . . . . . . . A.2.1 Basic Numbers . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Common Types of Graphs . . . . . . . . . . . . . . . . . A.2.3 Graph Constructions . . . . . . . . . . . . . . . . . . . . A.3 Graph Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Connectivity and Subgraph Numbers . . . . . . . . . . . A.3.2 Distance Numbers . . . . . . . . . . . . . . . . . . . . . A.3.3 Covering, Packing, Independence, and Matching Numbers A.3.4 Core Domination Numbers . . . . . . . . . . . . . . . . . A.3.5 Domatic Partitions . . . . . . . . . . . . . . . . . . . . . A.3.6 Perfect and Efficient Dominating Sets . . . . . . . . . . . A.3.7 Enclaveless Sets . . . . . . . . . . . . . . . . . . . . . . A.3.8 Grid Graphs . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Hypergraph Terminology and Concepts . . . . . . . . . . . . . . B Books Containing Information on Domination in Graphs

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C Surveys Containing Information on Domination in Graphs

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Bibliography

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Index 623 Symbol index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

Chapter 1

In the Beginning: Roots of Domination in Graphs 1.1 Introduction While domination in graphs was first formally defined by Berge in 1958, the roots of domination can be traced back to defense strategies used by the Roman Empire in the fourth century AD, to a precursor of the game of chess in India in the sixth century AD, and later in the mid-to-late 1800s, to a variety of chess problems. Other sources of domination can be found in a wide array of real-world areas such as radio broadcasting, computer communication networks, systems of distinct representatives, school bus routing, electrical power networks, influence in social networks, surveying, resource allocation, and even transporting hazardous materials. In the 1900s, a variety of international researchers began to develop the mathematical foundations of domination in graphs, including the British mathematician, lawyer, and fellow at Trinity College Cambridge, W.W. Rouse Ball; the Hungarian mathematician who wrote the first book on graph theory, Dénes Kőnig; the English mathematician and statistician, Patrick Michael Grundy; the Hungarian-American mathematician, physicist, computer scientist, and engineer, John von Neumann; the German-American economist, Oskar Morgenstern; the French mathematician recognized as one of the founders of graph theory, Claude Berge; the Hungarian graph theorist Tibor Gallai; the Norwegian-American mathematician who worked in ring theory, Galois theory, and graph theory, Øystein Ore; the Soviet and Ukrainian graph theorist, Vadim Vizing; the Finnish mathematician, Juho Nieminen; and the Canadian graph theorists, Amram Meir, John Moon, and E.J. Cockayne. In this chapter, we discuss the many origins, both historical and mathematical, of domination in graphs and highlight some of the most significant contributions of these mathematicians to the theory of domination up to the year 1998 when the first two books on domination in graphs were produced by the American graph theorists, Teresa Haynes, Stephen Hedetniemi, and Peter Slater [416, 417]. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_1

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2

Chapter 1. In the Beginning: Roots of Domination in Graphs

Before delving into the roots of domination in graphs, we give some basic definitions and notation in Section 1.2 that will be used throughout the book. To avoid repeating terminology in every chapter, we also provide a glossary in Appendix A including these basic terms and other definitions and refer the reader to it for terminology not defined on the spot.

1.2

Basic Terminology

A graph 𝐺 = (𝑉, 𝐸) consists of a finite nonempty set 𝑉 (𝐺) of objects called vertices together with a possibly empty set 𝐸 (𝐺) of 2-element subsets of 𝑉 (𝐺) called edges. Throughout, unless otherwise stated, the graphs in this book are simple graphs with no loops or multiple edges and 𝐺 is a graph with vertex set 𝑉 and edge set 𝐸. The number of vertices 𝑛 = |𝑉 | is called the order of 𝐺 and the number of edges 𝑚 = |𝐸 | is the size of 𝐺. An edge {𝑢, 𝑣} is denoted by 𝑢𝑣. If 𝑢𝑣 ∈ 𝐸, then 𝑢 and 𝑣 are adjacent vertices. The vertex 𝑢 (respectively, 𝑣) and edge 𝑢𝑣 are said to be incident to each other. Two distinct edges are adjacent if they are incident to a common vertex. The graph consisting of a single vertex is called the trivial graph; a nontrivial graph has order 𝑛 ≥ 2. Given a graph 𝐺 = (𝑉, 𝐸), the complement 𝐺 of 𝐺 is the graph 𝐺 = (𝑉, 𝐸), where 𝑢𝑣 ∈ 𝐸 if and only if 𝑢𝑣 ∉ 𝐸. The complete graph 𝐾𝑛 is a graph of order 𝑛 in which every two vertices are adjacent, while its complement 𝐾 𝑛 is an empty graph, that is, a graph on 𝑛 vertices with no edges. Note that 𝐾1 is the trivial graph. The open neighborhood of a vertex 𝑣 ∈ 𝑉 is the set N𝐺 (𝑣) = {𝑢 : 𝑢𝑣 ∈ 𝐸 } of vertices adjacent to 𝑣, called the neighbors of 𝑣, and its closed neighborhood is the set N𝐺 [𝑣] = N Ð𝐺 (𝑣) ∪ {𝑣}. The open neighborhood of a set 𝑆 ⊆ 𝑉 of vertices is N𝐺 (𝑆) = 𝑣 ∈𝑆 N𝐺 (𝑣), while the closed neighborhood of a set 𝑆 is Ð the set N𝐺 [𝑆] = 𝑣 ∈𝑆 N𝐺 [𝑣]. The degree of a vertex 𝑣 is deg𝐺 (𝑣) = |N𝐺 (𝑣)|. If the graph 𝐺 is clear from the context, then we omit the 𝐺 subscript in the above expressions. A vertex 𝑣 ∈ 𝑉 is called an isolated vertex if deg(𝑣) = 0, and is called a leaf if deg(𝑣) = 1. In a graph 𝐺 of order 𝑛, a vertex 𝑣 for which deg(𝑣) = 𝑛 − 1 is called a dominating vertex. A graph 𝐺 is called 𝑘-regular if every vertex 𝑣 ∈ 𝑉 has deg(𝑣) = 𝑘. We say that a graph is isolate-free if it has no isolated vertices. The largest degree among the vertices of 𝐺 is the maximum degree Δ(𝐺) and the smallest degree is the minimum degree 𝛿(𝐺). A graph 𝐺 ′ = (𝑉 ′ , 𝐸 ′ ) is a subgraph of a graph 𝐺 = (𝑉, 𝐸) if 𝑉 ′ ⊆ 𝑉 and 𝐸 ′ ⊆ 𝐸 and 𝐺 ′ is a spanning subgraph of 𝐺 if 𝑉 ′ = 𝑉. For a nonempty subset 𝑆 ⊆ 𝑉, the subgraph 𝐺 [𝑆] of 𝐺 induced by 𝑆 has 𝑆 as its vertex set and two vertices 𝑢 and 𝑣 are adjacent in 𝐺 [𝑆] if and only if 𝑢 and 𝑣 are adjacent in 𝐺. A clique is a complete subgraph. A set 𝑆 of vertices of a graph 𝐺 is a dominating set if every vertex in 𝑉 \ 𝑆 has a neighbor in 𝑆, that is, N[𝑆] = 𝑉. The domination number 𝛾(𝐺) equals the minimum cardinality of a dominating set of 𝐺 and a dominating set with cardinality 𝛾(𝐺) is called a 𝛾-set of 𝐺.

Section 1.2. Basic Terminology

3

A set 𝑆 of vertices of an isolate-free graph 𝐺 is a total dominating set, abbreviated TD-set, if every vertex in 𝑉 is adjacent to at least one vertex in 𝑆. Thus, a subset 𝑆 ⊆ 𝑉 is a TD-set of 𝐺 if N(𝑆) = 𝑉. Note that since every vertex must have a neighbor in 𝑆, total domination is only defined for isolate-free graphs. The total domination number 𝛾t (𝐺) equals the minimum cardinality of a TD-set of 𝐺 and a TD-set with cardinality 𝛾t (𝐺) is called a 𝛾t -set of 𝐺. A minimal dominating set in a graph 𝐺 is a dominating set that contains no dominating set of 𝐺 as a proper subset, and a minimal TD-set of 𝐺 is a TD-set that contains no TD-set of 𝐺 as a proper subset. The upper domination number Γ(𝐺) equals the maximum cardinality of a minimal dominating set in 𝐺. Similarly, the upper total domination number Γt (𝐺) equals the maximum cardinality of a minimal TD-set of 𝐺. A set 𝑆 ⊆ 𝑉 is independent if no two vertices in 𝑆 are adjacent in 𝐺, and an independent set 𝑆 is called maximal if no proper superset of 𝑆 is independent. The vertex independence number, or just independence number, 𝛼(𝐺) equals the maximum cardinality of an independent set of 𝐺. A set 𝑆 ⊆ 𝑉 is an independent dominating set, abbreviated ID-set, if it is both independent and dominating. The independent domination number 𝑖(𝐺) equals the minimum cardinality of any ID-set of 𝐺 and an ID-set with cardinality 𝑖(𝐺) is called an 𝑖-set of 𝐺. We note that 𝑖(𝐺) is the minimum cardinality of any maximal independent set of 𝐺. A set 𝑀 ⊆ 𝐸 is independent if no two edges in 𝑀 are adjacent in 𝐺, and a set of independent edges is called a matching. The matching number 𝛼′ (𝐺) equals the maximum number of edges in a matching of 𝐺. A set 𝑆 ⊆ 𝑉 is a packing in 𝐺 if for any two vertices 𝑢, 𝑣 ∈ 𝑆, N[𝑢] ∩ N[𝑣] = ∅. The packing number 𝜌(𝐺) equals the maximum cardinality of a packing of 𝐺. A vertex cover is a set 𝑆 of vertices such that every edge in 𝐸 is incident to at least one vertex in 𝑆. The vertex covering number 𝛽(𝐺), also denoted 𝜏(𝐺), equals the minimum cardinality of a vertex cover of 𝐺. An edge cover is a set 𝐹 of edges such that every vertex in 𝑉 is incident to at least one edge in 𝐹. The edge covering number 𝛽′ (𝐺) equals the minimum cardinality of an edge cover of 𝐺. These concepts will be explored in more detail in Chapters 2 and 4. A graph 𝐺 is bipartite if its vertex set 𝑉 can be partitioned into two sets 𝑋 and 𝑌 such that every edge in 𝐺 joins a vertex in 𝑋 and a vertex in 𝑌 . The sets 𝑋 and 𝑌 are called the partite sets of 𝐺. We note that the partite sets of a bipartite graph are independent sets. The complete bipartite graph 𝐾𝑟 ,𝑠 is a bipartite graph with partite sets 𝑋 and 𝑌 , where |𝑋 | = 𝑟, |𝑌 | = 𝑠, and every vertex in 𝑋 is adjacent to every vertex in 𝑌 . The union 𝐺 = 𝐺 1 ∪ 𝐺 2 of two graphs 𝐺 1 and 𝐺 2 has vertex set 𝑉 (𝐺) = 𝑉 (𝐺 1 ) ∪ 𝑉 (𝐺 2 ) and edge set 𝐸 (𝐺 1 ) ∪ 𝐸 (𝐺 2 ). If 𝐺 is a union of 𝑘 copies of a graph 𝐹, we write 𝐺 = 𝑘 𝐹. For an integer 𝑘 ≥ 1, let [𝑘] = {1, 2, . . . , 𝑘 } and [𝑘] 0 = [𝑘] ∪ {0} = {0, 1, . . . , 𝑘 }. A walk in a graph 𝐺 from a vertex 𝑢 to a vertex 𝑣, called a (𝑢, 𝑣)-walk, is a finite alternating sequence of vertices and edges, starting with the vertex 𝑢 and ending with the vertex 𝑣, in which each edge of the sequence joins the vertex that precedes it in the sequence to the vertex that follows it in the sequence. A (𝑢, 𝑣)-trail is a (𝑢, 𝑣)-walk containing no repeated edges and a (𝑢, 𝑣)-path is a (𝑢, 𝑣)-walk

Chapter 1. In the Beginning: Roots of Domination in Graphs

4

containing no repeated vertices. A cycle is a closed (𝑢, 𝑣)-trail. The length of a path (respectively, cycle) equals the number of edges in the path (respectively, cycle). A graph of order 𝑛 which itself is a path is called the path 𝑃𝑛 . Thus, the path 𝑃𝑛 is the graph of order 𝑛 whose vertices can be labeled 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 and whose edges are 𝑣 𝑖 𝑣 𝑖+1 for 𝑖 ∈ [𝑛 − 1]. For an integer 𝑛 ≥ 3, the cycle 𝐶𝑛 is the graph of order 𝑛 whose vertices can be labeled 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 and whose edges are 𝑣 1 𝑣 𝑛 and 𝑣 𝑖 𝑣 𝑖+1 for 𝑖 ∈ [𝑛 − 1]. The cycle 𝐶𝑛 is also referred to as an 𝑛-cycle. We write 𝑃𝑛 : 𝑣 1 𝑣 2 . . . 𝑣 𝑛 and 𝐶𝑛 : 𝑣 1 𝑣 2 . . . 𝑣 𝑛 𝑣 1 to denote the paths and cycles, respectively, with vertex sequence (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ). Two vertices 𝑢 and 𝑣 are connected if there is a (𝑢, 𝑣)-path in 𝐺, and a graph 𝐺 is said to be connected if every two of vertices in 𝑉 are connected. The distance 𝑑 (𝑢, 𝑣) = 𝑑𝐺 (𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 is the minimum length of a (𝑢, 𝑣)-path in 𝐺. The eccentricity ecc(𝑣) = ecc𝐺 (𝑣) of a vertex 𝑣 in a connected graph 𝐺 is themaximum of the distances from 𝑣 to the other vertices of 𝐺; that is, ecc(𝑣) = max 𝑑 (𝑢, 𝑣) : 𝑢 ∈ 𝑉 . The diameter diam(𝐺) is the maximum eccentricity taken over all vertices of 𝐺 and the radius rad(𝐺) is the minimum eccentricity taken over all vertices of 𝐺. A vertex of 𝐺 with eccentricity equal to rad(𝐺) is called a central vertex. Abusing notation slightly, we refer to a central vertex as simply a center and say that a graph having exactly one central vertex 𝑥 is centered at 𝑥. A tree is an acyclic connected graph. A star is a tree with at most one vertex that is not a leaf, that is, a star is a tree with diameter at most 2. Thus, stars consist of complete bipartite graphs 𝐾1,𝑠 for 𝑠 ≥ 1 along with the trivial graph 𝐾1 . A double star 𝑆(𝑟, 𝑠), for 1 ≤ 𝑟 ≤ 𝑠, is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having 𝑟 leaf neighbors and the other 𝑠 leaf neighbors. The subdivision of edge 𝑢𝑣 ∈ 𝐸 consists of deleting the edge 𝑢𝑣 from 𝐸, adding a new vertex 𝑤 to 𝑉, and adding the new edges 𝑢𝑤 and 𝑤𝑣 to 𝐸. In this case, we say that the edge 𝑢𝑣 has been subdivided. In general, for an edge 𝑢𝑣 ∈ 𝐸 to be subdivided 𝑘 ≥ 1 times, we mean that edge 𝑢𝑣 is removed and replaced by a (𝑢, 𝑣)-path of length 𝑘 + 1. The subdivision graph 𝑆(𝐺) is the graph obtained from 𝐺 by subdividing every edge of 𝐺 exactly once.

1.3

Origins

In this section, we present the origins of domination in military tactics and chessboard problems.

1.3.1

Defensive and Offensive Strategies of the Roman Empire

In the fourth century AD, the Roman Empire dominated large areas of three continents, Europe, Africa, and Asia Minor. But it had begun to lose its power and it became increasingly difficult to secure all of its conquered regions. During the reign of Emperor Constantine the Great, who ruled between 306 and 337 AD, the Roman Empire controlled Britain, Gaul, Iberia (Spain and Portugal), southern Central Europe

Section 1.3. Origins

5

(including Italy), Asia Minor (including Turkey and Constantinople, a city named after the Emperor), and North Africa (including Egypt). Under Emperor Constantine, the Roman army was reorganized to consist of mobile field units and garrison soldiers, or local militia, capable of countering internal threats and barbarian invasions. A region was secured by armies being stationed there, and a region without an army was protected by sending mobile armies from neighboring regions. But Emperor Constantine decreed that a mobile field army could not be sent to defend a region if doing so left its original region unsecured. This defense strategy suggests a type of domination in graphs in which there are three types of vertices: unsecured (no armies), secured with one army (usually composed of local militia, which are not mobile armies), and secured with two armies (one being a highly trained, mobile army). The condition to be met is that every unsecured vertex must be adjacent to at least one vertex at which two armies are stationed. In this way, the set of vertices having one or two armies is a dominating set of the set of vertices having no armies. This defense strategy inspired the papers of Stewart [691] in 1999 and ReVelle and Rosing [658] in 2000, and then was formally defined as a type of domination in graphs for the first time in 2004 by Cockayne, Dreyer, Hedetniemi, and Hedetniemi [183].

1.3.2 Chaturanga Chaturanga is a war-oriented board game generally considered to have been developed in India during the sixth century AD. The name is a Sanskrit word meaning “four arms,” which stood for the four arms of the military, being the chariots, the cavalry, the elephants, and the infantry. Considered to be the precursor to the modern game of chess, chaturanga is a chesslike, two-player game played on a board of 8 × 8 squares, and with pieces very similar to those in chess: 1. Raja (king): moves one square in any direction. 2. Mantri (early form of queen): moves one square diagonally in any direction. 3. Ratha (rook): moves across any number of unoccupied squares either vertically or horizontally. 4. Gaja (elephant, early form of bishop): moves two squares diagonally but can jump over the first square. 5. Ashva (horse, knight): moves like the knight in chess, either two squares horizontally and then one square vertically, or two squares vertically and then one square horizontally, jumping over all intermediate squares. 6. Padáti (foot soldier, pawn): moves only one unoccupied square vertically, but can capture one square diagonally, as in chess. A capture in chaturanga consists of a piece of any type moving to a square, according to the rules for that piece, on which an opponent’s piece is found. The opponent’s piece is captured and removed from the game, and the piece that was moved to that square and made the capture remains on that square. In this way, every piece is said to dominate all squares it can reach in one move. Thus, the set of squares dominated by the pieces of one of the two players consists of all the squares occupied by the pieces plus all the squares which can be reached in one move by all

6

Chapter 1. In the Beginning: Roots of Domination in Graphs

of the pieces. Although there were other board games that preceded chaturanga, they are generally called race games in which the objective is to reach some designated location before your opponent. Chaturanga is one of the first games to consider the concept of capturing an opponent’s pieces, and hence the concept of domination first appears.

1.3.3 Eight Queens Problem A German chess player, named Max Bezzel [75], posed the following problem in the September 1848 issue of the chess journal Berliner Schachzeitung: Eight Queens Problem. In how many ways can 8 queens be placed on the squares of the 8 × 8 chessboard so that no two queens can attack each other, that is, no two queens lie on the same row, or the same column, or the same diagonal? A chess piece is said to cover (attack or dominate) any square on a chessboard that it can reach in a single move. For example, in one move a queen can move any number of unoccupied squares horizontally, vertically, or diagonally. Thus, a queen covers all of the squares in the same row, column, or diagonal as the square on which it is located, as illustrated in Figure 1.1. Figure 1.2 illustrates one placement of 8 pairwise non-attacking queens.

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Z Z

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Z Z

Z Z Z Z5XqZ Z Z Z Z Z Z Z Z Z Z

Figure 1.1 Moves of a queen on an 8 × 8 chessboard

The 8-Queens Problem quickly generalizes to the 𝑛-Queens Problem of placing 𝑛 queens on an 𝑛 × 𝑛 board so that no two queens attack each other. In graph theory terminology the 𝑛-Queens Problem is easily stated as that of finding a maximum independent set 𝑆 of 𝑛 vertices in the queens graph Q𝑛 . The queens graph Q𝑛 has a vertex set 𝑉 consisting of the 𝑛2 squares of an 𝑛 × 𝑛 chessboard, and two vertices are adjacent if and only if the corresponding squares lie on a common row, a common column, or a common diagonal. The vertex independence number 𝛼(Q𝑛 ) of the queens graph Q𝑛 , therefore, equals the maximum number of queens which can be placed on the 𝑛 × 𝑛 chessboard so that no two queens attack

Section 1.3. Origins

7

Z Z 5™Xq Z Z Z Z5XqZ Z 5X Z Z Z Zq ™Xq 5Z Z Z Z Z Z Z 5™Xq Z Z5XqZ Z Z Z Z5XqZ Z Z Z ™Xq Z Z 5 Figure 1.2 Maximum independent set of 8 queens

each other. It is obvious that 𝛼(Q𝑛 ) ≤ 𝑛, since any set of more than 𝑛 queens would have to contain two queens that lie on a common row, column, or diagonal. It remains to be shown that for any 𝑛, 𝛼(Q𝑛 ) = 𝑛. In 1910 Ahrens [9] was the first person to prove that for every positive integer 𝑛 ≥ 4, 𝛼(Q𝑛 ) = 𝑛, that is, one can always place 𝑛 queens on an 𝑛 × 𝑛 chessboard so that no two queens attack each other. The 8-Queens Problem, posed by Max Bezzel, was reported to have attracted the attention of the famous mathematician Gauss, but it was Dr. Franz Nauck [608, 609] who in 1850 pointed out, apparently without proof, that there were 92 different ways to place 8 non-attacking queens on the standard chessboard. These solutions fell into 12 classes, 11 of which yield 8 solutions by rotations and reflections, and the 12th solution generates another 4 solutions. In 1874 Pauls [630] was the first to prove that 92 is indeed the total number of solutions to the 8-Queens Problem. In 1892, although no proofs were given, W.W. Rouse Ball [51] correctly reported that for boards of sizes 4, 5, 6, 7, 8, 9, and 10, there are altogether 2, 10, 4, 40, 92, 342, and 724 solutions, respectively, to the 𝑛-Queens Problem.

1.3.4

Five Queens Problem Five Queens Problem. Show that 5 queens can be placed on the squares of the 8 × 8 chessboard so that every square is either occupied by a queen or is attacked by a queen. In how many ways can this be done?

It was known from the earliest times that five queens were sufficient to cover or dominate every square of the 8 × 8 chessboard; see for example Figure 1.3, in which the five queens mutually cover one another, and Figure 1.4, in which the five queens form an independent set. But this was quickly generalized to the following. Queens Domination Problem. What is the minimum number of queens which can be placed on an 𝑛 × 𝑛 chessboard so that every square is either occupied by a queen or is attacked by a queen?

8

Chapter 1. In the Beginning: Roots of Domination in Graphs

5™Xq Z Z Z Z Z Z Z 5™Xq Z Z Z Z 5™Xq Z Z 5™Xq Z Z Z Z Z Z Z Z ™Xq Z Z Z 5Z Z Z Z Z Z

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Figure 1.3 Five queens covering an 8 × 8 chessboard

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Z Z Z 5X Z Z Z Zq Z Z Z5XqZ 5™Xq Z Z Z Z Z5XqZ Z Z Z Z5XqZ Z Z Z Z Z Figure 1.4 Five independent queens covering an 8 × 8 chessboard

According to Gibbons and Webb [335], this problem was first stated by Abbe Durand in 1861, but was also given in 1862 by C.F. de Jaenisch [218], a Finnish and Russian chess player (1813–1872) and theorist, who in the 1840s was among the top chess players in the world. In graph theory terminology the Queens Domination Problem is to determine the queens domination number 𝛾(Q𝑛 ), that is, the minimum number of queens necessary to cover, or dominate, every square of an 𝑛 × 𝑛 chessboard. Although it proved to be relatively easy to determine the queens independence number 𝛼(Q𝑛 ) = 𝑛, after all these years since 1861, the determination of the value of 𝛾(Q𝑛 ) for all 𝑛 ≥ 1, remains an unsolved, and quite difficult, problem. In 1862 De Jaenisch [218] determined the queens domination number 𝛾(Q𝑛 ), for 𝑛 ∈ [8], to be 1, 1, 1, 2, 3, 3, 4, 5. In particular, he showed that 𝛾(Q8 ) = 5; see Figure 1.4. The values 𝛾(Q9 ) = 𝛾(Q10 ) = 𝛾(Q11 ) = 5 were correctly reported by Ahrens [9] in 1910; see for example Figure 1.5. These values have since been verified by computer programs.

Section 1.3. Origins

9

Z

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Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z ™Xq Z Z Z Z 5Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z Figure 1.5 Five queens covering an 11 × 11 chessboard

1.3.5

Queens Independent Domination Problem

The independent domination number 𝑖(Q𝑛 ) of the queens graph Q𝑛 was identified as an interesting problem by De Jaenisch [218], who in 1862 correctly gave the first eight values of 𝑖(Q𝑛 ), which are 1, 1, 1, 3, 3, 4, 4, 5; see Figure 1.4 for 𝑛 = 8. These have been verified by computer. It is interesting to note that 𝛾(Q5 ) = 3 < 𝑖(Q5 ) = 4 and 𝛾(Q6 ) = 3 < 𝑖(Q6 ) = 4, while 𝛾(Q7 ) = 4 = 𝑖(Q7 ) and 𝛾(Q8 ) = 5 = 𝑖(Q8 ). Determining the domination numbers and independent domination numbers of the queens graph seem to be extremely difficult problems. As noted in [446] and [626], only relatively few exact values of these two domination numbers of the queens graph are known. The value of 𝛾(Q𝑛 ) is either known, or known to be one of two consecutive values, for all 𝑛 ≤ 120 (see [626]). The known values of 𝛾(Q𝑛 ) and 𝑖(Q𝑛 ), for 4 ≤ 𝑛 ≤ 20, are summarized in Table 1.1; the values 𝛾(Q20 ) = 11 and 𝑖(Q19 ) = 𝑖(Q20 ) = 11 were discovered in the 2017 PhD thesis [77] of Bird at the University of Victoria; Bird [77] also found five other new values: 𝛾(Q22 ) = 12, 𝛾(Q24 ) = 13, 𝑖(Q22 ) = 12, 𝑖(Q23 ) = 13, and 𝑖(Q24 ) = 13. An independent covering of five queens for an 11 × 11 chessboard is illustrated in Figure 1.5.

4

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8

9 10 11 12 13 14 15 16 17 18 19 20

𝛾(Q𝑛 ) 2 𝑖(Q𝑛 ) 3

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𝑛

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Table 1.1 First 20 values of 𝛾(Q𝑛 ) and 𝑖(Q𝑛 )

9 9 10 11 9 10 11 11

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Chapter 1. In the Beginning: Roots of Domination in Graphs

1.3.6 Queens Total Domination Problem Still another interesting variant of the above three types of problems was formally introduced in 1892 by W.W. Rouse Ball [51]. Queens Total Domination Problem. What is the minimum number of queens which can be placed on an 𝑛 × 𝑛 chessboard so that every square is attacked by a queen, including the squares occupied by a queen? This is, of course, the total domination number 𝛾t (Q𝑛 ). Notice for example that Figure 1.3 shows five queens dominating the standard 8 × 8 chessboard, all of which lie on a common diagonal, and thus this set of five queens induces a connected subgraph, and thus this set is both a total dominating set and a connected dominating set. Hence, for 𝑛 = 8, 𝛾(Q8 ) = 𝛾t (Q8 ) = 5. At this point we have seen examples of dominating sets of queens of several different types, for example, dominating sets, maximum and minimum independent dominating sets, and total dominating sets. We next discuss these types of domination for different chess pieces.

1.3.7

Generalizations to Other Chess Pieces

Figure 1.6 W.W. Rouse Ball

In 1939 W.W. Rouse Ball [52] listed these three basic types of problems that were being studied on chessboards at the time. A photograph of Rouse Ball is given in Figure 1.6. • Covering: Determine the minimum number of chess pieces of a given type that are required to cover every square of an 𝑛 × 𝑛 chessboard (domination number). • Independent Covering: Determine the minimum number of mutually non-attacking chess pieces of a given type that are required to cover every square of an 𝑛 × 𝑛 chessboard (independent domination number).

Section 1.3. Origins

11

• Independence: Determine the maximum number of chess pieces of a given type that can be placed on an 𝑛 × 𝑛 chessboard such that no two pieces attack (cover) each other (independence number).

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Z Z Z Z Z2UnZ –Un2UnZ2Un2–Un 2Z Z Z 2 –Un Z Z Z Z Z Z 2–Un Z Z 2–Un2UnZ2Un–Un 2Z Z Z Z2UnZ Z Z Z Z Z Z Figure 1.7 Twelve knights covering an 8 × 8 chessboard

2Un2–Un Z2UnZ Z Z Z Z Z2UnZ Z Z2Un2–Un Z Z 2 –Un Z Z Z Z Z Z Z 2–Un Z 2–Un2UnZ Z Z Z Zn 2U Z Z Z Z Z Z2Un2–Un Z2Un Figure 1.8 Fourteen independent knights covering an 8 × 8 chessboard

For example, Figure 1.7 shows a minimum set of 12 knights that dominate the 8×8 chessboard, while Figure 1.8 shows a minimum set of 14 knights that independently dominate the 8 × 8 chessboard. In fact, 𝛾(N8 ) = 12 < 𝑖(N8 ) = 14, where N8 is the knights graph defined as expected by the moves of a knight on an 8 × 8 chessboard. As another example, Figure 1.9 shows a minimum set of 8 bishops dominating the 8 × 8 chessboard. In 1954 and 1964, two Russian mathematicians and twin brothers, Akiva Moiseevich Yaglom (1921–2007) and Isaak Moiseevich Yaglom (1921–1988) produced a comprehensive collection of results about a wide variety of independent, dominating, and independent dominating solutions for a variety of chess pieces [758].

12

Chapter 1. In the Beginning: Roots of Domination in Graphs

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Z Z Z 4˜Wb4WbZ Z Z4WbZ Z Z Z Z4Wb Z Z Z 4˜Wb Z 4˜Wb4Wb4˜Wb Z Z Z Z Z Z Z Z Z Z Z Z Figure 1.9 Eight bishops covering an 8 × 8 chessboard

1.4

Application Driven Origins

In this section, we present a sample of applications that have prompted the growth of domination as a popular area of graph theory.

1.4.1

Radio Broadcasting

In his 1968 book, C.L. Liu [566] briefly discusses the notion of dominance as it applies to communication networks, in which a dominating set represents a set of designated nodes from which broadcast messages can be transmitted to all other nodes in the network. In his model, however, it was assumed that a broadcast station could only transmit messages to adjacent nodes. A more general graph theory model was presented by Erwin in 2004 [261] and subsequently by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi in 2006 [248], in which broadcast stations could be assigned varying amounts of broadcast power, which would enable them to transmit messages to nodes at greater distances. In order to explain this model we will need a few definitions. Recall that ecc(𝑣) denotes the eccentricity denotes the diameter of a graph 𝐺.  of a vertex 𝑣 and diam(𝐺) A function 𝑏 : 𝑉 → 0, 1, . . . , diam(𝐺) is called a broadcast if for every vertex 𝑣 ∈ 𝑉, 𝑏(𝑣) 𝑣 for which 𝑏(𝑣) > 0 are called broadcast vertices  ≤ ecc(𝑣). Vertices and 𝑉 + = 𝑣 : 𝑏(𝑣) > 0 . We say that a vertex 𝑣 hears a broadcast of a vertex 𝑤 ∈ 𝑉 +  if 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑤), and 𝐻 (𝑣) = 𝑤 ∈ 𝑉 + : 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑤) is the set of vertices which vertex 𝑣 can hear. of a broadcast vertex 𝑣 is defined as 𝑁 𝑏 [𝑣] =  The broadcast neighborhood 𝑤 : 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑣) , which is the set of vertices that can hear Í a broadcast from vertex 𝑣. The cost of a broadcast 𝑏 is defined to be 𝑏(𝑉) = 𝑣 ∈𝑉 + 𝑏(𝑣), the sum of the broadcast powers of all broadcast vertices. A broadcast 𝑏 is a dominating broadcast if 𝑁 𝑏 [𝑉 + ] = 𝑉, which means that every vertex hears at least one broadcast, that is, 𝐻 (𝑣) ≥ 1 for all 𝑣 ∈ 𝑉. Finally, the broadcast domination number 𝛾𝑏 (𝐺) of a graph 𝐺 equals the minimum cost of a dominating broadcast in 𝐺.

Section 1.4. Application Driven Origins

13

A dominating set 𝑆 in a graph 𝐺 = (𝑉, 𝐸) is said to be efficient if for every vertex 𝑣 ∈ 𝑉, |N[𝑣] ∩ 𝑆| = 1, that is, every vertex 𝑤 ∈ 𝑉 \ 𝑆 is adjacent to exactly one vertex in 𝑆 and no vertex in 𝑆 is adjacent to any other vertex in 𝑆. Not all graphs have efficient dominating sets, for example, the cycle 𝐶5 , and in general, most graphs do not have efficient dominating sets. However, for broadcast domination the situation is different. We say that a broadcast is efficient if every vertex hears exactly one broadcast, that is, if |𝐻 (𝑣)| = 1 for every 𝑣 ∈ 𝑉. Dunbar et al. [248] proved that every graph 𝐺 has an efficient dominating broadcast. Theorem 1.1 ([248]) The broadcast domination number 𝛾𝑏 (𝐺) of every connected graph 𝐺 can be achieved by an efficient dominating broadcast.

1.4.2

Computer Communication Networks

Most essential services for networked distributed systems, such as mobile or wired, are provided by maintaining a global predicate over the entire network, that is defined by some invariant components of the global state of the network. For example, a spanning tree of the network is maintained as each node of the network maintains links to its neighbors in the spanning tree in order to minimize latency and bandwidth requirements of multicast or broadcast messages, or to implement echo-based distributed algorithms [41]. In such a distributed computing environment, we are often faced with the problem of allocating a minimal number of a scarce or valuable hardware or software resources, such as high-performance graphics workstations, very large databases or file servers, to certain nodes in the network, in such a way that every node in the network not having such a resource has efficient access to each resource at a neighboring node. The nodes containing these resources, therefore, form a minimal dominating set. If a dominating set in a communication network represents a set of servers which provide an acceptable level of service and resources, then a total dominating set represents a similar set of servers with the added capability that each server is adjacent to at least one other server. Thus, each server has a backup, such that if it suffers a fault and its capability as a server is disabled, it can obtain backup from an adjacent server with a minimum delay. In this way, total dominating sets are more fault tolerant than dominating sets. Another useful set of nodes in a communication network is a connected dominating set 𝑆, which can serve as a communication backbone, because every node is either in the set 𝑆 or is adjacent to some node in 𝑆, and the fact that the nodes in 𝑆 form a connected subnetwork guarantees that any two nodes in 𝑆 can send messages to each other via a path of nodes in 𝑆. We say that a computer at a node 𝑢 can effectively access a resource at a node 𝑣 if 𝑑 (𝑢, 𝑣) ≤ 𝑑 for some suitably small distance 𝑑. If 𝑑 = 1, then the minimum number of copies of a given resource, which can be allocated to some of the nodes in a network so that all nodes either have the resource or have efficient access to a copy of the resource, is simply the domination number of the network. Suppose, furthermore, that we need to allocate many different resources, but all nodes have a fixed capacity 𝑟, which prevents us from assigning more than 𝑟 resources

14

Chapter 1. In the Beginning: Roots of Domination in Graphs

to any node. We wish to determine the maximum number 𝑅 (r,d) (𝐺) of different resources we can allocate to the nodes of a network 𝐺 such that (i) no more than 𝑟 different resources are allocated to any one node and (ii) every node has efficient 𝑑-access to every resource. An allocation that achieves this maximum is called an (𝑟, 𝑑)-configuration. For more information on (𝑟, 𝑑)-configurations, the reader is referred to the work of Fujita, Yamashita, and Kameda [314].

1.4.3

Sets of Representatives

Let the vertices of a graph represent the people in some organization. An edge between two people means that they know each other. We wish to form a committee with as few members as possible such that everyone not on the committee knows at least one member of the committee. Thus, we seek a minimum cardinality dominating set 𝑆 of this organization. The set 𝑆 could have another property of interest if every member of the committee knows at least one other member of the committee. In this case we would seek to find a minimum total dominating set of this organization. This brings to mind the following challenging problem from The World’s Hardest IQ Test by Scott Morris [600]. 51. One third of the members of a parliamentary body are elected every two years. The body has six committees. Each member of the body is a member of at least one committee, and no member is a member of more than two committees. No committee has more than eleven members. Each pair of committees has exactly two members in common. The Chairman is a member of the Rules Committee and of no other committee. Each member of the Budget Committee is also a member of another committee. The last digit of the number of members of the parliamentary body is: a: 2, b: 3, c: 4, d: 6, e: It cannot be determined from the information given. Next, let the vertices of a graph be of two kinds: (i) vertices representing the members of an organization and (ii) vertices representing areas of expertise. In this graph, an edge between a person 𝑢 and an area of expertise 𝑣 means that person 𝑢 has expertise in area 𝑣. In order to save costs, we want to select as few people as possible for a set 𝑆 such that for every area of expertise there is at least once person in 𝑆 who has expertise in this area. Thus, in this bipartite graph, we seek a minimum subset of the vertices/people that dominates all of the vertices/areas of expertise needed. One real world application of this type of domination occurs in making personnel assignments in the U.S. Navy. For example, the U.S. Navy has approximately 326,000 active personnel, many of whom periodically qualify for reassignments. In any given month, from 20,000 to 30,000 personnel are reassigned to fill a similar number of available positions requiring differing types of expertise.

Section 1.4. Application Driven Origins

15

1.4.4 School Bus Routing and Bus Stop Selection The School Bus Routing Problem, a complex and multi-faceted generalization of the well-known Vehicle Routing Problem, involves the routing, planning, and scheduling of public school bus transportation. The problem is typically divided into several subproblems, including (i) data preparation, which involves the determination of the road network, the location(s) of the school(s), locations of the homes of all students, and possible locations of bus stops, (ii) bus stop selection, in which bus stop locations are determined and all students are assigned to bus stops, and then (iii) determining the actual bus routes and their schedules, which must meet the constraints that for each student the distance travelled on the bus does not exceed the shortest distance from their home to the school by more than a given threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In bus stop selection, for schools in rural surroundings the students are assumed to be picked up at their homes. However, in urban areas the students are required to walk to a bus stop from their homes, this distance being no more than some specified maximum, and the actual walk must meet safety requirements, such as not crossing extremely busy highways. It must also be the case that a limit be placed on the number of students assigned to a given bus stop or series of consecutive bus stops, so as not to exceed the capacity of the bus. Considerations such as this give rise to the concept of capacitated domination as introduced in 2010 by Goddard, Hedetniemi, Huff, and McRae [345]. An 𝑟capacitated dominating set in a graph 𝐺 = (𝑉, 𝐸) is a set 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } for which there exists a partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } satisfying the following three conditions, for all 𝑖 ∈ [𝑘]: 1. 𝑣 𝑖 ∈ 𝑉𝑖 , 2. |𝑉𝑖 | ≤ 𝑟 + 1, and 3. 𝑣 𝑖 is adjacent to all vertices in 𝑉𝑖 \ {𝑣 𝑖 }. In an 𝑟-capacitated dominating set 𝑆, each vertex in 𝑆 dominates at most 𝑟 vertices in 𝑉 \ 𝑆, which corresponds to the maximum capacity of a school bus. The 𝑟-capacitated domination number 𝛾rc (𝐺) equals the minimum cardinality of an 𝑟-capacitated dominating set in 𝐺. In the special case that 𝑟 = 1, each vertex dominates at most one vertex in 𝑉 \ 𝑆, which means that the 1-capacitated domination number equals the minimum number of isolated vertices 𝐾1 and edges 𝐾2 into which the vertices can be partitioned; this equals what is known as the edge covering number 𝛽′ (𝐺). Two sources for information on school bus routing problem are by Park and Kim [629] and Bögl, Doerner, and Parragh [83].

1.4.5 Electrical Power Domination Electric power companies must continually monitor the state of their electric power networks. This includes such things as monitoring voltage magnitudes and machine phase angles at generators. One type of monitoring is to place phase measurement units, or PMUs, at selected locations in the network. Since PMUs are expensive, it is desirable to minimize their number, while still being able to monitor the entire network. A network is said to be completely observed if all of the state variables

16

Chapter 1. In the Beginning: Roots of Domination in Graphs

(e.g. voltages and currents) can be determined from the set of measurements being monitored. Let 𝐺 = (𝑉, 𝐸) be a graph representing an electric power network, where a vertex represents an electrical node (a substation bus where transmission lines, loads, and generators are connected), and an edge represents a transmission power line joining two electrical nodes. The problem of locating a smallest set of PMUs to monitor the entire network can be stated as follows. A PMU measures/observes the state variables (voltage and phase angle) at the vertex at which it is placed, along with the variables of all incident edges and the other vertices incident to these edges. These incident edges and neighboring vertices are also observed, according to the following rules: 1. Any vertex incident with an observed edge is observed. 2. Any edge between two observed vertices is observed. 3. If a vertex is incident to 𝑘 > 1 edges and if 𝑘 − 1 of these edges are observed, then all 𝑘 of these edges are observed. To illustrate this, let 𝑆 ⊂ 𝑉 be a set of vertices at which PMUs are located. Let 𝑆 = 𝑆0 ⊆ 𝑆1 ⊆ 𝑆2 ⊆ · · · be the sequence of successive sets defined as follows: (i) 𝑆1 = N[𝑆0 ], where N[𝑆0 ] equals the set of all vertices which are either in 𝑆0 or are adjacent to at least one vertex in 𝑆0 , (ii) for 𝑘 ≥ 2, 𝑆 𝑘 is obtained from 𝑆 𝑘−1 by adding to 𝑆 𝑘−1 all vertices 𝑤 ∈ 𝑉 \ 𝑆 𝑘−1 for which there exists a vertex 𝑣 ∈ 𝑆 𝑘−1 whose only neighbor in 𝑉 \ 𝑆 𝑘−1 is 𝑤. If there exists a 𝑘 such that 𝑆 𝑘 = 𝑉, then we say that 𝑆 is a power dominating set of 𝐺. The minimum cardinality of a power dominating set in 𝐺 is called the power domination number 𝛾 𝑃 (𝐺). This type of domination was introduced in 2002 by Haynes, Hedetniemi, Hedetniemi, and Henning [411].

1.4.6

Influence in Social Networks

The parameter which we introduce in this section is motivated by its applicability to the study of influence in social networks. Associated with each vertex 𝑣 ∈ 𝑉 in a social network, modeled by a graph 𝐺 = (𝑉, 𝐸) of order 𝑛, is an influence threshold 𝑡 (𝑣), where 𝑡 : 𝑉 → {0, 1, . . . , 𝑛 − 1} such that for every vertex 𝑣 ∈ 𝑉, 0 ≤ 𝑡 (𝑣) ≤ deg(𝑣). Let 𝑆 ⊂ 𝑉 be an arbitrary subset of 𝑉. We say that a vertex 𝑣 ∈ 𝑉 \ 𝑆 is influenced by the set 𝑆 if |N(𝑣) ∩ 𝑆| ≥ 𝑡 (𝑣), that is, 𝑣 has at least 𝑡 (𝑣) neighbors in 𝑆. The threshold 𝑡 (𝑣) is used as an indicator of the likelihood that 𝑣 will adopt a given product if a sufficient number 𝑡 (𝑣) of neighbors of 𝑣 also adopt a given product. The goal is to find a relatively small number of vertices in a network such that if the vertices in this set adopt a given product, then ultimately every vertex in the graph will also adopt the product. This influence threshold can be applied to many things, such as the likelihood that someone will vote for something if sufficiently many of their neighbors vote for it. It could also indicate the likelihood that someone will get a virus, if sufficiently many neighbors have the virus. In general, the influence value 𝑡 (𝑣) is used to predict that 𝑣 will make some decision if at least 𝑡 (𝑣) of the

Section 1.4. Application Driven Origins

17

neighbors of 𝑣 make the same decision. Note that the threshold 𝑡 (𝑣) can vary from vertex to vertex. For any sequence 𝜋 of vertices of a graph 𝐺, let 𝜋 𝑗 denote the set containing the first 𝑗 vertices in the sequence 𝜋. A set 𝑆 ⊆ 𝑉 is an influence set for a graph 𝐺 = (𝑉, 𝐸) of order 𝑛 = |𝑉 | if there exists a vertex sequence 𝜋 = (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ) such that for every 𝑗 ∈ [𝑛], either vertex 𝑣 𝑗 ∈ 𝑆 or 𝑣 𝑗 is influenced by 𝜋 𝑗 −1 . The minimum cardinality of an influence set for a graph 𝐺 with a threshold function 𝑡 is the influence domination number 𝛾(𝐺, 𝑡). In the early papers on this form of influence in social networks, the focus has been on studying the computational complexity of determining, or even approximating, this minimum number of initially selected vertices. The interested reader is referred to the 2014 paper by Bazgan, Chopin, Nichterlein, and Sikora [63].

1.4.7

Topographic Maps

In the field of surveying, a typical task is to produce a topographic map of a tract of land that records the positions and elevations of a carefully selected set of control points. A grid can be used in areas where the topography is fairly regular. The tract of land is divided uniformly into squares or rectangles by two sets of lines running in perpendicular directions and spaced uniformly apart. Once stakes are set at the intersections of these grid lines, it is then necessary to determine the elevations of all grid points. This is typically done with the use of a transit, which can measure horizontal and vertical distances between the transit and the stakes. From this information drafting instruments can automatically generate the contour lines. Since these are line-of-sight distance measurements, any obstruction, such as a building structure, a tree, a hill, a steep ravine or gully, can prevent a measurement of a grid point from being taken. In this case, the transit must be moved to another observation point from which no line-of-sight obstruction exists to the given grid point. Thus, surveyors seek to find a minimum number of control points, from at least one of which a line of sight measurement can be taken of any grid point. This is equivalent to finding a minimum, or at least minimal, dominating set of the line-of-sight graph for the given tract of land, the vertices of which correspond 1-1 with the grid points, and two vertices are adjacent if and only if the corresponding grid points permit a line-of-sight measurement.

1.4.8 Transporting Hazardous Materials The transportation of hazardous materials (hazmats) is an increasingly important activity, because of the steadily increasing number of shipments per day, more than one million a day in North America in 2007 [259], and the corresponding increase in the associated risks, including injuries, deaths, and millions of dollars in property damage resulting from incidents. Consequently, there are increasingly strict government regulations for the transportation of hazardous materials [637].

18

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.10 Dénes Kőnig

One of the more important restrictions is that all such vehicles must be inspected at one or more intermediate locations while enroute to their destination, in order to ensure that the security of materials being transported is in continual compliance with regulations. This raises the problem of deciding where to locate inspection stations in a transportation network [69], which gives rise to this general problem. Assume that no hazardous materials shipment can travel a distance of 𝑘 without reaching at least one inspection station. A set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is a 𝑘-path vertex cover if every path of order 𝑘 contains at least one vertex in 𝑆, or equivalently if no path 𝑃 𝑘 is contained in the subgraph 𝐺 [𝑉 \ 𝑆] induced by 𝑉 \ 𝑆. The 𝑘-path covering number 𝜓 𝑘 (𝐺) equals the minimum cardinality of a 𝑘-path vertex cover in 𝐺. By definition, if 𝑆 is a 𝑘-path vertex cover, then the subgraph 𝐺 [𝑉 \ 𝑆] does not contain a path of length 𝑘. Thus, if the vertices in 𝑆 are the inspection stations, then it is not possible for anyone to illegally transport hazardous materials over a distance of 𝑘 without being detected. The 𝑘-path covering number was introduced in 2011 by Brešar, Kardoš, Katrenič, and Semanišin [115].

1.5

Early Chronology of Domination in Graph Theory

As we have seen, the concept of domination has appeared under several different guises as early as the fourth century. Although domination in graphs was not formally defined in mathematics until the 1960s, the basic idea appeared in digraphs some thirty years earlier. In this section, we briefly review the sequence of books and papers in graph theory, which have served to provide the foundations of the theory of domination in graphs. We also give examples of some early results that shaped the field. These results will be presented in more detail in subsequent chapters.

Section 1.5. Early Chronology of Domination in Graph Theory

19

1936 Dénes Kőnig (1884–1944)

Although elements of graph theory can be found in a number of earlier sources, it is generally agreed that the first book written specifically about graph theory was by Dénes Kőnig [535] in 1936, entitled Theorie der Endlichen und Unendlichen Graphen, Kombinatorische Topologie der Streckenkomplexe. Kőnig was at that time professor of mathematics at the Royal Joseph University in Budapest (today known as Budapest University of Technology and Economics). See Figure 1.10 for a photograph of Kőnig. It is interesting to note that according to the Mathematics Genealogy Project, Kőnig’s PhD advisors, in 1907, were József Kürschák and Hermann Minkowski, while Kőnig himself had but one PhD student, the well-known Tibor Gallai (shown in Figure 1.14), who in turn had two PhD students, Jenö Lehel and László Lovász. In this book, Kőnig was perhaps the first person to formally define independent domination in digraphs. In brief, a digraph 𝐷 = (𝑉, 𝐴) consists of a set 𝑉 of vertices and a set 𝐴 of arcs or ordered pairs of vertices (𝑢, 𝑣), the arc being directed from vertex 𝑢 to vertex 𝑣, sometimes denoted by 𝑢 → 𝑣, in which case we say that vertex 𝑣 is an out-neighbor of vertex 𝑢 and vertex 𝑢 is an in-neighbor of vertex 𝑣. A directed path from a vertex 𝑢 to a vertex 𝑣 is a vertex sequence of the form 𝑢 = 𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑘 = 𝑣 such that for every 𝑖 ∈ [𝑘], (𝑣 𝑖−1 , 𝑣 𝑖 ) is an arc in 𝐴. In [535] Kőnig defined a basis of a directed graph 𝐷 = (𝑉, 𝐴) to be a set 𝐵 ⊂ 𝑉 having the following two properties: (a) For every vertex 𝑣 ∈ 𝑉 \ 𝐵, there exists a vertex 𝑢 ∈ 𝐵 and a directed path from 𝑢 to 𝑣. (b) For every pair of vertices 𝑢, 𝑣 ∈ 𝐵, there is no directed path from 𝑢 to 𝑣. Notice that by (b), every basis of a directed graph 𝐷 is an independent set. Theorem 1.2 ([535]) Every finite directed graph 𝐷 = (𝑉, 𝐴) has a basis. Kőnig then defined a basis of the second kind to be a vertex set 𝐵 satisfying the following two conditions: (a) If 𝑣 is a vertex in 𝑉 \ 𝐵, then there is an arc (𝑢, 𝑣) from a vertex 𝑢 ∈ 𝐵 to 𝑣. (b) There is no arc between two vertices in 𝐵. In the case where a digraph 𝐷 is symmetric, meaning that whenever there is an arc (𝑢, 𝑣) ∈ 𝐴, then the arc (𝑣, 𝑢) is also in 𝐴, Kőnig’s basis of the second kind appears to be the first time in the literature where an independent dominating set is defined in an undirected graph. It also, of course, defines an independent dominating set in a digraph for the first time. Kőnig is also very well known for the following theorem. Theorem 1.3 ([535]) A graph 𝐺 = (𝑉, 𝐸) is bipartite if the vertex set 𝑉 can be partitioned into two independent sets, or equivalently, if 𝐺 contains no cycles of odd length. Recall that 𝛼′ (𝐺) and 𝛽(𝐺) denote the matching number and the vertex cover number of 𝐺, respectively. Theorem 1.4 ([535]) If 𝐺 is a bipartite graph, then 𝛼′ (𝐺) = 𝛽(𝐺).

20

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.11 John von Neumann

Figure 1.12 Oskar Morgenstern

In terms of domination, a maximum matching 𝑀 is both a dominating set of edges, in that every edge not in 𝑀 must be adjacent to an edge in 𝑀, and an independent dominating set of edges. We should note that Theorem 1.4, proved in 1931 by Kőnig, was also independently proved in 1931 in the more general case of weighted graphs by Jenö Egerváry. Thus, this is often referred to as the Kőnig-Egerváry Theorem. 1939 Patrick Michael Grundy (1917–1958)

In 1939 Grundy [366] defined the following type of function 𝑔 : 𝑉 → {0, 1, . . . , 𝑛 − 1} on a digraph 𝐷 = (𝑉, 𝐴) of order 𝑛 = |𝑉 |. We say that such a function 𝑔 is a Grundy function if for every vertex 𝑢 ∈ 𝑉, 𝑔(𝑢) is the value not assigned 
smallest to an out-neighbor of 𝑢, that is, not in the set 𝑔 N(𝑢) = 𝑔(𝑣) : (𝑢, 𝑣) ∈ 𝐴 .  For any Grundy function 𝑔 it is easy to see that the set 𝑆 = 𝑢 : 𝑔(𝑢) = 0 is an independent dominating set, since no two vertices 𝑢 and 𝑣 that are adjacent can have 𝑔(𝑢) = 𝑔(𝑣) = 0, and any vertex 𝑤 ∈ 𝑉 \ 𝑆 must have at least one out-neighbor 𝑢 with 𝑤𝑢 ∈ 𝐴 and 𝑔(𝑢) = 0. 1944 John von Neumann (1903–1957) and Oskar Morgenstern (1902–1977)

In 1944 Hungarian-American mathematician, physicist, and computer scientist, John von Neumann and economist Oskar Morgenstern published their well-known book, Theory of Games and Economic Behavior [739], which founded the interdisciplinary field of research called game theory. They introduced the notion of a kernel into the theory of games under the name solution. We include photographs of von Neumann and Morgenstern in Figs. 1.11 and 1.12, respectively. A kernel in a digraph 𝐷 = (𝑉, 𝐴) is a set 𝑆 of vertices having the property that (i) for every vertex 𝑣 ∈ 𝑆, there is no arc (𝑣, 𝑤) to another vertex 𝑤 ∈ 𝑆, and (ii) for every vertex 𝑤 ∈ 𝑉 \ 𝑆, there is an arc (𝑤, 𝑣) from 𝑤 to some vertex 𝑣 ∈ 𝑆. In undirected graphs, kernels are nothing more than independent dominating sets, first discussed by Kőnig in 1936. Von Neumann and Morgenstern [739] described the following scenario. Suppose a set of 𝑛 players are given a set 𝑋 of situations to consider in terms of preference.

Section 1.5. Early Chronology of Domination in Graph Theory

21

Given two situations 𝑢 and 𝑣, if a subset of the players prefer situation 𝑢 to 𝑣 and can, by some means, make their point of view prevail, then we add an arc from 𝑣 to 𝑢, by which it is indicated that 𝑢 is preferred to 𝑣. In this way, the players construct a digraph 𝐷 = (𝑉, 𝐴) in which the out-neighbors 𝑢 of a vertex 𝑣, indicated by arcs (𝑣, 𝑢), are all the situations which have been decided to be more preferable than 𝑣. Independent dominating sets occur in the study of 2-person games, where they are called kernels. They are used to represent winning positions 𝑆 in a game, where an opponent makes a move starting from a winning position, but must always move to a non-winning position in 𝑉 \ 𝑆. Since the set 𝑆 is independent, the player cannot move from a winning position to another winning position. The player with a winning strategy, therefore always has a move from a non-winning position in 𝑉 \ 𝑆 to a winning position in 𝑆. Now, if a kernel can be found for this digraph 𝐷, then since 𝑆 is an independent set, no situation in 𝑆 is preferred to any other, and since 𝑆 is a dominating set, for any situation 𝑥 not in 𝑆, there must be a situation 𝑢 ∈ 𝑆 that is more preferable and hence there is an arc (𝑥, 𝑢) from 𝑥 to 𝑢. Research therefore focuses on the question: does a directed graph or undirected graph have a kernel? Subsequently many theorems emerged with either necessary or sufficient conditions under which a kernel will or will not exist. For example, from the previous subsection about Grundy functions, it is clear that if a digraph has a Grundy function, then it has a kernel. 1958 Claude Berge (1926–2002)

In 1958 French mathematician, Claude Berge, at Maître de Recherches au Centre National de la Recherche Scientifique, published the second book on graph theory, The Theory of Graphs and its Applications [67], after Kőnig’s book in 1936. Berge was the first to formally define and study the vertex independence number, which he called the coefficient of internal stability and the domination number, which he called the coefficient of external stability. Berge is pictured in Figure 1.13. Berge also defined a counterpart for domination in digraphs called the absorption number of a digraph 𝐷, to equal the minimum cardinality of an absorbant set, which is a vertex set 𝑆 having the property that for every vertex in 𝑣 ∈ 𝑉 \ 𝑆, there is an arc (𝑣, 𝑤) from 𝑣 to a vertex 𝑤 ∈ 𝑆, or equivalently, every vertex in 𝑉 \ 𝑆 is adjacent to a vertex in 𝑆. He gave as examples of minimum dominating sets the 5 minimum dominating queens in Figure 1.4, the 12 minimum dominating knights in Figure 1.7, and the 8 minimum dominating bishops in Figure 1.9. We will explore other results of Berge in subsequent chapters. Berge [67] was also the first to observe the following result. Theorem 1.5 ([67]) A set 𝑆 of vertices in a graph 𝐺 is an independent and dominating set of 𝐺 if and only if 𝑆 is maximal independent. 1959 Tibor Gallai (1912–1992)

In 1959 Hungarian mathematician Tibor Gallai (pictured in Figure 1.14), a student of Dénes Kőnig, proved the following classic theorem, where 𝛼(𝐺), 𝛽(𝐺), 𝛼′ (𝐺), and 𝛽′ (𝐺) denote the independence number, the vertex covering number, the matching number, and the edge covering number, respectively.

22

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.13 Claude Berge

Figure 1.14 Tibor Gallai

Figure 1.15 Øystein Ore

Theorem 1.6 (Gallai’s Theorem [324]) If 𝐺 is a connected graph of order 𝑛 ≥ 2, then 𝛼(𝐺) + 𝛽(𝐺) = 𝛼′ (𝐺) + 𝛽′ (𝐺) = 𝑛. Since Gallai’s Theorem, many theorems of the general form A (𝐺) + B (𝐺) = 𝑛 have appeared in the literature, and are called Gallai theorems. See for example Cockayne, Hedetniemi, and Laskar [195]. 1962 Øystein Ore (1899–1968)

In 1962 Øystein Ore (see Figure 1.15), a Norwegian-American mathematician at Yale University, published what can be considered the third graph theory book, after those by Kőnig and Berge. In Chapter 13 Dominating Sets, Covering Sets, and Independent Sets, Ore [622] provides the first use of the word “domination,” for what Kőnig had previously referred to as a basis of the second kind and Berge had referred to as the coefficient of external stability. From Ore’s use forward the word ‘domination’ became the accepted terminology. Ore’s wording for a dominating set

Section 1.5. Early Chronology of Domination in Graph Theory

23

Figure 1.16 Vadim Vizing

applied to both directed and undirected graphs, as follows: “A subset 𝐷 of 𝑉 is a dominating set for 𝐺 when every vertex not in 𝐷 is the endpoint of some edge from a vertex in 𝐷.” Ore [622] provided the following well-known upper bound on the domination number. Theorem 1.7 (Ore’s Theorem [622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 12 𝑛. 1963 Vadim Vizing (1937–2017)

In 1963 Ukrainian (former Soviet) graph theorist Vadim Vizing, pictured in Figure 1.16, raised a question about the domination number of the Cartesian product of two graphs 𝐺 and 𝐻, which has become the most famous conjecture in domination theory. We need to give the following definition. The Cartesian product of
two graphs 𝐺 = (𝑉, 𝐸) and 𝐻 = (𝑊, 𝐹) is the graph 𝐺 □ 𝐻 = 𝑉 × 𝑊, 𝐸 (𝐺 □ 𝐻) , where two vertices (𝑥 1 , 𝑦 1 ) and (𝑥2 , 𝑦 2 ) are adjacent in 𝐺 □ 𝐻 if and only if 𝑥 1 = 𝑥2 and 𝑦 1 𝑦 2 ∈ 𝐸 (𝐻), or 𝑥 1 𝑥2 ∈ 𝐸 (𝐺) and 𝑦 1 = 𝑦 2 . Conjecture 1.8 (Vizing’s Conjecture [732]) For every pair of graphs 𝐺 and 𝐻, 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺)𝛾(𝐻). Several hundred papers have now been written on Vizing’s Conjecture; the reader is referred to an entire chapter by Hartnell and Rall [396] in 1998, and a survey paper by Brešar, Dorbec, Goddard, Hartnell, Henning, Klavžar, and Rall [113] in 2012. We devote Chapter 18 of this book to this conjecture. 1974 Juho Nieminen

One of the earliest and most basic theorems about the domination number of a graph is the following, due to the Finnish mathematician Juho Nieminen (Finnish Academy, Helsinki). A pendant edge in a graph 𝐺 is any edge, one of whose vertices has degree one. Let 𝜀 𝑓 (𝐺) denote the maximum number of pendant edges in a spanning forest of 𝐺 (a forest is an acyclic graph). It can be seen that 𝜀 𝑓 (𝐺) equals

24

Chapter 1. In the Beginning: Roots of Domination in Graphs

the maximum number of edges in what is called a spanning star forest, that is, a disjoint collection of stars. In 1974 Nieminen [613] proved the following Gallai type theorem. Theorem 1.9 ([613]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛. 1975 Amram Meir and John Moon

In 1975 Meir and Moon (University of Alberta) [589] published a key paper relating packing and covering numbers of trees, as follows. Recall that 𝛼(𝐺) denotes the independence number and 𝜌(𝐺) denotes the packing number of a graph 𝐺. Theorem 1.10 ([589]) If 𝑇 is a tree of order 𝑛 ≥ 2, then 𝛼(𝑇) + 𝛾(𝑇) ≤ 𝑛. Corollary 1.11 ([589]) If 𝑇 is a tree of order 𝑛 ≥ 2, then 1 ≤ 𝛾(𝑇) ≤ 𝛼(𝑇) ≤ 𝑛 − 1.

𝑛 2

≤

We note that both Theorem 1.10 and its corollary generalize to all isolate-free graphs. Theorem 1.12 ([589]) For any nontrivial tree 𝑇, 𝜌(𝑇) = 𝛾(𝑇). Theorem 1.12 has become one of the best known results in domination theory. 1975 Ernest J. Cockayne, Seymour Goodman, and Stephen T. Hedetniemi

In 1975 Cockayne (University of Victoria) along with Goodman and Hedetniemi (University of Virginia) [187] published the first algorithm for computing the domination number of a tree, which executes in linear O (𝑛) time. 1977 E.J. Cockayne and S.T. Hedetniemi

In 1977 Cockayne and Hedetniemi (University of Oregon), published a paper, Towards a Theory of Domination in Graphs [194], which was one of the first to bring focus to domination in graphs as a field of study within graph theory, while citing the earlier work of Berge and Ore, mentioned previously. This paper, now having been cited more than 600 times, was noteworthy for introducing the notation 𝛾(𝐺) for the domination number and 𝑖(𝐺) for the independent domination number of a graph. The authors also introduced the concept of the domatic number dom(𝐺) and idomatic number idom(𝐺) of a graph, which are defined to equal the maximum orders of a partition of the vertices of a graph into dominating sets and independent dominating sets, respectively. There are by now nearly 250 papers on various aspects of domatic numbers in graphs. Domatic numbers and related partitions are discussed in Chapter 12. 1978 E.J. Cockayne, S.T. Hedetniemi, and Donald J. Miller

In 1978 Cockayne, Hedetniemi, and Miller (University of Victoria), published a paper, Properties of Hereditary Hypergraphs and Middle Graphs [196], which for the first time defined the irredundance numbers of a graph and presented what is called the Domination Chain of inequalities involving domination parameters, as follows.

Section 1.5. Early Chronology of Domination in Graph Theory

25

A set 𝑆 is called irredundant if for every vertex 𝑣 ∈ 𝑆, N[𝑣] \N[𝑆 \ {𝑣}] ≠ ∅. The lower and upper irredundance numbers ir(𝐺) and IR(𝐺) equal the minimum cardinality of a maximal irredundant set and the maximum cardinality of an irredundant set in 𝐺, respectively. The parameters ir(𝐺), IR(𝐺), and Γ(𝐺) were first defined in this paper. The domination number 𝛾(𝐺), the independent domination number 𝑖(𝐺), the independence number 𝛼(𝐺), and the upper domination number Γ(𝐺) have already been defined. The following inequalities are referred to as The Domination Chain. Theorem 1.13 ([196]) For any graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). The authors also observed the following result, which is analogous to Berge’s Theorem 1.5. Theorem 1.14 ([196]) A set 𝑆 of vertices of a graph 𝐺 is an irredundant and dominating set of 𝐺 if and only if 𝑆 is minimal dominating. Indeed, it was later observed that if a set 𝑆 is a maximal independent set, then it is a minimal dominating set, and if a set 𝑆 is a minimal dominating set, then it is a maximal irredundant set. One final result in [196] is interesting to observe. The well-studied chromatic number 𝜒(𝐺) of a graph 𝐺 = (𝑉, 𝐸) equals the minimum order of a partition of 𝑉 into independent sets. One can define the independence graph 𝐼 (𝐺) of any graph 𝐺 to be the intersection graph of the set of all independent sets of vertices in 𝐺, that is, the vertices of 𝐼 (𝐺) correspond to the independent sets of vertices in 𝐺 and two vertices are adjacent in 𝐼 (𝐺) if and only if the corresponding independent sets have a vertex in common. Theorem 1.15 ([196]) For any graph 𝐺, 𝜒(𝐺) = 𝛾(𝐼 (𝐺)). 1980 E.J. Cockayne, Robyn Mason Dawes (1936–2010), and S.T. Hedetniemi

In 1980 Cockayne, Hedetniemi, and Dawes (Professor of Psychology at the University of Oregon) published a short note [182] which for the first time formally defined the total domination number of a graph, giving it the notation 𝛾t (𝐺). Clearly, many examples of total dominating sets of queens and other chess pieces had been observed, even some for more than 100 years, but the formal definition which applies to arbitrary graphs had not been defined. Today this paper has been cited more than 600 times, and more than 850 papers have been published on total domination alone. The reader is referred to the comprehensive 2013 book on total domination by Henning and Yeo [490]. 1998 Teresa W. Haynes, S.T. Hedetniemi, and Peter J. Slater (1946–2016)

In 1998 Haynes, Hedetniemi, and Slater published the first comprehensive treatment of domination in graphs in the two books: Fundamentals of Domination in Graphs [417] and Domination in Graphs: Advanced Topics [416]. The fundamentals book [417] has been cited more than 5000 times to date. Peter Slater is pictured

26

Chapter 1. In the Beginning: Roots of Domination in Graphs

in Figure 1.17. A list of other books containing information on domination can be found in Appendix B and a list of survey papers on domination can be found in Appendix C.

Figure 1.17 Peter J. Slater

Chapter 2

Fundamentals of Domination 2.1 Introduction As we have seen in Chapter 1, domination in graphs has roots in many sources, including defense strategies, games such as chess, computer communication networks, and network surveillance and security. In this chapter, we discuss the graph theoretical core concepts of domination and equivalent definitions for the domination number, thereby setting the foundation for the remaining chapters in the book. In order to explain the core concepts in domination in graphs, we need only a few definitions, which are given in the glossary in Appendix A and in Chapter 1.

2.2

Core Concepts

In this section, we discuss the core concepts of domination and develop what is called the Domination Chain, which was introduced in 1978 by Cockayne et al. [196]. The Domination Chain expresses relationships that exist among independent sets, dominating sets, and irredundant sets in graphs. This inequality sequence has become one of the major focal points in the study of domination in graphs, inspiring much interest and serving as a source for several hundred papers. Prior to stating the chain, we give a brief discussion of how this chain is developed using maximality and minimality conditions.

2.2.1 Independent Sets Definition 2.1 A set 𝑆 ⊆ 𝑉 of vertices in a graph 𝐺 is independent if no two vertices in 𝑆 are adjacent. The concept of domination in graphs can be said to originate in the definition of a maximal independent set, as follows. Let P denote any property of sets of vertices in a graph 𝐺 or a property of a graph 𝐺. A subset 𝑆 ⊆ 𝑉 having some property P is called a P-set, for example, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_2

27

28

Chapter 2. Fundamentals of Domination

the property P0 of being an independent set, or the property PΔ𝑘 of being a set 𝑆 of vertices whose induced subgraph 𝐺 [𝑆] has maximum degree Δ(𝐺) ≤ 𝑘, for some positive integer 𝑘. A property P is called hereditary if every subset of a P-set is also a P-set, for example, the property P0 of being an independent set is hereditary, and so is property PΔ𝑘 . A property P is called superhereditary if every superset of a P-set is also a P-set, for example, the property of having a vertex that has at least 𝑘 neighbors in the set is superhereditary. A set 𝑆 is a maximal P-set if 𝑆 has property P but no proper superset 𝑆 ′′ , 𝑆 ⊂ 𝑆 ′′ , is a P-set. A set 𝑆 is a 1-maximal P-set if 𝑆 has property P but for every vertex 𝑤 ∈ 𝑉 \ 𝑆, the set 𝑆 ∪ {𝑤} does not have property P. A set 𝑆 is a minimal P-set if 𝑆 has property P but no proper subset 𝑆 ′ , 𝑆 ′ ⊂ 𝑆, is a P-set. A set 𝑆 is a 1-minimal P-set if 𝑆 has property P but for every vertex 𝑤 ∈ 𝑆, the set 𝑆 \ {𝑤} does not have property P. In 1997 Cockayne et al. [190] provided straightforward proofs of the following two results. Proposition 2.2 ([190]) For any graph 𝐺 and any superhereditary property P, a set 𝑆 ⊆ 𝑉 is a minimal P-set if and only if 𝑆 is a 1-minimal P-set. Proof By definition, every minimal P-set is a 1-minimal P-set. For the converse, let 𝑆 be a 1-minimal P-set, and let 𝑆 ′ be a proper subset of 𝑆. We wish to show that 𝑆 ′ is not a P-set, which would imply that 𝑆 is a minimal P-set. If |𝑆 ′ | = |𝑆| − 1, then 𝑆 ′ = 𝑆 \ {𝑣} for some vertex 𝑣 ∈ 𝑆, and the result follows from the definition of a 1-minimal P-set. Hence, we may assume |𝑆 ′ | ≤ |𝑆| − 2. Suppose, to the contrary, that 𝑆 ′ is a P-set. Let 𝑆 ′′ ⊂ 𝑆 be a superset of 𝑆 ′ such that |𝑆 ′′ | = |𝑆| − 1. We note that 𝑆 ′ ⊂ 𝑆 ′′ ⊂ 𝑆. Since P is a superhereditary property, the superset 𝑆 ′′ is a P-set. But this contradicts the supposition that 𝑆 is a 1-minimal P-set. Proposition 2.3 ([190]) For any graph 𝐺 and any hereditary property P, a set 𝑆 ⊆ 𝑉 is a maximal P-set if and only if 𝑆 is a 1-maximal P-set. Proof By definition, every maximal P-set is a 1-maximal P-set. For the converse, let 𝑆 be a 1-maximal P-set. If 𝑆 = 𝑉, then the result holds vacuously, so let 𝑆 ′′ be a proper superset of 𝑆. We wish to show that 𝑆 ′′ is not a P-set, which would imply that 𝑆 is a maximal P-set. If |𝑆 ′′ | = |𝑆| + 1, then the result follows from the definition of a 1-maximal P-set. Hence, we may assume |𝑆 ′′ | ≥ |𝑆| + 2. Suppose, to the contrary, that 𝑆 ′′ is a P-set. Let 𝑆 ′ be a subset of 𝑆 ′′ such that 𝑆 ⊂ 𝑆 ′ and |𝑆 ′ | = |𝑆| + 1. We note that 𝑆 ⊂ 𝑆 ′ ⊂ 𝑆 ′′ . Since P is a hereditary property, the subset 𝑆 ′ of 𝑆 ′′ is a P-set. But this contradicts the supposition that 𝑆 is a 1-maximal P-set. Since the property of being an independent set is hereditary, we can say by Proposition 2.3 that an independent set 𝑆 is maximal if and only if 𝑆 is a 1-maximal independent set. This is equivalent to saying that for every vertex 𝑣 ∈ 𝑉 \ 𝑆, the set 𝑆 ∪ {𝑣} is not an independent set. But this, in turn, is equivalent to the following property:

Section 2.2. Core Concepts

29

P ′ : for every vertex 𝑣 ∈ 𝑉 \ 𝑆, the induced subgraph 𝐺 [𝑆 ∪ {𝑣}] contains an edge between vertex 𝑣 and a vertex in 𝑆, which means that every vertex 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to at least one vertex in 𝑆. Property P ′ then motivates the definition of dominating sets.

2.2.2 Dominating Sets Definition 2.4 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if every vertex 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to at least one vertex in 𝑆. This, in turn, is equivalent to the following definition. Definition 2.5 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if N[𝑆] = 𝑉, that is, every vertex 𝑣 ∈ 𝑉 is either an element of 𝑆 or is in 𝑉 \ 𝑆 and is adjacent to at least one vertex in 𝑆. Definition 2.6 If 𝑋, 𝑌 ⊆ 𝑉, where the sets 𝑋 and 𝑌 are not necessarily disjoint, then 𝑋 dominates 𝑌 if 𝑌 ⊆ N[𝑋], that is, every vertex in 𝑌 belongs to 𝑋 or has a neighbor in 𝑋. In particular, if 𝑋 dominates 𝑉, then 𝑋 is a dominating set of 𝐺. Definition 2.7 The domination number 𝛾(𝐺) equals the minimum cardinality of a dominating set in 𝐺. The upper domination number Γ(𝐺) equals the maximum cardinality of a minimal dominating set in 𝐺. We say that any dominating set 𝑆 for which |𝑆| = 𝛾(𝐺) is a 𝛾-set of 𝐺, and any minimal dominating set 𝑆 for which |𝑆| = Γ(𝐺) is a Γ-set. Thus, the maximality condition for an independent set is identical to the condition that a set be a dominating set. And from this it follows immediately that every maximal independent set must also be a dominating set. Indeed, it was Berge [67, 68], as early as 1962, who first observed the following. Theorem 2.8 ([67, 68]) A subset of vertices in a graph 𝐺 is maximal independent if and only if it is independent and minimal dominating. Definition 2.9 The vertex independence number 𝛼(𝐺) equals the maximum cardinality of an independent set in 𝐺. The independent domination number 𝑖(𝐺) equals the minimum cardinality of a maximal independent set in 𝐺. An independent set of 𝐺 with cardinality 𝛼(𝐺) is called an 𝛼-set of 𝐺, while any maximal independent set of cardinality 𝑖(𝐺) is called an 𝑖-set of 𝐺. Thus, by definition, for any graph 𝐺, 𝑖(𝐺) ≤ 𝛼(𝐺). Notice that the property of being a dominating set 𝑆 is superhereditary, meaning that every proper superset of 𝑆 is also a dominating set. Thus, since every maximal independent set is also a minimal dominating set, we have the following inequalities, for any graph 𝐺: 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺). For example, the tree 𝑇 in Figure 2.1 has maximal independent sets of three cardinalities: {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 }, {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 }, and {𝑣 4 , 𝑣 6 , 𝑣 7 }. Thus, 𝑖(𝑇) = 3 and

Chapter 2. Fundamentals of Domination

30 𝑣3

𝑣7

𝑣2 𝑣4

𝑣5 𝑣6

𝑣1 Figure 2.1 A tree 𝑇

𝛼(𝑇) = 5. This tree 𝑇 has exactly four minimal dominating sets: {𝑣 4 , 𝑣 5 }, {𝑣 4 , 𝑣 6 , 𝑣 7 }, {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 }, and {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 }. Hence, 𝛾(𝑇) = 2 and Γ(𝑇) = 5. Since the property of being a dominating set is superhereditary, it follows from Proposition 2.2 that a set 𝑆 is a minimal dominating set if and only if it is a 1-minimal dominating set. Thus, for every vertex 𝑣 ∈ 𝑆, the set 𝑆 \ {𝑣} is not a dominating set. But this means that every minimal dominating set 𝑆 satisfies the following property:
P ′′ : for every vertex 𝑣 ∈ 𝑆, there exists a vertex 𝑤 ∈ 𝑉 \ 𝑆 \ {𝑣} which is not dominated by the set 𝑆 \ {𝑣}. This undominated vertex could be the vertex 𝑣 itself or a vertex 𝑤 ∈ 𝑉 \ 𝑆. In either case, it means that vertex 𝑣 dominates some vertex that no other vertex in 𝑆 dominates. This gives rise to the notion of a private neighbor with respect to a set of vertices, first introduced in 1978 by Cockayne et al. [196]. Definition 2.10 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the 𝑆-private neighborhood of 𝑣 is the set N[𝑣] \ N[𝑆 \ {𝑣}] and is denoted by pn[𝑣, 𝑆]. That is, pn[𝑣, 𝑆] equals the set of vertices that are in the closed neighborhood of 𝑣 but not in the closed neighborhood of the set 𝑆 \ {𝑣}. Equivalently, pn[𝑣, 𝑆] = 𝑤 ∈ 𝑉 : N[𝑤] ∩ 𝑆 = {𝑣} . If pn[𝑣, 𝑆] ≠ ∅, then we say that every vertex in pn[𝑣, 𝑆] is an 𝑆-private neighbor of 𝑣. Definition 2.11 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the open 𝑆-private neighborhood of 𝑣 is the set N(𝑣) \ N(𝑆 \ {𝑣}) and is denoted by pn(𝑣, 𝑆). That is, pn(𝑣, 𝑆) equals the set of vertices that are in the open neighborhoodof 𝑣 but not in the open neighborhood of the set 𝑆 \ {𝑣}. Equivalently, pn(𝑣, 𝑆) = 𝑤 ∈ 𝑉 : N(𝑤) ∩ 𝑆 = {𝑣} . If pn(𝑣, 𝑆) ≠ ∅, then we say that every vertex in pn(𝑣, 𝑆) is an open 𝑆-private neighbor of 𝑣. Definition 2.12 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the sets pn[𝑣, 𝑆] \ 𝑆 and pn(𝑣, 𝑆) \ 𝑆 are equal and we define the 𝑆-external private neighborhood of 𝑣 to be this set, abbreviated epn[𝑣, 𝑆] or epn(𝑣, 𝑆). Thus, epn[𝑣, 𝑆] = epn(𝑣, 𝑆). The 𝑆-internal private neighborhood of 𝑣 is defined by ipn[𝑣, 𝑆] = pn[𝑣, 𝑆] ∩ 𝑆 and its open 𝑆-internal private neighborhood is defined by ipn(𝑣, 𝑆) = pn(𝑣, 𝑆) ∩ 𝑆. We note that pn[𝑣, 𝑆] = ipn[𝑣, 𝑆] ∪ epn[𝑣, 𝑆] and pn(𝑣, 𝑆) = ipn(𝑣, 𝑆) ∪ epn(𝑣, 𝑆). To illustrate the private neighbor concept, we consider the tree 𝑇 in Figure 2.1. If 𝑆 = {𝑣 4 , 𝑣 5 }, then pn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 } since epn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 } and ipn[𝑣 4 , 𝑆] = ∅, and pn[𝑣 5 , 𝑆] = epn[𝑣 5 , 𝑆] = {𝑣 6 , 𝑣 7 }, while ipn[𝑣 5 , 𝑆] = ∅. Furthermore, pn(𝑣 4 , 𝑆) = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 } since ipn(𝑣 4 , 𝑆) = {𝑣 5 } and epn(𝑣 4 , 𝑆) =

Section 2.2. Core Concepts

31

epn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 }. Similarly, pn(𝑣 5 , 𝑆) = {𝑣 4 , 𝑣 6 , 𝑣 7 } since ipn(𝑣 5 , 𝑆) = {𝑣 4 } and epn(𝑣 5 , 𝑆) = epn[𝑣 5 , 𝑆] = {𝑣 6 , 𝑣 7 }. If 𝑆 ′ = {𝑣 4 , 𝑣 6 , 𝑣 7 }, then pn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 } since epn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 } and ipn[𝑣 4 , 𝑆 ′ ] = {𝑣 4 }. Also, pn[𝑣 6 , 𝑆 ′ ] = {𝑣 6 } and pn[𝑣 7 , 𝑆 ′ ] = {𝑣 7 } since ipn[𝑣 6 , 𝑆 ′ ] = {𝑣 6 }, ipn[𝑣 7 , 𝑆 ′ ] = {𝑣 7 }, and epn[𝑣 6 , 𝑆 ′ ] = epn[𝑣 7 , 𝑆 ′ ] = ∅. Furthermore, pn(𝑣 4 , 𝑆 ′ ) = {𝑣 1 , 𝑣 2 , 𝑣 3 } as ipn(𝑣 4 , 𝑆) = ∅ and epn(𝑣 4 , 𝑆 ′ ) = epn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 }, and pn(𝑣 6 , 𝑆 ′ ) = ipn(𝑣 6 , 𝑆 ′ ) = epn(𝑣 6 , 𝑆 ′ ) = ∅. Similarly, pn(𝑣 7 , 𝑆 ′ ) = ipn(𝑣 7 , 𝑆 ′ ) = epn(𝑣 7 , 𝑆 ′ ) = ∅.

2.2.3

Irredundant Sets

The earlier observation that every vertex in a minimal dominating set dominates some vertex that no other vertex in 𝑆 dominates gives rise to the concept of irredundant sets. Irredundant sets were first defined by Cockayne et al. [196]. Definition 2.13 A set 𝑆 ⊆ 𝑉 is irredundant if for every vertex 𝑣 ∈ 𝑆, pn[𝑣, 𝑆] ≠ ∅. Definition 2.14 The irredundance number ir(𝐺) of a graph 𝐺 equals the minimum cardinality of a maximal irredundant set in 𝐺. The upper irredundance number IR(𝐺) equals the maximum cardinality of an irredundant set in 𝐺. For example, consider the graph 𝐺 in Figure 2.2. Each of the sets 𝑆 = {𝑣 1 , 𝑣 4 }, 𝑆 ′ = {𝑣 1 , 𝑣 5 }, and 𝑆 ′′ = {𝑣 1 , 𝑣 2 , 𝑣 3 } is a maximal irredundant set. Note that for 𝑆, pn[𝑣 1 , 𝑆] = {𝑣 2 , 𝑣 3 }, pn[𝑣 4 , 𝑆] = {𝑣 5 , 𝑣 6 }; for 𝑆 ′ , pn[𝑣 1 , 𝑆 ′ ] = {𝑣 1 , 𝑣 3 }, pn[𝑣 5 , 𝑆 ′ ] = {𝑣 5 , 𝑣 6 }; and for 𝑆 ′′ , pn[𝑣 1 , 𝑆 ′′ ] = {𝑣 4 }, pn[𝑣 2 , 𝑆 ′′ ] = {𝑣 5 }, and pn[𝑣 3 , 𝑆 ′′ ] = {𝑣 6 }. It can be shown that ir(𝐺) = 2 and IR(𝐺) = 3. 𝑣3

𝑣6 𝑣2

𝑣1

𝑣5 𝑣4

Figure 2.2 Graph 𝐺 with maximal irredundant sets of cardinality 2 and 3

We note that the domination number of the graph 𝐺 in Figure 2.2 is also 2, and so ir(𝐺) = 𝛾(𝐺) = 2. Slater provided one of the first known examples of a graph having irredundance number strictly less than its domination number. This graph, known as the Slater graph 𝐻, is illustrated in Figure 2.3. It can be shown that the set {𝑣 2 , 𝑣 3 , 𝑣 8 , 𝑣 9 } is a minimum maximal irredundant set for 𝐻 and the set {𝑣 2 , 𝑣 4 , 𝑣 6 , 𝑣 8 , 𝑣 10 } is a minimum dominating set of 𝐻, and so 4 = ir(𝐻) < 𝛾(𝐻) = 5. We note that attaching a leaf to a vertex of 𝐻 that is not in N[𝑣 6 ] creates another tree 𝑇 with ir(𝑇) = 4 < 5 = 𝛾(𝑇). It can easily be observed that the property of being an irredundant set is hereditary, namely, every subset of an irredundant set is also irredundant.

Chapter 2. Fundamentals of Domination

32 𝑣7

𝑣8

𝑣9

𝑣 10

𝑣 11

𝑣4

𝑣5

𝑣6 𝑣1

𝑣2

𝑣3

Figure 2.3 The Slater graph 𝐻

Proposition 2.15 ([196]) A subset of vertices in a graph 𝐺 is a minimal dominating set if and only if it is dominating and maximal irredundant. The following inequalities, called the Domination Chain, follow from the definitions, since every maximal independent set is a minimal dominating set and every minimal dominating set is a maximal irredundant set. Theorem 2.16 ([196]) For every graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). As we have seen with the Slater graph, strict inequality between ir(𝑇) and 𝛾(𝑇) is possible for trees. On the other hand, in 1981 Cockayne et al. [185] proved that the three upper parameters of the Domination Chain are equal for bipartite graphs. Hence, the Domination Chain for bipartite graphs can be stated as follows. Theorem 2.17 ([185]) If 𝐺 is a bipartite graph, then ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) = Γ(𝐺) = IR(𝐺). Much research has focused on when equality is achieved between two parameters in the Domination Chain. In Chapter 5, we give a characterization of the trees having 𝛾(𝑇) = 𝑖(𝑇).

2.3

Parameters Suggested by the Definition of a Dominating Set

In this section, we define several types of dominating sets which naturally arise from different but equivalent definitions of a dominating set.

2.3.1

Total Dominating Sets

Perhaps the simplest and most natural variant of Definition 2.4, first formally defined in 1980 by Cockayne et al. [182], is the following, where all we do is change N[𝑆] = 𝑉 to N(𝑆) = 𝑉. Definition 2.18 A set 𝑆 ⊆ 𝑉 is a total dominating set, abbreviated TD-set, of a graph 𝐺 if N(𝑆) = 𝑉, that is, every vertex 𝑣 ∈ 𝑉 has at least one neighbor in 𝑆. The

Section 2.3. Parameters Suggested by the Definition of a Dominating Set

33

total domination number 𝛾t (𝐺) and the upper total domination number Γt (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal TD-set in 𝐺. A TD-set 𝑆 for which |𝑆| = 𝛾t (𝐺) is a 𝛾t -set of 𝐺, and a TD-set 𝑆 for which |𝑆| = Γt (𝐺) is a Γt -set of 𝐺. Definition 2.19 If 𝑋, 𝑌 ⊆ 𝑉, where 𝑋 and 𝑌 are not necessarily disjoint, then the set 𝑋 totally dominates the set 𝑌 if every vertex in 𝑌 has a neighbor in 𝑋, that is, 𝑌 ⊆ N(𝑋). In particular, if 𝑋 totally dominates 𝑉, then 𝑋 is a TD-set of 𝐺. For example, the tree 𝑇 in Figure 2.1 has only one minimal TD-set, namely, the set {𝑣 4 , 𝑣 5 }, so 𝛾t (𝑇) = Γt (𝑇) = 2. Notice that if 𝑆 is a TD-set, then the induced subgraph 𝐺 [𝑆] is isolate-free. For that matter, the graph 𝐺 itself cannot have any isolated vertices, because every vertex in 𝑉 must have a neighbor in 𝑆. Thus, when speaking of the total domination number of a graph 𝐺, we must assume that the graph 𝐺 in question is isolate-free. The following observation is worth noting. Observation 2.20 For every isolate-free graph 𝐺, 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ Γt (𝐺). However, Γ(𝑇) and Γt (𝐺) are not comparable. For a star 𝐾1,𝑘 with 𝑘 ≥ 3, we have Γ(𝐾1,𝑘 ) = 𝑘 > Γt (𝐾1,𝑘 ) = 2, but for a complete graph 𝐾𝑛 , we have Γ(𝐾𝑛 ) = 1 < Γt (𝐾𝑛 ) = 2. Recall that a double star 𝑆(𝑟, 𝑠), for 1 ≤ 𝑟 ≤ 𝑠, is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having 𝑟 leaf neighbors and the other 𝑠 leaf neighbors. The double star 𝑆(2, 3) is illustrated in Figure 2.1 for example. If 𝐺 is a double star 𝑆(𝑟, 𝑠), then 𝛾(𝐺) = 𝛾t (𝐺) = 2, while Γ(𝐺) = 𝑟 + 𝑠 ≥ 2 = Γt (𝐺). The domination and total domination numbers of a path 𝑃𝑛 and cycle 𝐶𝑛 on 𝑛 ≥ 3 vertices are given by the following closed formulas, which will be discussed in more detail in Chapter 4. Observation 2.21 For 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) =

𝑛 3

and 𝛾t (𝑃𝑛 ) = 𝛾t (𝐶𝑛 ) =

𝑛 2

+

𝑛 4

−

𝑛

.

4

By Observation 2.21, for example, we have 𝛾(𝑃12𝑘 ) = 4𝑘 < 6𝑘 = 𝛾t (𝑃12𝑘 ) for 𝑘 ≥ 1. For another example of strict inequality between the domination number and the total domination number, let 𝑇𝑘 be the graph formed from a star 𝐾1,𝑘 , for 𝑘 ≥ 2, by subdividing each edge exactly twice. For these trees 𝑇𝑘 , we have 𝛾(𝑇𝑘 ) = 𝑘 + 1 < 2𝑘 = 𝛾t (𝑇𝑘 ); see for example the tree 𝑇5 in Figure 2.4.

2.3.2

𝒌-Dominating Sets

Another definition equivalent to Definition 2.4 is the following.

Chapter 2. Fundamentals of Domination

34

Figure 2.4 𝛾(𝑇5 ) = 6 < 10 = 𝛾t (𝑇5 )

Definition 2.22 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| ≥ 1. This definition suggests the following generalization, first introduced by Fink and Jacobson [296] in 1985. Definition 2.23 A set 𝑆 ⊆ 𝑉 is a 𝑘-dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| ≥ 𝑘. The 𝑘-domination number 𝛾 𝑘 (𝐺) and upper 𝑘-domination number Γ𝑘 (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal 𝑘-dominating set in 𝐺. The reader is referred to a chapter on 𝑘-domination by Haynes et al. [413].

2.3.3

𝑯-forming Dominating Sets

Another definition equivalent to Definition 2.4 is the following. Definition 2.24 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if every vertex 𝑣 ∈ 𝑉 \ 𝑆 is a vertex in a 𝐾2 -subgraph with at least one vertex in 𝑆. This definition suggests the following generalization, introduced in 2003 by Haynes et al. [415]. Definition 2.25 For a given graph 𝐻, a vertex set 𝑆 is an 𝐻-forming set if every vertex in 𝑉 \ 𝑆 is contained in a copy of 𝐻 with a subset of vertices in 𝑆. Equivalently, a set 𝑆 ⊆ 𝑉 is an 𝐻-forming set of 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a subset 𝑅 ⊆ 𝑆, where |𝑅| = |𝑉 (𝐻)| − 1, such that 𝐺 [𝑅 ∪ {𝑣}] contains 𝐻 as a subgraph (not necessarily an induced subgraph). The minimum cardinality of a minimal 𝐻-forming set of 𝐺 is the 𝐻-forming number 𝛾 𝐻 (𝐺).

2.3.4

Perfect and Efficient Dominating Sets

The special case of Definition 2.22 when |N(𝑣) ∩ 𝑆| = 1 is worth consideration, as originally introduced by Bange et al. [56] in 1988, and independently by Dejter and Weichsel [220] in 1993 (see also [189] and [749]). Definition 2.26 A set 𝑆 ⊆ 𝑉 is called a perfect dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| = 1, and is called an efficient dominating set if for every 𝑣 ∈ 𝑉, |N[𝑣] ∩ 𝑆| = 1.

Section 2.3. Parameters Suggested by the Definition of a Dominating Set

35

It should be pointed out, however, that not all graphs have an efficient dominating set, for example, the cycle 𝐶5 does not have an efficient dominating set, but any three consecutive vertices of 𝐶5 form a perfect dominating set. In addition, there are graphs whose only perfect dominating sets are the entire set 𝑉, such as the complete tripartite graph 𝐾2,2,2 or the smaller (in size) graph 𝐾 2 + 2𝐾2 , as given in [189]. We present more on perfect and efficient dominating sets in Chapter 9.

2.3.5

Distance-𝒌 Dominating Sets

Still another definition equivalent to Definition 2.4 is the following: Definition 2.27 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑣, 𝑤) = 1. This definition suggests the following generalization, first introduced in 1976 by Slater [678]. Definition 2.28 A set 𝑆 ⊆ 𝑉 is a distance-𝑘 dominating set of 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑣, 𝑤) ≤ 𝑘. The distance𝑘 domination number 𝛾 ≤ 𝑘 (𝐺) and upper distance-𝑘 domination number Γ≤ 𝑘 (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal distance-𝑘 dominating set in 𝐺. The reader is referred to a chapter on distance domination by Henning in [413].

2.3.6

Fractional Domination

If 𝑆 ⊆ 𝑉 is a set of vertices of a graph 𝐺, we can define the characteristic function of 𝑆 to be the function 𝑓𝑆 : 𝑉 → {0, 1}, such that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 (𝑣) = 1 if 𝑣 ∈ 𝑆, and 𝑓𝑆 (𝑣) = 0 if 𝑣 ∈ 𝑉 \ 𝑆. Given the characteristic function 𝑓𝑆 of a set 𝑆 ⊆ 𝑉, we can define
∑︁ 𝑓𝑆 N(𝑣) = 𝑓𝑆 (𝑤). 𝑤 ∈N(𝑣)

This gives rise to the next equivalent definition of a dominating set of a graph 𝐺 = (𝑉, 𝐸). Definition 2.29 A set 𝑆 ⊆ 𝑉 is a dominating set of 𝐺 if the characteristic function 𝑓𝑆
satisfies the condition that for every vertex 𝑣 ∈ 𝑉 \ 𝑆, 𝑓𝑆 N(𝑣) ≥ 1. Several variations are suggested by this definition. Definition 2.30 A set 𝑆 ⊆ 𝑉 is a dominating set of 𝐺 if the
characteristic function 𝑓𝑆 satisfies the condition that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 N[𝑣] ≥ 1. Definition 2.31 A set 𝑆 ⊆ 𝑉 is a total dominating set of 𝐺 if the characteristic
function 𝑓𝑆 satisfies the condition that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 N(𝑣) ≥ 1. The following definition of fractional domination was first given in 1987 by Hedetniemi et al. [444].

Chapter 2. Fundamentals of Domination

36

Definition 2.32 A function 𝑓 : 𝑉 → [0, 1], which assigns to each vertex 𝑣 ∈ 𝑉 a a fractional dominating rational number in the closed unit interval [0, 1] is called
N[𝑣] ≥ 1. The fractional dominafunction if for every vertex 𝑣 ∈ 𝑉, we have 𝑓 Í tion number 𝛾f (𝐺) equals the minimum 𝑣 ∈𝑉 𝑓 (𝑣) over all fractional dominating functions 𝑓 on 𝐺. The fractional version of total domination is defined as follows. Definition 2.33 A function 𝑓 : 𝑉 → [0, 1] is called a
fractional total dominating function if for every vertex 𝑣 ∈ 𝑉, we have 𝑓ÍN(𝑣) ≥ 1. The fractional total domination number 𝛾tf (𝐺) equals the minimum 𝑣 ∈𝑉 𝑓 (𝑣) over all fractional total dominating functions 𝑓 on 𝐺. The reader is referred to a chapter on fractional domination and fractional total domination by Goddard and Henning in [356]. We note that there are many other variations of domination that are not presented here. There is a discusison in [417] of parameters that arise from placing conditions on a dominating set and/or on the method of dominating. For example, if the subgraph 𝐺 [𝑆] induced by a dominating set 𝑆 is connected, then 𝑆 is a connected dominating set, while if 𝐺 [𝑆] has a perfect matching, then 𝑆 is a paired dominating set. Since our focus in this book is on domination, total domination, and independent domination, we refer the reader to the companion books [413, 414] for information on other types of domination in graphs.

2.4

Equivalent Formulations of Domination

In this section, we show that the concept of a dominating set in a graph has several equivalent formulations.

2.4.1 Pendant Edges in Spanning Forests A pendant edge in a graph 𝐺 = (𝑉, 𝐸) is any edge 𝑢𝑣 ∈ 𝐸 for which 𝑢 or 𝑣 is a leaf, that is, a vertex of degree 1. A graph 𝐺 is a forest if it is acyclic, that is, contains no cycles. Definition 2.34 Let 𝜀 𝑓 (𝐺) equal the maximum number of pendant edges in a spanning forest of 𝐺. In 1974 Nieminen [613] proved the following fundamental result relating the domination number of a graph 𝐺 to the maximum number of pendant edges in a spanning forest of 𝐺. Theorem 2.35 ([613]) For any graph 𝐺 of order 𝑛, 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛. Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } ⊆ 𝑉 be any 𝛾-set of 𝐺. Construct a spanning forest 𝐹 of 𝐺 as follows. Begin by letting 𝐹 consist only of 𝑛 isolated vertices. For every vertex 𝑤 ∈ 𝑉 \ 𝑆, arbitrarily select an edge between 𝑤 and a vertex in 𝑆 and add

Section 2.4. Equivalent Formulations of Domination

37

it to 𝐹; since 𝑆 is a dominating set, every vertex in 𝑉 \ 𝑆 must be adjacent to at least one vertex in 𝑆, so there must be at least one such edge. The graph thus constructed is acyclic and has precisely |𝑉 \ 𝑆| edges, each of which is a pendant edge. Therefore, 𝜀 𝑓 (𝐺) ≥ |𝑉 \ 𝑆| = 𝑛 − 𝛾(𝐺). Conversely, let 𝐹 be a spanning forest of a graph 𝐺 containing 𝜀 𝑓 (𝐺) = 𝑘 pendant edges, 𝑒 1 , 𝑒 2 , . . . , 𝑒 𝑘 . For each pendant edge 𝑒 𝑖 , select a leaf 𝑤 𝑖 in 𝐹 incident to this edge, and let 𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑘 be the set 𝑆 of 𝑘 selected leaves. It follows that the set 𝑉 \ 𝑆 is a dominating set of 𝐺, since every vertex in 𝑆 is a leaf in 𝐹 that is adjacent to a vertex in 𝑉 \ 𝑆. Thus, 𝛾(𝐺) ≤ |𝑉 \ 𝑆| = 𝑛 − 𝜀 𝑓 (𝐺), or 𝜀 𝑓 (𝐺) ≤ 𝑛 − 𝛾(𝐺).

2.4.2

Enclaveless Sets

We next define a type of set first introduced in 1977 by Slater [679]. Definition 2.36 For a graph 𝐺 = (𝑉, 𝐸) and a set 𝑆 ⊆ 𝑉, a vertex 𝑣 ∈ 𝑆 is called an enclave of 𝑆 if N[𝑣] ⊆ 𝑆. A set is said to be enclaveless if it does not contain any enclaves. In other words, every vertex in 𝑆 has a neighbor in 𝑉 \ 𝑆. The enclaveless number Ψ(𝐺) is the maximum cardinality of an enclaveless set of 𝐺 and 𝜓(𝐺) is the minimum cardinality of a maximal enclaveless set. Having the definition of an enclaveless set, one can prove the following, first shown in [679]. Theorem 2.37 ([679]) For any graph 𝐺 of order 𝑛, 𝛾(𝐺) + Ψ(𝐺) = 𝑛. Proof Let 𝐺 be a graph of order 𝑛, and let 𝑆 be a 𝛾-set in 𝐺. Consider the complement 𝑆 = 𝑉 \ 𝑆. It follows that 𝑆 is an enclaveless set, for if 𝑆 contains the closed neighborhood of some vertex 𝑣 ∈ 𝑆, then the vertex 𝑣 is not adjacent to any vertex in 𝑆, contradicting the assumption that 𝑆 is a dominating set. Since 𝑆 is an enclaveless set, it follows that Ψ(𝐺) ≥ |𝑆| = |𝑉 \ 𝑆| = 𝑛 − 𝛾(𝐺). Conversely, let 𝑆 ′ be a Ψ-set of 𝐺, that is, a maximum cardinality enclaveless set in 𝐺. Consider the set 𝑆 = 𝑉 \ 𝑆 ′ . Since 𝑆 ′ is enclaveless, every vertex in 𝑆 ′ has a neighbor in 𝑉 \ 𝑆 ′ = 𝑆. Thus, the set 𝑆 dominates the set 𝑆 ′ , and so 𝑆 is a dominating set of 𝐺. Therefore, 𝛾(𝐺) ≤ |𝑆| = |𝑉 \ 𝑆 ′ | = 𝑛 − Ψ(𝐺), or Ψ(𝐺) ≤ 𝑛 − 𝛾(𝐺). The simple proof of this theorem enables one to also prove the following two results. Corollary 2.38 For any graph 𝐺, the complement of any (minimal) dominating set is a (maximal) enclaveless set, and conversely, the complement of any (maximal) enclaveless set is a (minimal) dominating set. Corollary 2.39 For any graph 𝐺 of order 𝑛, Γ(𝐺) + 𝜓(𝐺) = 𝑛.

2.4.3

Spanning Star Partitions

Still another equivalent formulation of the domination number can be given, due to Hedetniemi [445] in 1983.

38

Chapter 2. Fundamentals of Domination

Definition 2.40 A spanning star partition of 𝐺 is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of the vertex set of 𝐺 such that for all 𝑖 ∈ [𝑘], the induced subgraph 𝐺 [𝑉𝑖 ] contains a nontrivial spanning star, that is, 𝐺 [𝑉𝑖 ] has order 𝑛 ≥ 2 and contains a vertex which is adjacent to every other vertex in 𝐺 [𝑉𝑖 ]. Let 𝛼★ max (𝐺) denote the maximum order of a spanning star partition of 𝐺, and 𝛼★ min (𝐺) denote the minimum order of a spanning star partition of 𝐺. We note that the spanning star partition defined in Definition 2.40 is different than the definition of Acharya and Walikar [7], who in 1979 defined a star partition to be a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } such that for all 𝑖 ∈ [𝑘], the induced subgraph 𝐺 [𝑉𝑖 ] is a star, that is a graph of the form 𝐾1,𝑛𝑖 , where 𝑛𝑖 = 0 is allowed (𝐾1,0 is the trivial graph 𝐾1 ). Recall that 𝛼′ (𝐺) is the matching number of 𝐺, that is, the maximum cardinality of an independent set of edges in 𝐺. Theorem 2.41 ([445]) For any isolate-free graph 𝐺, 𝛼′ (𝐺) = 𝛼★ max (𝐺). Theorem 2.42 ([445]) For any isolate-free graph 𝐺, 𝛾(𝐺) = 𝛼★ min (𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a 𝛾-set of 𝐺. For each vertex 𝑣 𝑖 ∈ 𝑆, create a set 𝑉𝑖 containing vertex 𝑣 𝑖 , for 𝑖 ∈ [𝑘], as follows. For each vertex 𝑤 ∈ 𝑉 \ 𝑆, select one of its neighbors 𝑣 𝑗 ∈ 𝑆 (since 𝑆 is a dominating set, there must be at least one such neighbor) and place 𝑤 ∈ 𝑉 𝑗 . This will create a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉, such that each set 𝑉𝑖 contains a vertex 𝑣 𝑖 ∈ 𝑆 that is adjacent to every other vertex in the set. Among all such partitions, let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be one having a minimum number of subsets 𝑉𝑖 for which |𝑉𝑖 | = 1. If |𝑉𝑖 | ≥ 2 for each 𝑉𝑖 ∈ 𝜋, then 𝜋 defines a spanning star partition of 𝐺 having order 𝛾(𝐺). If not, renaming vertices in 𝑆 if necessary, assume without loss of generality that 𝑉1 = {𝑣 1 }. Since 𝐺 has no isolated vertices, 𝑣 1 must be adjacent to at least one other vertex 𝑤. If 𝑤 ∈ 𝑆, then the set 𝑆 \ {𝑣 1 } is a dominating set, contradicting the minimality of 𝑆. Hence, 𝑤 ∈ 𝑉 \ 𝑆. By the construction of the partition 𝜋, vertex 𝑤 must be in some 𝑉𝑖 of 𝜋 for 𝑖 ∈ [𝑘] \ {1}. Renaming vertices 𝑤 ∈ 𝑉2 , that is, both 𝑤 and 𝑣 2 are in the in 𝑆 if necessary, we may assume that
set 𝑉2 . If |𝑉2 | = 2, then 𝑆 \ {𝑣 1 , 𝑣 2 } ∪ {𝑤} is a dominating set of 𝐺, contradicting the minimality of 𝑆. Hence, |𝑉2 | ≥ 3, and  we can therefore form a new partition 𝜋 ′ = 𝑉1 ∪ {𝑤}, 𝑉2 \ {𝑤}, 𝑉3 , 𝑉4 , . . . , 𝑉𝑘 having fewer sets consisting of a single vertex, contradicting our choice of 𝜋. Hence, from an (arbitrary) 𝛾-set 𝑆, we can always construct a spanning star partition of order 𝛾(𝐺), and so 𝛼★ min (𝐺) ≤ 𝛾(𝐺). Conversely, let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be a spanning star partition of order 𝛼★ min (𝐺). The set 𝑆 consisting of the central vertices of the spanning stars in the induced subgraphs 𝐺 [𝑉𝑖 ] forms a dominating set. Thus, 𝛾(𝐺) ≤ 𝛼★ min (𝐺). We summarize the results of Theorems 2.35, 2.37, and 2.42 as follows. Corollary 2.43 If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) = 𝑛 − 𝜀 𝑓 (𝐺) = 𝑛 − Ψ(𝐺) = 𝛼★ min (𝐺).

Section 2.4. Equivalent Formulations of Domination

39

2.4.4 Non-dominating Partitions The domination and total domination numbers of any graph also have equivalent formulations in terms of what are called non-dominating partitions of a graph 𝐺 and its complement 𝐺, as introduced in 2016 by Desormeaux et al. [228]. Definition 2.44 A set of vertices that does not dominate 𝐺 is called a non-dominating set of 𝐺. In other words, for any non-dominating set 𝑆, there exists a vertex 𝑣 ∈ 𝑉 \ 𝑆 that has no neighbor in 𝑆, which means that 𝑣 is an enclave of 𝑉 \ 𝑆. A partition of the vertices of 𝐺 into non-dominating sets is a non-dominating partition. Let 𝜋nd (𝐺) equal the minimum order of a non-dominating partition of 𝐺. Definition 2.45 A set of vertices that does not totally dominate 𝐺 is called a nontotal dominating set of 𝐺, abbreviated non-TD-set. Equivalently, for any non-TD-set 𝑆, either 𝑆 is a non-dominating set or there is an isolated vertex in the subgraph 𝐺 [𝑆] induced by 𝑆. A partition of the vertices of 𝐺 into non-total dominating sets is a non-total dominating partition. Let 𝜋ntd (𝐺) equal the minimum order of a non-total dominating partition of 𝐺. If the domination number of a graph 𝐺 equals 1, then clearly, 𝐺 has a dominating vertex and hence, 𝐺 does not have a non-dominating partition. But the identity  partition, 𝜋 = {𝑣 1 }, {𝑣 2 }, . . . , {𝑣 𝑛 } , is a non-dominating partition for every graph 𝐺 with 𝛾(𝐺) ≥ 2. Also, since 𝑉 is a dominating set of 𝐺, any non-dominating partition must have a least two sets. Hence, we have the following observation. Observation 2.46 A graph 𝐺 of order 𝑛 has a non-dominating partition if and only if 𝐺 has no dominating vertex, in which case 2 ≤ 𝜋nd (𝐺) ≤ 𝑛. The next result relates the minimum order of a non-dominating partition of a graph 𝐺 to the total domination number of its complement 𝐺. Theorem 2.47 ([228]) If 𝐺 is a graph with no dominating vertex, then 𝛾t (𝐺) = 𝜋nd (𝐺). Proof Let 𝐺 be a graph with no dominating vertex, and let 𝜋 = {𝐴1 , 𝐴2 , . . . , 𝐴 𝑘 } be a non-dominating partition of 𝐺 of minimum order 𝜋nd (𝐺) = 𝑘. By Observation 2.46, we have 2 ≤ 𝜋nd (𝐺) ≤ 𝑛. We first show that 𝛾t (𝐺) ≤ 𝜋nd (𝐺). Since 𝜋 is a non-dominating partition of 𝐺, the set 𝐴𝑖 does not dominate 𝐺 for each 𝑖 ∈ [𝑘], implying that there exists a vertex 𝑎 𝑖 ∈ 𝑉 \ 𝐴𝑖 such that N(𝑎 𝑖 ) ∩ 𝐴𝑖 = ∅. Moreover, since 𝜋 has minimum cardinality, 𝑎 𝑖 is dominated by the set 𝐴 𝑗 for every 𝑗 ∈ [𝑘] \ {𝑖}; otherwise, the partition formed from 𝜋 by removing 𝐴𝑖 and 𝐴 𝑗 and adding 𝐴𝑖 ∪ 𝐴 𝑗 is a non-dominating partition of 𝐺 with order less than 𝜋nd (𝐺), a contradiction. Ð Hence, the non-dominated vertices 𝑎 𝑖 𝑘 are distinct for each 𝑖 ∈ [𝑘]. Note that 𝐴 = 𝑖=1 {𝑎 𝑖 } is a dominating set of 𝐺 of cardinality 𝑘, since 𝑎 𝑖 dominates every vertex in 𝐴𝑖 in the complement 𝐺. If 𝐴 is a TD-set of 𝐺, then 𝛾t (𝐺) ≤ 𝑘. If not, then it follows that there exists an 𝑎 𝑖 ∈ 𝐴 such that 𝑎 𝑖 is an isolated vertex of 𝐺 [ 𝐴]. By our selection of 𝑎 𝑖 , it follows that 𝑎 𝑖 ∉ 𝐴𝑖 . Hence, 𝑎 𝑖 ∈ 𝐴 𝑗 for some 𝑗 ∈ [𝑘] \ {𝑖}. Since the vertex 𝑎 𝑗 ∈ 𝐴 has no neighbor

40

Chapter 2. Fundamentals of Domination

in 𝐴 𝑗 in 𝐺, we note, in particular, that 𝑎 𝑗 is not adjacent to 𝑎 𝑖 in 𝐺. Hence, in 𝐺, the vertices 𝑎 𝑖 and 𝑎 𝑗 are adjacent, contradicting the fact that 𝑎 𝑖 is an isolate in 𝐺 [ 𝐴]. Thus, 𝐴 is a TD-set for 𝐺, and so 𝛾t (𝐺) ≤ | 𝐴| = 𝑘 = 𝜋nd (𝐺). Since 𝐺 has no dominating vertex, 𝐺 has no isolated vertex, that is, the total domination number of 𝐺 is defined. To see that 𝛾t (𝐺) ≥ 𝜋nd (𝐺), let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 ℓ } be a 𝛾t -set of 𝐺, and so ℓ = 𝛾t (𝐺). For 𝑖 ∈ [ℓ], let 𝐵𝑖 = N𝐺 (𝑣 𝑖 ). Since 𝑆 is a TD-set of 𝐺, every vertex of 𝑉 belongs to some 𝐵𝑖 . Moreover, since 𝑆 is a 𝛾t -set of 𝐺, we have pn𝐺 (𝑣 𝑖 , 𝑆) ≠ ∅ and pn𝐺 (𝑣 𝑖 , 𝑆) ⊆ 𝐵𝑖 for each 𝑖 ∈ [ℓ]. We partition the vertices of 𝑉 as follows: let 𝐵1′ = 𝐵1 . For each 𝑗 ≥ 2, form 𝐵′𝑗 by removing the vertices Ð 𝑗 −1 from 𝐵 𝑗 that are contained in 𝑖=1 𝐵𝑖 . Note that pn𝐺 (𝑣 𝑖 , 𝑆) ⊆ 𝐵𝑖′ , and so 𝐵𝑖′ ≠ ∅ for 𝑖 ∈ [ℓ]. Note also that the vertex 𝑣 𝑖 ∉ 𝐵𝑖′ and 𝑣 𝑖 is not dominated by 𝐵𝑖′ in 𝐺 for 𝑖 ∈ [ℓ]. Hence, each 𝐵𝑖′ is a non-dominating set of 𝐺. Thus, 𝜋 = {𝐵1′ , 𝐵2′ , . . . , 𝐵ℓ′ } is a partition of 𝑉 into non-dominating sets of 𝐺, implying that 𝜋nd (𝐺) ≤ |𝜋| = ℓ = 𝛾t (𝐺). Consequently, 𝛾t (𝐺) = 𝜋nd (𝐺). A proof similar to the one used to prove Theorem 2.47 yields the following result for the domination number. Theorem 2.48 ([228]) For any graph 𝐺, 𝛾(𝐺) = 𝜋ntd (𝐺).

2.4.5

Total Domination and Splitting Graphs

Still another equivalent formulation of the total domination number in terms of the domination number can be given as follows. Definition 2.49 The splitting graph 𝐺 2 = (𝑉 ∪ 𝑉 ′ , 𝐸 ∪ 𝐸 ′ ) of an isolate-free graph 𝐺 = (𝑉, 𝐸) is the graph obtained from 𝐺 by adding to it a set 𝑉 ′ of vertices consisting of one copy 𝑣 ′ of every vertex 𝑣 ∈ 𝑉, and adding a set 𝐸 ′ of edges of the form 𝑣 ′ 𝑤, where 𝑣 ′ ∈ 𝑉 ′ , 𝑤 ∈ 𝑉, and 𝑣𝑤 ∈ 𝐸. Thus, the copy 𝑣 ′ ∈ 𝑉 ′ of the vertex 𝑣 ∈ 𝑉 is joined in 𝐺 2 to all neighbors of 𝑣 in 𝐺, that is, N𝐺2 (𝑣 ′ ) = N𝐺 (𝑣). We note that the set 𝑉 ′ is an independent set. Theorem 2.50 For any isolate-free graph 𝐺, 𝛾(𝐺 2 ) = 𝛾t (𝐺). Proof Let 𝑆 be a 𝛾t -set of 𝐺. Since 𝑆 is a TD-set of 𝐺, every vertex in 𝑉 has a neighbor in the set 𝑆 in 𝐺. Thus, the set 𝑆 totally dominates the set 𝑉 in 𝐺, and therefore the set 𝑉 in 𝐺 2 (noting that 𝐺 2 [𝑉] = 𝐺). Adopting the notation in Definition 2.49, let 𝑣 ′ ∈ 𝑉 ′ be the copy of the vertex 𝑣 ∈ 𝑉. Let 𝑥 be a neighbor of 𝑣 in 𝐺 that belongs to the TD-set 𝑆 of 𝐺. By construction, the vertices 𝑥 and 𝑣 ′ are adjacent in 𝐺 2 , implying that the set 𝑆 totally dominates the set 𝑉 ′ in 𝐺 2 . Hence, the set 𝑆 totally dominates the set 𝑉 ∪ 𝑉 ′ in 𝐺 2 , that is, 𝑆 is a TD-set of 𝐺 2 , and so 𝛾(𝐺 2 ) ≤ 𝛾t (𝐺 2 ) ≤ |𝑆| = 𝛾t (𝐺). Conversely, we must show that 𝛾t (𝐺) ≤ 𝛾(𝐺 2 ). Among all 𝛾-sets of 𝐺 2 , let 𝑆 have a minimum number of vertices in 𝑉 ′ . We show firstly that 𝑆 ⊆ 𝑉, that is, 𝑆 has ′ ′ ′ no vertices in 𝑉 ′ . Suppose, to the contrary,
that 𝑆 contains a vertex 𝑣 ∈ 𝑉 . If 𝑣 has ′ a neighbor 𝑥 ∈ 𝑆, then the set 𝑆 \ {𝑣 } ∪ {𝑣} is also a 𝛾-set of 𝐺 2 having fewer

Section 2.4. Equivalent Formulations of Domination

41

vertices in 𝑉 ′ than 𝑆, contradicting the choice of 𝑆. Hence, 𝑣 ′ has no neighbor in 𝑆, implying that the vertex 𝑣 has no neighbor in 𝑆. Thus, 𝑣 must be in 𝑆 and both 𝑣 ′ and 𝑣 are isolated vertices in 𝐺 [𝑆]. Now since 𝐺 is isolate-free, the vertices 𝑣 ′ and
𝑣 must have a common neighbor 𝑥 ∉ 𝑆. We note that 𝑥 ∈ 𝑉. It follows that 𝑆 \ {𝑣 ′ } ∪ {𝑥} is a 𝛾-set of 𝐺 2 having fewer vertices in 𝑉 ′ than 𝑆, a contradiction. Therefore, 𝑆 ⊆ 𝑉. We show next that there are no isolated vertices in 𝐺 [𝑆]. Suppose, to the contrary, that there exists an isolated vertex 𝑣 in 𝐺 [𝑆]. Since no neighbor of 𝑣 is in 𝑆, then no neighbor of 𝑣 ′ is in 𝑆 either. Thus, either 𝑣 ′ ∈ 𝑆, which contradicts our earlier observation that 𝑆 ⊆ 𝑉, or 𝑣 ′ ∉ 𝑆, which contradicts the fact that 𝑆 is a dominating set of 𝐺 2 . Hence, there are no isolated vertices in 𝐺 [𝑆], implying that 𝑆 is a TD-set of 𝐺, and hence 𝛾t (𝐺) ≤ |𝑆| = 𝛾(𝐺 2 ).

2.4.6

Dominating Sets and (1, 4 : 3)-Sets

Definition 2.51 A set 𝑆 ⊆ 𝑉 is called an (𝑖, 𝑗)-set in a graph 𝐺 if every vertex in 𝑉 \ 𝑆 is at most distance 𝑖 from at least one vertex in 𝑆 and is at most distance 𝑗 from a second, distinct vertex in 𝑆. The (𝑖, 𝑗)-domination number 𝛾 (𝑖, 𝑗 ) (𝐺) equals the minimum cardinality of an (𝑖, 𝑗)-set in 𝐺. Notice that by definition every (1, 𝑗)-set is a dominating set, and therefore, for any nontrivial, connected graph 𝐺, we have 𝛾(𝐺) ≤ 𝛾 (1, 𝑗 ) (𝐺). Definition 2.52 A set 𝑆 ⊆ 𝑉 is an (𝑖, 𝑗 : 𝑘)-set of 𝐺 if 𝑆 is an (𝑖, 𝑗)-set and every vertex in 𝑆 is at most distance 𝑘 from a second vertex in 𝑆. The next theorem was proved in 2008 by Hedetniemi et al. [443]. Theorem 2.53 ([443]) Every nontrivial dominating set of a connected graph 𝐺 is a (1, 4 : 3)-set. Proof Let 𝑆 ⊆ 𝑉 be any dominating set in a connected graph 𝐺, where we assume that |𝑆| ≥ 2. We will first show that for any vertex 𝑢 ∈ 𝑆, there exists another vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑢, 𝑤) ≤ 3. Suppose, to the contrary, that there exists a vertex 𝑢 ∈ 𝑆 such that the minimum distance between 𝑢 and another vertex in 𝑆 \ {𝑢} is at least 4. Let 𝑢, 𝑥, 𝑦 be the first three vertices on a shortest path from vertex 𝑢 to some vertex 𝑤 ∈ 𝑆, where necessarily 𝑥, 𝑦 ∈ 𝑉 \ 𝑆 and 𝑑 (𝑢, 𝑤) > 3. Notice that vertex 𝑢 cannot be adjacent to vertex 𝑦, else this is not a shortest path from 𝑢 to 𝑤. But since 𝑆 is a dominating set, vertex 𝑦 must be adjacent to at least one vertex 𝑧 ∈ 𝑆. It follows therefore that 𝑑 (𝑢, 𝑧) ≤ 3, contradicting our assumption. Thus, 𝑑 (𝑢, 𝑤) ≤ 3 for some 𝑤 ∈ 𝑆. Since 𝑆 is a dominating set, every vertex in 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to some vertex in 𝑆. Further, since every vertex in 𝑆 is at most distance 3 to some other vertex in 𝑆, 𝑣 is at most distance 4 to a second vertex in 𝑆. It follows that the set 𝑆 is (1, 4 : 3)-set. It is important to observe that the (1, 4 : 3) bounds in Theorem 2.53 can be achieved, and thus the values of 𝑗 = 4 and 𝑘 = 3 cannot be decreased for arbitrary dominating sets in connected graphs. The set 𝑆 = {𝑣 2 , 𝑣 5 } in the path 𝑃6 in Figure 2.5 can be seen to be a (1, 4 : 3)-set.

Chapter 2. Fundamentals of Domination

42

Corollary 2.54 For any connected graph 𝐺, 𝛾(𝐺) = 𝛾 (1,4) (𝐺).

𝑣1

𝑣2

𝑣3

𝑣4

𝑣5

𝑣6

Figure 2.5 A (1, 4 : 3)-set of 𝑃6 Notice that the path 𝑃6 has only one 𝛾-set, namely the set 𝑆 = {𝑣 2 , 𝑣 5 }, and since this set 𝑆 is a (1, 4)-set, we can conclude that 𝛾(𝑃6 ) = 2 = 𝛾 (1,4) (𝑃6 ). We can also conclude that the path 𝑃6 does not have a (1, 3)-set of cardinality 2. However, the set 𝑆 ′ = {𝑣 1 , 𝑣 4 , 𝑣 6 } is both a (1, 3)-set and a (1, 2)-set, and the set 𝑆 ′′ = {𝑣 1 , 𝑣 3 , 𝑣 5 , 𝑣 6 } is a smallest (1, 1)-set. Notice that a (1, 1)-set is, in fact, a 2-dominating set in that every vertex in 𝑉 \ 𝑆 has at least two neighbors in 𝑆. Notice also that a (1, 4)-set is also a (1, 𝑗)-set for every 𝑗 ≥ 5. Thus, when we say that every dominating set in a graph 𝐺 is a (1, 4)-set, it might also be a (1, 3)-set, a (1, 2)-set, or a (1, 1)-set as well; but every (1, 3)-set, (1, 2)-set, and (1, 1)-set is always a (1, 4)-set. In 2012 Hedetniemi et al. [438] proved the following: (i) Every independent dominating set is a (1, 4 : 3)-set. (ii) Every total dominating set is a (1, 2 : 1)-set. (iii) For every graph 𝐺, 𝛾(𝐺) = 𝛾 (1,4) (𝐺) ≤ 𝛾 (1,3) (𝐺) ≤ 𝛾 (1,2) (𝐺) ≤ 𝛾 (1,1) (𝐺) = 𝛾2 (𝐺).

2.5

Domination in Terms of Perfection in Graphs

To the definitions of independent sets, dominating sets, and irredundant sets, we now introduce several concepts and parameters having to do with what is called perfection in graphs. These concepts were first introduced by Fricke et al. [310] in 1999 and later developed in detail by Hedetniemi et al. [439] in 2013. In the remainder of this section, we review the results given in [439]. Definition 2.55 Given a set 𝑆 ⊆ 𝑉 in a graph 𝐺, a vertex 𝑣 ∈ 𝑉 is said to be 𝑆-perfect if |N[𝑣] ∪ 𝑆| = 1, that is, the closed neighborhood N[𝑣] contains exactly one vertex in 𝑆. Definition 2.56 Given a set 𝑆 ⊆ 𝑉 in a graph 𝐺, a vertex 𝑣 is almost 𝑆-perfect if it is either 𝑆-perfect or is adjacent to an 𝑆-perfect vertex. When a set 𝑆 has been given and is assumed, we simply say that a vertex is perfect or almost perfect without referring to the set 𝑆. Definition 2.57 A set 𝑆 ⊆ 𝑉 is internally perfect if every vertex 𝑣 ∈ 𝑆 is perfect, and is internally almost perfect if every vertex 𝑣 ∈ 𝑆 is almost perfect; for brevity we say that an internally almost perfect set is an ap-set. Let 𝜃 ap (𝐺) and Θap (𝐺) equal the minimum and maximum cardinality, respectively, of a maximal ap-set in 𝐺.

Section 2.5. Domination in Terms of Perfection in Graphs

43

Definition 2.58 A set 𝑆 ⊆ 𝑉 is externally perfect if every vertex in 𝑉 \ 𝑆 is perfect, and is externally almost perfect if every vertex in 𝑉 \ 𝑆 is either perfect or adjacent to a perfect vertex; for brevity we say that an externally almost perfect set is an eap-set. Let 𝜃 ap (𝐺) and Θap (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal eap-set in 𝐺. Similarly, a set 𝑆 is completely perfect if every vertex 𝑣 ∈ 𝑉 is perfect, that is, if 𝑆 is both internally and externally perfect. In the graph in Figure 2.6, given in [439], a vertex labeled “p” is perfect, while a vertex labeled “ap” is almost perfect with respect to the three red vertices forming a set 𝑆, two of which are almost perfect while the third is perfect. This set 𝑆 is also externally almost perfect since every vertex in 𝑉 \ 𝑆 is either perfect with respect to 𝑆 or adjacent to a perfect vertex. p p

ap p

ap ap

ap

p ap

p p

ap ap

Figure 2.6 A perfect neighborhood set

Definition 2.59 A set 𝑆 ⊆ 𝑉 is a perfect neighborhood set if every vertex 𝑣 ∈ 𝑉 is either perfect or is adjacent to a perfect vertex (see [191, 284, 310, 440]). The perfect neighborhood number 𝜃 (𝐺) is the minimum cardinality of a perfect neighborhood set in 𝐺, while the upper perfect neighborhood number Θ(𝐺) is the maximum cardinality ap ap of a perfect neighborhood set in 𝐺. Let 𝜃 p (𝐺) and Θp (𝐺) equal the minimum and maximum cardinality, respectively, of an independent perfect neighborhood set in 𝐺. The following somewhat surprising theorem was first proved in [310]. We give a proof of this result in Chapter 14, where a more detailed discussion of the upper domination number is given. Theorem 2.60 ([310]) For any graph 𝐺, Γ(𝐺) = Θ(𝐺). Definition 2.61 A set 𝑆 ⊆ 𝑉 is an eap irredundant set, eap dominating set, or an eap independent set if it is a maximal irredundant, minimal dominating, or maximal independent set that is also an eap-set. Thus, every vertex 𝑣 ∈ 𝑉 \ 𝑆 is either perfect or is adjacent to a perfect vertex. Let irap (𝐺), 𝛾 ap (𝐺), 𝑖 ap (𝐺), 𝛼ap (𝐺) Γap (𝐺), and IRap (𝐺) denote the minimum and maximum cardinalities of such sets. Given these definitions, we can relate them to independent, dominating, and irredundant sets. For example, the concept of a set 𝑆 being internally perfect is equivalent to the concept of a set being independent.

44

Chapter 2. Fundamentals of Domination

Proposition 2.62 ([439]) A set 𝑆 is internally perfect if and only if it is independent. Proof A vertex 𝑢 ∈ 𝑆 is perfect if and only if |N[𝑢] ∩ 𝑆| = 1, that is, it is an isolated vertex in the induced subgraph 𝐺 [𝑆]. Thus, if every vertex in 𝑆 is perfect, then 𝑆 is an independent set. Conversely, if 𝑆 is an independent set, then clearly every vertex 𝑣 ∈ 𝑆 satisfies the condition that |N[𝑣] ∩ 𝑆| = 1. Therefore, 𝑆 is internally perfect. ap

Corollary 2.63 ([439]) For any graph 𝐺, 𝜃 p (𝐺) ≤ 𝑖(𝐺) = 𝑖 ap (𝐺). Proof We first show that 𝑖(𝐺) = 𝑖 ap (𝐺). Let 𝑆 be a maximal independent set of minimum cardinality, and so |𝑆| = 𝑖(𝐺). By Proposition 2.62, the set 𝑆 is internally perfect. By definition, 𝑆 is also a minimal dominating set. Thus, every vertex in 𝑆 is perfect, and every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex. Thus, 𝑆 is an internally perfect set whose complement 𝑉 \ 𝑆 is an ap-set, and therefore, 𝑖 ap (𝐺) ≤ |𝑆| = 𝑖(𝐺). Conversely, since every 𝑖 ap -set of 𝐺 is maximal independent, it is therefore an independent dominating set, implying that 𝑖(𝐺) ≤ 𝑖 ap (𝐺). ap The fact that 𝜃 p (𝐺) ≤ 𝑖(𝐺) follows from the observation that every maximal independent set is an independent perfect neighborhood set. ap

Corollary 2.64 ([439]) For any graph 𝐺, 𝛼(𝐺) = 𝛼ap (𝐺) = Θp (𝐺). Proof As discussed previously in this chapter, every 𝛼-set 𝑆 of 𝐺 is both a maximal independent set and a minimal dominating set. Therefore, from Proposition 2.62, 𝑆 is an internally perfect set and an eap-set, since every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex in 𝑆. It is therefore an independent perfect neighborhood set, and ap ap thus, 𝛼(𝐺) ≤ Θp (𝐺). But every Θp -set of 𝐺 is an independent set, and therefore by ap definition, Θp (𝐺) ≤ 𝛼(𝐺). Similarly, every 𝛼-set 𝑆 of 𝐺 is a maximal independent set that is also externally almost perfect, since every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex in 𝑆. Thus, 𝛼(𝐺) ≤ 𝛼ap (𝐺). But every 𝛼ap -set of 𝐺 is an independent set, and therefore by definition, 𝛼ap (𝐺) ≤ 𝛼(𝐺). The next results follow directly from the definitions. Proposition 2.65 ([439]) A set 𝑆 is externally perfect if and only if 𝑆 is a perfect dominating set. Proposition 2.66 ([439]) A set 𝑆 is completely perfect if and only if 𝑆 is an efficient dominating set. We remark that an efficient dominating set is also called a perfect code in the literature, and efficient dominating sets will be covered in Chapter 9. Next, we show that the concept of being internally almost perfect (ap) is equivalent to the concept of being irredundant. Proposition 2.67 ([439]) A set 𝑆 is internally almost perfect if and only if 𝑆 is irredundant. Proof If a set 𝑆 is internally almost perfect, then every vertex 𝑢 ∈ 𝑆 is either perfect or adjacent to a perfect vertex. Either 𝑢 is an isolated vertex in 𝐺 [𝑆], in which case it

Section 2.5. Domination in Terms of Perfection in Graphs

45

is perfect and is its own 𝑆-private neighbor or 𝑢 is adjacent to a perfect vertex 𝑤. Thus, 𝑤 ∈ 𝑉 \ 𝑆 and |N[𝑤] ∩ 𝑆| = |{𝑢}| = 1, and so 𝑤 is an 𝑆-external private neighbor of 𝑢. Therefore, every vertex 𝑢 ∈ 𝑆 has an 𝑆-private neighbor and hence 𝑆 is irredundant. Conversely, if 𝑆 is irredundant, then every vertex 𝑢 ∈ 𝑆 is either its own 𝑆-private neighbor, in which case 𝑢 is perfect or has an 𝑆-external private neighbor 𝑤. But in this case 𝑤 is perfect and therefore 𝑢 is adjacent to a perfect vertex. Therefore, 𝑆 is internally almost perfect. Corollary 2.68 ([439]) For any graph 𝐺, ir(𝐺) = 𝜃 ap (𝐺) ≤ Θap (𝐺) = IR(𝐺). The Domination Chain, restated below as Theorem 2.69, can now be considerably expanded in terms of the concept of perfection. Theorem 2.69 ([196]) For any graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). Theorem 2.70 For any graph 𝐺, the following system of inequalities holds: ap

ap

𝜃 ap (𝐺) ≤ 𝜃 (𝐺) ≤ 𝜃 p (𝐺) ≤ Θp (𝐺) ≤ Θ(𝐺) ≤ Θap (𝐺) =

=

=

≤

≤

𝜃 ap (𝐺) = ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺) ≥

≥

=

=

≤

≤

𝜃 ap (𝐺) ≤ irap (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖 ap (𝐺) ≤ 𝛼ap (𝐺) ≤ Γ(𝐺) ≤ IRap (𝐺) ≥ 𝜃 (𝐺) ap

Let 𝛾d (𝐺) equal the minimum cardinality of a dominating set that is externally almost perfect. Let 𝛾 ap (𝐺) equal the minimum cardinality of a minimal dominating set that is externally almost perfect. These two parameters lead to the following refinement. ap

Proposition 2.71 ([439]) For any graph 𝐺, 𝛾(𝐺) ≤ 𝛾d (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖(𝐺). As given in [439], the fact that each of these inequalities can be strict is illustrated by the unicyclic graph 𝐺 in Figure 2.7. For this graph, 𝛾(𝐺) = 4 and {𝑣 1 , 𝑣 3 , 𝑣 4 , 𝑣 6 } ap ap is a 𝛾-set of 𝐺, 𝛾d (𝐺) = 6 and {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6 } is a 𝛾d -set of 𝐺, 𝛾 ap (𝐺) = 8 and {𝑣 1 , 𝑣 6 , 𝑣 7 , 𝑣 8 , 𝑣 9 , 𝑣 10 , 𝑣 11 , 𝑣 12 } is a 𝛾 ap -set of 𝐺, and 𝑖(𝐺) = 9 and {𝑣 1 , 𝑣 4 , 𝑣 7 , 𝑣 8 , 𝑣 9 , 𝑣 13 , 𝑣 14 , 𝑣 15 , 𝑣 16 } is an 𝑖-set of 𝐺. The concept of perfection in graphs provides a framework for unifying the concepts of independent sets, dominating sets, irredundant sets, perfect and efficient dominating sets, and perfect neighborhood sets. For example: (a) A set is independent if and only if it is an internally perfect set. (b) The independence parameters 𝑖(𝐺) and 𝛼(𝐺) can be expressed as maximal independent sets whose complements are internally almost perfect, that is, 𝑖(𝐺) = 𝑖 ap (𝐺) and 𝛼(𝐺) = 𝛼ap (𝐺). (c) A set is an irredundant set if and only if it is an internally almost perfect set.

Chapter 2. Fundamentals of Domination

46 𝑣 20 𝑣 19

𝑣7 𝑣2

𝑣8

𝑣 18 𝑣1

𝑣3

𝑣9

𝑣6

𝑣4

𝑣 10

𝑣 17 𝑣 16 𝑣 15 𝑣 14 𝑣 13

𝑣5

𝑣 11 𝑣 12

ap

Figure 2.7 𝛾(𝐺) < 𝛾d 𝐺 < 𝛾 ap (𝐺) < 𝑖(𝐺)

(d) The parameters ir(𝐺) and IR(𝐺) can be expressed in terms of internally almost perfect sets, namely, ir(𝐺) = 𝜃 ap (𝐺) and IR(𝐺) = Θap (𝐺). (e) Theorem 2.60, that Θ(𝐺) = Γ(𝐺), establishes an equality between two seemingly unrelated parameters. This result is now clearer. In particular, the inequality chain: 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺) can now be stated equivalently ap ap as: Θp (𝐺) ≤ Θ(𝐺) ≤ Θap (𝐺), since Θp (𝐺) = 𝛼(𝐺), Θ(𝐺) = Γ(𝐺), and Θap (𝐺) = IR(𝐺). (f) An expanded inequality chain exists between the domination and independence parameters: ap

𝛾(𝐺) ≤ 𝛾d (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γap (𝐺) ≤ Γ(𝐺).

2.6

Ore’s Lemmas and Their Implications

The property of being a dominating set is superhereditary, that is, every superset of a dominating set in a graph 𝐺 is also a dominating set of 𝐺. Thus, by Proposition 2.2, a dominating set 𝐷 is minimal if and only if for every vertex 𝑣 ∈ 𝐷, the set 𝐷 \ {𝑣} is not a dominating set. Therefore, it suffices, in order to show minimality of a dominating set 𝐷, for us only to consider 𝐷 \ {𝑣} for each vertex 𝑣 ∈ 𝐷 rather than all possible nonempty subsets of 𝐷. In 1962 Ore [622] proved the following properties of minimal dominating sets in graphs. Lemma 2.72 ([622]) A dominating set 𝐷 in a graph 𝐺 is a minimal dominating set of 𝐺 if and only if every vertex 𝑣 ∈ 𝐷 either is an internal private neighbor or has an external private neighbor, that is, if and only if ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. Proof Suppose, firstly, that 𝐷 is a minimal dominating set of a graph 𝐺. Let 𝑣 be an arbitrary vertex of 𝐷. By the minimality of 𝐷, the set 𝐷 \ {𝑣} is not a dominating set of 𝐺. Let 𝑤 be a vertex of 𝐺 not dominated by 𝐷 \ {𝑣}. If 𝑤 ∈ 𝐷, then 𝑤 = 𝑣, implying that 𝑣 is isolated in 𝐺 [𝐷] and therefore ipn[𝑣, 𝐷] = {𝑣}. If 𝑤 ∈ 𝑉 \ 𝐷, then 𝑤 is adjacent to 𝑣 but to no other vertex of 𝐷, implying that 𝑤 ∈ epn[𝑣, 𝐷]. Thus, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. Conversely, if ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for

Section 2.6. Ore’s Lemmas and Their Implications

47

every vertex 𝑣 ∈ 𝐷, then the set 𝐷 \ {𝑣} is not a dominating set of 𝐺 for each such vertex 𝑣. Lemma 2.73 ([622]) If 𝐺 = (𝑉, 𝐸) is an isolate-free graph, then the complement 𝑉 \ 𝐷 of any minimal dominating set 𝐷 is a dominating set. Proof Let 𝐷 be a minimal dominating set of 𝐺. We show that every vertex in 𝐷 has a neighbor in 𝑉 \ 𝐷. Let 𝑣 be an arbitrary vertex of 𝐷. Since 𝐺 is isolate-free, the vertex 𝑣 has at least one neighbor. By Lemma 2.72, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. If ipn[𝑣, 𝐷] ≠ ∅, then 𝑣 is isolated in 𝐺 [𝐷] and therefore all neighbors of 𝑣 belong to 𝑉 \ 𝐷. If epn[𝑣, 𝐷] ≠ ∅, then 𝑣 has a neighbor in 𝑉 \ 𝐷. Thus, 𝑉 \ 𝐷 dominates every vertex in 𝐷 and is a dominating set of 𝐺. As a consequence of Lemma 2.73, we have the following result. Corollary 2.74 ([622]) In any nontrivial connected graph 𝐺 = (𝑉, 𝐸), there is a partition of the vertex set 𝑉 into two sets, each of which is a dominating set; furthermore, one of the sets can be chosen to be either (i) a minimal dominating set, (ii) a 𝛾-set, (iii) a Γ-set, (iv) a maximal independent set, (v) an 𝑖-set, or (vi) an 𝛼-set. It was Ore’s Lemma 2.73 that prompted Cockayne and Hedetniemi [194] to introduce the concept of the domatic number of a graph in 1977. Definition 2.75 The domatic number dom(𝐺) of a graph 𝐺 equals the maximum order 𝑘 of a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } such that each subset 𝑉𝑖 is a dominating set of 𝐺. As an immediate consequence of Ore’s Lemma 2.73, the domatic number of a nontrivial connected graph is at least 2. Corollary 2.76 ([622]) If 𝐺 is a connected graph of order 𝑛 ≥ 2, then dom(𝐺) ≥ 2. We shall see more implications of Ore’s Lemmas in Chapter 4 when we discuss general bounds on the domination number, in Chapter 6 when we discuss upper bounds on the domination number, and in Chapter 12 when we discuss dominating partitions.

Chapter 3

Complexity and Algorithms for Domination in Graphs 3.1 Introduction In this chapter, we provide an overview of some of the core results on NP-completeness and algorithms for domination, independent domination, and total domination in graphs. First, we consider the following general question: given an arbitrary graph 𝐺, how difficult is it to determine, or compute, the value of any of the following six basic domination parameters, which satisfy the following well-known inequalities: 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) and 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ Γt (𝐺). In the best cases, there is either a simple method to determine the value of any one of these domination parameters, or at least a polynomial algorithm will exist which enables these values to be determined with relatively little computer time. Unfortunately, these best cases are quite rare. It is well known that all of these problems are NP-complete when the input is an arbitrary graph. This means that the general expectation is that it will require an amount of time that is exponential, as a function of the order and size of the graph, in order to compute any of these domination numbers for an arbitrary graph. Therefore, the general expectation is that we can compute these numbers in a reasonable amount of time only for graphs of relatively small order, say 𝑛 ≤ 100, with a few notable exceptions, of course. If, on the other hand, we restrict the types of graphs for which we would like to compute the value of these parameters, then we can expect some degree of success. Indeed, a significant amount of research has been dedicated to finding classes of graphs for which these problems can be solved in polynomial time. An almost equal amount of time has been spent determining classes of graphs for which these problems are NP-complete. As a very simple example, it is known that a simple linear algorithm © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_3

49

50

Chapter 3. Complexity and Algorithms for Domination in Graphs

exists for computing the value of 𝛾(𝑇) when the input is an arbitrary tree 𝑇. But the class of trees is clearly a proper subclass of the class of bipartite graphs, and it is known that the domination problem is NP-complete for arbitrary bipartite graphs. The recognition of this fact has therefore led researchers to find classes of graphs that properly contain trees as a subclass, but which are themselves properly contained within the class of bipartite graphs, such that the domination problem can either be computed in polynomial time for this class or that the problem is still NP-complete for this subclass of bipartite graphs. The possibilities for research, either in finding polynomial algorithms or NPcompleteness results in this area, are almost unlimited. In the book, Graph Classes, A Survey by Brandstädt et al. [102], definitions are given of close to 150 different classes of graphs that have been studied. And in the paper, An Annotated Glossary of Graph Theory Parameters with Conjectures by Gera et al. [333], definitions are given of some 300 graph theory parameters, including approximately 80 parameters related to domination in graphs. Thus, from the perspective of possible algorithms and NP-completeness results related to domination in graphs, there are at least 150 × 80 = 12,000 possible combinations. The possibilities are even larger than this because one can construct: (i) an exact polynomial-time sequential algorithm, (ii) an exact exponential-time sequential algorithm, (iii) an exact, parallel 𝑛-processor algorithm, (iv) a sequential approximation algorithm, (v) a self-stabilizing algorithm, (vi) a parallel distributed algorithm, (vii) a greedy algorithm using heuristics, (viii) a mixed integer-linear program, (ix) a fixed parameter tractable algorithm, or (x) a genetic algorithm. And for each type of algorithm, the vertices and/or edges can be weighted or unweighted, the edges can be directed or undirected, and there may or may not be loops and multiple edges. In this chapter we present, without attempting to be comprehensive, a brief review of some of the classes of graphs for which domination algorithms or NPcompleteness proofs have been constructed, focusing only the basic six domination parameters 𝛾(𝐺), 𝑖(𝐺), 𝛼(𝐺), Γ(𝐺), 𝛾t (𝐺), and Γt (𝐺).

3.2

Brief Review of NP-Completeness

Basically, NP-completeness has to do with three classes of computational problems: the class P, the class NP, and the class NPc. A computation problem is in the class P if there exists an algorithm for solving any input to problem in O (𝑛 𝑘 ) time, for some fixed integer 𝑘 ≥ 1, where 𝑛 denotes the length of the input. Several common examples of computational problems that are known to be in the class P are: (a) finding a maximum or minimum of a set of 𝑛 integers (b) computing the sum or product of 𝑛 integers (c) sorting a set of 𝑛 integers (d) finding a shortest path between two vertices in a connected graph (e) finding a maximum matching in a graph (f) finding the maximum or minimum degree of a vertex in a graph (g) finding a spanning tree of a graph

Section 3.2. Brief Review of NP-Completeness

51

deciding if a graph is connected deciding if a graph is bipartite computing the diameter of a graph deciding if a graph is planar deciding if two trees, planar graphs, interval graphs, or permutation graphs are isomorphic. A computation problem is in the class NP if it can be solved in polynomial time by a nondeterministic Turing machine. Thus, the letters NP stand for Nondeterministic Polynomial time. So, what is a Turing machine? What is a nondeterministic Turing machine? Without getting overly technical, let us illustrate this with an intuitive example. Suppose we are given an arbitrary graph 𝐺 and we wish to know if 𝐺 has a dominating set 𝑆 of cardinality at most 𝑘 for some 𝑘 ≥ 1. Thus, we are given as input to this nondeterministic algorithm: (i) a graph 𝐺, and (ii) the question: does 𝐺 have a dominating set 𝑆 of cardinality at most 𝑘? A nondeterministic algorithm, or nondeterministic Turing machine, has the ability, and is granted permission, to examine in polynomial time, one-by-one, every vertex in the graph 𝐺 and make a ‘guess’ as to whether this vertex is in a dominating set of cardinality at most 𝑘. Having reviewed all vertices, it selects a set 𝑆 of cardinality at most 𝑘. It then must be able to determine, again in polynomial time, whether this set is a dominating set. If it is a dominating set, then the nondeterministic algorithm answers the given question, ‘yes.’ We say that a graph 𝐺 is a ‘yes’ instance to this problem if it does have a dominating set of cardinality at most 𝑘, otherwise, it is a ‘no’ instance. The nondeterministic algorithm is required to say ‘yes’ to at least one execution if the graph is a ‘yes’ instance, and never to say ‘yes’ to a ‘no’ instance, and never take more than a polynomial amount of time for any instance. An equivalent, and perhaps easier, formulation of nondeterministic algorithms can be given in terms of polynomial-time verification. All that is required is to have a deterministic algorithm, called a verification algorithm, which can be given an instance of the problem, a graph 𝐺, and a set 𝑆, called a candidate solution, and be able to verify or decide in polynomial time whether 𝑆 is a solution for the graph 𝐺. In this view, the class NP is the class of all problems which can be verified in polynomial time. This leaves us to define the class NPc of NP-complete problems. We say that a computation problem 𝐴 is polynomial-time reducible to a computation problem 𝐵, we can write this as 𝐴 → 𝐵, if (i) there exists a function 𝑓 which maps any instance 𝐼 of problem 𝐴 to an instance 𝑓 (𝐼) of problem 𝐵, such that 𝐼 is a ‘yes’ instance of problem 𝐴 if and only if 𝑓 (𝐼) is a ‘yes’ instance of problem 𝐵, and (ii) for any instance 𝐼 of problem 𝐴, the corresponding instance 𝑓 (𝐼) of problem 𝐵 can be constructed in polynomial time. Note that it follows from this definition, that if problem 𝐴 is polynomial-time reducible to problem 𝐵, and problem 𝐵 can be solved in polynomial time, then problem 𝐴 can also be solved in polynomial time. In order to solve an instance 𝐼 of problem 𝐴, simply construct in polynomial time the instance 𝑓 (𝐼) of problem 𝐵 and (h) (i) (j) (k) (l)

Chapter 3. Complexity and Algorithms for Domination in Graphs

52

use the polynomial algorithm for problem 𝐵 to determine whether 𝑓 (𝐼) is a ‘yes’ or ‘no’ instance. We define a problem 𝐴 to be NP-complete if (i) 𝐴 is in the class NP, which can be shown by constructing a polynomial verification algorithm for 𝐴, and (ii) for any problem 𝑋 ∈ NPc, 𝑋 → 𝐴. We say that a problem is NP-hard if it satisfies condition (ii) but not necessarily condition (i). It is easy to see that the relation 𝐴 → 𝐵 is transitive. Thus, if 𝐴 → 𝐵 and 𝐵 → 𝐶, then 𝐴 → 𝐶. This means that an algorithm for solving problem 𝐶 can be used not only to solve problem 𝐵 but also to solve problem 𝐴. And it follows from this that any algorithm for solving a problem in NPc can be used to solve all problems in NPc. Thus, in order to show that a computation problem 𝐴 is NP-complete, that is, that 𝐴 ∈ NPc, one must: (i) Show that 𝐴 ∈ NP, that is, construct a polynomial-time verification algorithm for 𝐴. (ii) Find a known NP-complete problem 𝑋 ∈ NPc and show that 𝑋 → 𝐴, that is, an algorithm for solving 𝐴 can be used to solve the known NP-complete problem 𝑋. We illustrate this process of showing that a problem is NP-complete in the next section.

3.3

NP-Completeness of Domination, Independent Domination, and Total Domination

In this section, we consider the NP-completeness of domination, independent domination and total domination. NP-completeness results are presented in Section 3.3.1 for arbitrary graphs and Section 3.3.2 for bipartite graphs. A summary of NP-completeness results for some families of graphs is presented in Section 3.3.3.

3.3.1

NP-Completeness Results for Arbitrary Graphs

To the best of our knowledge, the first proof of the NP-completeness of the domination problem was due to David Johnson at a graph theory conference held in Qualicum Beach, Vancouver Island, British Columbia sometime during 1975–1976 [510], using a simple reduction, shown below, from the well-known NP-complete problem called 3SAT. This result was subsequently stated in the NP-completeness book by Garey and Johnson [325] as [GT2] DOMINATING SET. The basic statement of this problem, as a decision problem, is the following: DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a dominating set of cardinality at most 𝑘? Theorem 3.1 ([510]) DOMINATING SET is NP-complete.

Section 3.3. NP-Completeness of Domination Parameters

53

Proof We must first show that DOMINATING SET ∈ NP, that is, construct a polynomial-time verification algorithm for DOMINATING SET. This part of the proof is straightforward. Given a graph 𝐺 and a set 𝑆 of vertices, one can easily check in polynomial time to see that |𝑆| ≤ 𝑘 and that 𝑆 is or is not a dominating set of 𝐺, that is, check to see that for all 𝑣 ∈ 𝑉 \ 𝑆, N(𝑣) ∩ 𝑆 ≠ ∅. Now we must find a known NP-complete problem 𝑋 ∈ NPc and show that 𝑋 → DOMINATING SET. Johnson chose one of the most commonly known NPcomplete problems, 3-SATISFIABILITY (or 3SAT). 3SAT

Instance: A set 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } of 𝑛 Boolean variables and a set 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 𝑚 clauses, each being a disjunction of three variables or their complements, such as 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 5 ). Question: Does 𝐶 have a satisfying truth assignment, that is, a function 𝑓 : 𝑉 → {TRUE, FALSE} such that at least one variable or its complement in each clause is assigned the value TRUE? Thus, the clause 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 5 ) will be true if either 𝑓 (𝑣 1 ) = TRUE or 𝑓 (𝑣 2 ) = FALSE or 𝑓 (𝑣 5 ) = TRUE. We must construct a polynomial time reduction from an instance of 3SAT to an instance of DOMINATING SET. We do this as follows. Given an instance 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } and 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3SAT, we construct an instance of DOMINATING SET, that is, a graph 𝐺 (𝑉, 𝐶) and a positive integer 𝑘 = 𝑛 (see Figure 3.1). 𝑥1 𝑣1

𝑐1

𝑥3

𝑥2 𝑣1

𝑣2

𝑐2

𝑣2

𝑣3

𝑐3

𝑥5

𝑥4 𝑣3

𝑣4

𝑣4

𝑣5

𝑣5

𝑐4

Figure 3.1 Reduction from 3SAT to DOMINATING SET For each variable 𝑣 𝑖 ∈ 𝑉, construct a triangle with the top vertex labeled 𝑥𝑖 and the bottom two vertices labeled 𝑣 𝑖 and 𝑣 𝑖 . For each clause 𝐶 𝑗 ∈ 𝐶, construct a single vertex labeled 𝑐 𝑗 . Finally, for each clause, such as 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 3 ), add the three edges 𝑐 1 𝑣 1 , 𝑐 1 𝑣 2 , and 𝑐 1 𝑣 3 , as shown in Figure 3.1. We must show that (i) 𝑉, 𝐶 is a ‘yes’ instance of 3SAT if and only if the graph 𝐺 (𝑉, 𝐶) so constructed with the positive integer 𝑘 = 𝑛 is a ‘yes’ instance of

54

Chapter 3. Complexity and Algorithms for Domination in Graphs

DOMINATING SET, and (ii) that for any instance 𝑉, 𝐶 of 3SAT, we can construct the instance 𝐺 (𝑉, 𝐶) and 𝑘 in polynomial time. But this is easy to see. The size of an instance of 3SAT is O (𝑛 + 𝑚), that is, 𝑛 Boolean variables and 𝑚 clauses, each clause containing three variables. The size of the constructed instance of DOMINATING SET is 3𝑛 + 𝑚 vertices and 3𝑛 + 3𝑚 edges. Thus, the size of an instance of DOMINATING SET is at most a constant times the size of an instance of 3SAT. This brings us to (i). Assume that an instance 𝑉, 𝐶 of 3SAT has a satisfying truth assignment 𝑓 : 𝑉 → {TRUE, FALSE}. We construct a set 𝑆 of the graph 𝐺 (𝑉, 𝐶) as follows: if 𝑓 (𝑣 𝑖 ) = TRUE, then place vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑗 ) = FALSE, then place vertex 𝑣 𝑗 ∈ 𝑆. Notice that the set 𝑆 so constructed contains exactly one vertex from each of the 𝑛 triangles. Thus, each vertex in a triangle is either in 𝑆 or is adjacent to exactly one vertex in 𝑆. In addition, each clause vertex 𝑐 𝑖 is adjacent to at least one vertex in 𝑆 because 𝑓 is a satisfying truth assignment, which means that at least one variable in each clause is assigned the value TRUE. In other words, if 𝑓 (𝑣 𝑖 ) = TRUE, then vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑖 ) = FALSE, then 𝑣 𝑖 ∈ 𝑆. Thus, 𝑆 is a dominating set of the graph 𝐺 (𝑉, 𝐶) and the cardinality of 𝑆 is exactly 𝑛. Hence, if 𝑉, 𝐶 is a ‘yes’ instance of 3SAT, then the graph 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of DOMINATING SET. Conversely, assume that 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of DOMINATING SET, that is, 𝐺 has a dominating set 𝑆 such that |𝑆| ≤ 𝑘 = 𝑛. Since 𝑆 is a dominating set, it must contain at least one vertex in each of the 𝑛 triangles. Hence, |𝑆| ≥ 𝑛. If a vertex 𝑥 𝑗 ∈ 𝑆, then we can replace it with either 𝑣 𝑗 or 𝑣 𝑗 and still have a dominating set. Thus, we may assume 𝑥 𝑗 ∉ 𝑆 for 𝑗 ∈ [𝑛]. Therefore, |𝑆| = 𝑛 and the vertices in 𝑆 must dominate all clause vertices 𝑐 𝑖 . Hence, if a clause vertex 𝑐 𝑖 is dominated by a triangle vertex 𝑣 𝑗 , define a truth assignment such that 𝑓 (𝑣 𝑗 ) = TRUE. Similarly, if a clause vertex is dominated by a triangle vertex 𝑣 𝑗 , define 𝑓 (𝑣 𝑗 ) = FALSE. This produces a satisfying truth assignment for the 3SAT instance 𝑉, 𝐶.

Notice that the dominating set constructed in the NP-completeness proof of DOMINATING SET is an independent dominating set. This provides us as a corollary

a second NP-completeness proof to the following decision problem. INDEPENDENT DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘 Question: Does 𝐺 have a independent dominating set of cardinality at most 𝑘? Corollary 3.2 ([510]) INDEPENDENT DOMINATING SET is NP-complete. Proof The proof is identical, word-for-word, to the proof of the NP-completeness of DOMINATING SET, with the single exception that every occurrence of DOMINATING SET or dominating set is replaced by INDEPENDENT DOMINATING SET. A very simple change in the construction of the graph 𝐺 (𝑉, 𝐶) in the NP-completeness proof for DOMINATING SET will enable us construct an NP-completeness proof for the following decision problem. Recall that we abbreviate a total dominating set by TD-set.

Section 3.3. NP-Completeness of Domination Parameters

55

TOTAL DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a TD-set of cardinality at most 𝑘? Corollary 3.3 ([510]) TOTAL DOMINATING SET is NP-complete. Proof The proof is the same as the proof of the NP-completeness of DOMINATING SET, with the following changes. We must construct a polynomial time reduction from an instance of 3SAT to an instance of TOTAL DOMINATING SET. We do this as follows. Given an instance 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } and 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3SAT, we construct an instance of TOTAL DOMINATING SET, that is, a graph 𝐺 (𝑉, 𝐶) and a positive integer 𝑘 = 2𝑛. For each variable 𝑣 𝑖 ∈ 𝑉, construct a triangle with the top vertex labeled 𝑥𝑖 and the bottom two vertices labeled 𝑣 𝑖 , 𝑣 𝑖 . To this triangle add a fourth vertex 𝑦 𝑖 , a leaf which is adjacent only to vertex 𝑥𝑖 . For each clause 𝐶 𝑗 ∈ 𝐶, construct a single vertex labeled 𝑐 𝑗 . Finally, for each clause, such as 𝐶1 = (𝑣 1 ∧ 𝑣 2 ∧ 𝑣 3 ), add the three edges 𝑐 1 𝑣 1 , 𝑐 1 𝑣 2 , and 𝑐 1 𝑣 3 . We must show (i) that 𝑉, 𝐶 is a ‘yes’ instance of 3SAT if and only if the graph 𝐺 (𝑉, 𝐶) so constructed with the positive integer 𝑘 = 2𝑛 is a ‘yes’ instance of TOTAL DOMINATING SET. We must also show (ii) that for any instance 𝑉, 𝐶 of 3SAT, we can construct the instance 𝐺 (𝑉, 𝐶) and 𝑘 = 2𝑛 in polynomial time. But this is easy to see. The size of an instance of 3SAT is O (𝑛 + 𝑚), that is, 𝑛 Boolean variables and 𝑚 clauses, each containing three variables. The size of the constructed instance 𝐺 (𝑉, 𝐶) of TOTAL DOMINATING SET is 4𝑛 + 𝑚 vertices and 4𝑛 + 3𝑚 edges. Thus, the size of an instance of TOTAL DOMINATING SET is at most a constant times the size of an instance of 3SAT. This brings us to (i). Assume that an instance 𝑉, 𝐶 has a satisfying truth assignment 𝑓 : 𝑉 → {TRUE, FALSE}. We construct a set 𝑆 of the graph 𝐺 (𝑉, 𝐶) as follows: if 𝑓 (𝑣 𝑖 ) = TRUE, then place vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑗 ) = FALSE, then place vertex 𝑣 𝑗 ∈ 𝑆. In addition, place into the set 𝑆 all 𝑛 vertices of the form 𝑥𝑖 . Thus, |𝑆| = 2𝑛. Notice that the set 𝑆 so constructed contains exactly two vertices from each of the 𝑛 triangles. Thus, each vertex in a triangle is either in 𝑆 and has exactly one neighbor in 𝑆, or is adjacent to exactly two vertices in 𝑆. Each leaf 𝑦 𝑖 is adjacent to 𝑥𝑖 ∈ 𝑆. In addition, each clause vertex 𝑐 𝑖 is adjacent to at least one vertex in 𝑆 because 𝑓 is a satisfying truth assignment, which means that at least one variable in each clause is assigned the value TRUE, which means that if 𝑓 (𝑣 𝑖 ) = TRUE, then vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑖 ) = FALSE, then 𝑣 𝑖 ∈ 𝑆. Thus, 𝑆 is a TD-set of the graph 𝐺 (𝑉, 𝐶) and the cardinality of 𝑆 is exactly 2𝑛. Thus, if 𝑉, 𝐶 is a ‘yes’ instance of 3SAT, then the graph 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of TOTAL DOMINATING SET. Conversely, assume that 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of TOTAL DOMINATING SET, that it has a TD-set 𝑆 of cardinality |𝑆| ≤ 2𝑛. Since 𝑆 is a TD-set, it must contain at least two vertices in each of the 𝑛 subgraphs consisting of a triangle with an attached leaf. Furthermore, if 𝑆 were to contain the pair of vertices 𝑥𝑖 , 𝑦 𝑖 , it could be

Chapter 3. Complexity and Algorithms for Domination in Graphs

56

replaced by either the pair 𝑥𝑖 , 𝑣 𝑖 or the pair 𝑥𝑖 , 𝑣 𝑖 . Thus, we can assume that 𝑆 does not contain any 𝑦 𝑖 vertex. Therefore, |𝑆| ≥ 2𝑛. Consequently, |𝑆| = 2𝑛 and the vertices in 𝑆 must dominate all clause vertices 𝑐 𝑖 . Hence, if a clause vertex 𝑐 𝑖 is dominated by a triangle vertex 𝑣 𝑗 , define a truth assignment such that 𝑓 (𝑣 𝑗 ) = TRUE. Similarly, if a clause vertex is dominated by a triangle vertex 𝑣 𝑗 define 𝑓 (𝑣 𝑗 ) = FALSE. This produces a satisfying truth assignment for the 3SAT instance 𝑉, 𝐶.

3.3.2

NP-Completeness Results for Bipartite Graphs

The NP-completeness results in the previous section apply to arbitrary graphs. The great majority of NP-completeness results, however, show that some computation problem remains NP-complete even when restricted to some special class of graphs. We illustrate this with the following theorem of Chang and Nemhauser [144] and three corollaries that can be derived from it. BIPARTITE DOMINATING SET

Instance: A bipartite graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a dominating set of cardinality at most 𝑘? Theorem 3.4 ([144]) BIPARTITE DOMINATING SET is NP-complete, or equivalently, DOMINATING SET is NP-complete, even when restricted to bipartite graphs. Proof We first consider showing BIPARTITE DOMINATING SET ∈ NP. This part of the proof is quite straight forward. Given a bipartite graph 𝐺 and a set 𝑆 of vertices, one can easily check in polynomial time to see that |𝑆| ≤ 𝑘 and that 𝑆 is or is not a dominating set of 𝐺. Next we define a polynomial time reduction from DOMINATING SET for arbitrary graphs, which has already been shown to be NP-complete, to BIPARTITE DOMINATING SET. Given an instance of DOMINATING SET, that is, an arbitrary graph 𝐺 = (𝑉, 𝐸), with 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }, and a positive integer 𝑘, we construct an instance of BIPARTITE DOMINATING SET, that is, a bipartite graph 𝐺 ′ = (𝑉 ′ , 𝐸 ′ ) and a positive integer 𝑘 ′ = 𝑘 + 1, as follows. Let 𝑉 ′ = 𝑉 ∪ 𝑉 ′′ ∪ {𝑥, 𝑥 ′′ }, where 𝑉 ′′ = {𝑣 1′′ , 𝑣 2′′ , . . . , 𝑣 ′′ 𝑛 } is a copy of 𝑉, and 𝑥 and 𝑥 ′′ are two vertices not in 𝑉. Let 𝐸 ′ = {𝑢𝑣 ′′ , 𝑢 ′′ 𝑣 : 𝑢𝑣 ∈ 𝐸 } ∪ {𝑣𝑣 ′′ : 𝑣 ∈ 𝑉 } ∪ {𝑢𝑥 : 𝑢 ∈ 𝑉 } ∪ {𝑥𝑥 ′′ }. Finally, for any set 𝑆 ⊆ 𝑉 as part of an instance of DOMINATING SET, let 𝑆 ′ = 𝑆 ∪ {𝑥} be the corresponding set in the instance of BIPARTITE DOMINATING SET. For example, see Figure 3.2. We must show that (i) 𝐺, 𝑆 is a ‘yes’ instance of DOMINATING SET if and only if the 𝐺 ′ , 𝑆 ′ is a ‘yes’ instance of BIPARTITE DOMINATING SET. We must also show that (ii) for any instance 𝐺, 𝑘 of DOMINATING SET, we can construct the instance 𝐺 ′ , 𝑘 + 1 in polynomial time. But this is easy to see. The size of an instance of DOMINATING SET is O (𝑛 + 𝑚), where |𝑉 | = 𝑛 and |𝐸 | = 𝑚. The size of the constructed instance of BIPARTITE DOMINATING SET is 2𝑛 + 2 vertices and

Section 3.3. NP-Completeness of Domination Parameters

𝑎

57

𝑎 ′′

𝑎

𝑏 ′′

𝑏

𝑐′′

𝑐

𝑥 𝑥 ′′

𝑒

𝑏 𝑑 ′′ 𝑑 𝑐

𝑑

𝑒 ′′

(a) 𝐺

𝑒 (b) 𝐺 ′′

Figure 3.2 DOMINATING SET to BIPARTITE DOMINATING SET

2𝑛 + 2𝑚 + 1 edges. Thus, the size of an instance of BIPARTITE DOMINATING SET is at most a constant times the size of an instance of DOMINATING SET. This brings us to (i). Assume that an instance 𝐺, 𝑆 is a ‘yes’ instance of DOMINATING SET. We construct a set 𝑆 ′ = 𝑆 ∪ {𝑥} of the bipartite graph 𝐺 ′ . It is easy to see that if 𝑆 is a dominating set of 𝐺 of cardinality at most 𝑘, then 𝑆 ′ is a dominating set of 𝐺 ′ of cardinality at most 𝑘 + 1. Conversely, assume that 𝐺 ′ , 𝑆 ′ is a ‘yes’ instance of BIPARTITE DOMINATING SET. Since 𝑆 ′ is a dominating set of 𝐺 ′ , it must contain either vertex 𝑥 or vertex 𝑥 ′′ . We can assume, without loss of generality, that 𝑆 ′ contains 𝑥. Then 𝑥 dominates 𝑉 in 𝐺 ′ , and the remaining vertices in 𝑆 ′ dominate 𝑉 ′′ . It is easy to see that if there is some vertex 𝑣 ′′ ∈ 𝑆 ′ , then we can replace it with its corresponding vertex 𝑣 ∈ 𝑉 and still have a dominating set of 𝐺 ′ . Thus, we can make sure that there is a subset of vertices only in 𝑉 that dominates 𝑉 ′′ . This means that the same subset of vertices is a dominating set of 𝐺. Thus, a ‘yes’ instance of BIPARTITE DOMINATING SET corresponds to a ‘yes’ instance of DOMINATING SET. Notice that the dominating set of the bipartite graph 𝐺 ′ in the above proof is also a TD-set. From this we get an immediate corollary, using virtually the same proof. BIPARTITE TOTAL DOMINATING SET

Instance: A bipartite graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a TD-set of cardinality at most 𝑘? Corollary 3.5 BIPARTITE TOTAL DOMINATING SET is NP-complete, or equivalently, TOTAL DOMINATING SET is NP-complete, even when restricted to bipartite graphs. We have thus demonstrated the NP-completeness of the decision problems associated with DOMINATING SET, INDEPENDENT DOMINATING SET, and

58

Chapter 3. Complexity and Algorithms for Domination in Graphs

TOTAL DOMINATING SET. In the next section, we review a much wider collection of NP-completeness results for DOMINATING SET when restricted to many other

classes of graphs.

3.3.3

Summary of Complexity Results for Graph Families

In 1990 Corneil and Stewart [201] provided a summary of results on the complexity of the DOMINATING SET decision problem for some 35 classes of graphs, where, for example, trees [P] ⊂ chordal bipartite graphs [NPc] indicates that the class of trees is a proper subclass of the class of chordal bipartite graphs, and that the domination number can be computed in polynomial time [P] for trees, but for chordal bipartite graphs the problem is NP-complete [NPc]. Subset inclusions like this show in some sense the boundary between classes of graphs for which DOMINATING SET can be solved in polynomial time and a larger class of graphs for which the problem is NP-complete. • trees ⊂ 𝑘-trees, fixed 𝑘 ⊂ 𝑘-trees, arbitrary 𝑘 ⊂ chordal graphs • trees ⊂ directed-path graphs ⊂ undirected path graphs ⊂ chordal graphs • split graphs ⊂ chordal graphs ⊂ weakly chordal graphs • interval graphs ⊂ directed-path graphs ⊂ strongly chordal graphs ⊂ chordal graphs • cographs ⊂ permutation graphs ⊂ comparability graphs ⊂ perfectly orderable graphs • permutation graphs ⊂ cocomparibility ⊂ asteroidal triple-free • cographs ⊂ 𝑃4 -reducible graphs ⊂ permutation graphs ⊂ trapezoid graphs ⊂ cocomparability graphs • trees ⊂ chordal bipartite graphs ⊂ bipartite graphs ⊂ comparability graphs • trees ⊂ generalized series-parallel graphs ⊂ partial 2-trees ⊂ partial 𝑘-trees, fixed 𝑘 ⊂ 𝑘-trees, arbitrary 𝑘 We summarize the complexity results for domination, total domination, and independent domination of some graph families in the following table. Using subset inclusions, other complexity results can be deduced. For example, chordal graphs ⊂ weakly chordal graphs, and since chordal graphs are in the class NPc for DOMINATING SET, weakly chordal graphs are also in NPc. For each graph family, the complexity P or NPc for the decision problems associated with domination, total domination, and independent domination is listed under 𝛾(𝐺), 𝛾t (𝐺), and 𝑖(𝐺), respectively in Table 3.1. Citations follow the complexity result.

3.4

A Representative Sample of Domination Algorithms for Trees

In this section, we present linear algorithms for computing the domination number, independent domination number, and total domination number of an arbitrary tree.

Section 3.4. A Representative Sample of Domination Algorithms for Trees Graph Class

𝛾(𝐺)

general graphs bipartite graphs comparability graphs chordal graphs split graphs 𝑘-trees, arbitrary 𝑘 undirected path graphs chordal bipartite graphs trees permutation graphs 𝑘-trees, fixed 𝑘 directed path graphs cographs cocomparability graphs interval graphs strongly chordal graphs partial 𝑘-trees, fixed 𝑘 asteroidal triple-free graphs series-parallel graphs circular-arc graphs

NPc [325, 510] NPc [144, 233] NPc [233] NPc [87] NPc [71, 200] NPc [199] NPc [87] NPc [601] P [187] P [100, 270] P [199] P [87] P [200] P [545] P [87, 269] P [269] P [38] P [544] P [523] P [147]

𝛾t (𝐺)

59

𝑖(𝐺)

NPc [510, 634] NPc [634] NPc [634] NPc [554, 555] NPc [554]

NPc [325, 510] NPc [200] NPc [200] P [268] P [268] P [268] NPc [555] P [268] P [211] NPc [211] P [555] P [74] P [100, 201, 546] P [270] P [38] P [38] P [268] P [200] P [270] P [545] P [545] P [72, 519] P [268, 269] P [142] P [268, 269] P [37] P [37] P [544] P [122] P [635] P [635] P [147] P [147]

Table 3.1 NP-completeness results for some graph families

3.4.1

Minimum Dominating Set

The first polynomial algorithm for computing the domination number of any nontrivial class of graphs, in this case trees, was published in 1975 by Cockayne et al. [187]. Their algorithm actually solves a more general version of the domination problem. Assume that the vertices of a graph 𝐺 = (𝑉, 𝐸) are partitioned into three sets, 𝐹, 𝐵, and 𝑅, where vertices in 𝐹 are called free, vertices in 𝐵 are called bound, and vertices in 𝑅 are called required. In the optional domination problem, given a graph 𝐺 and a partition {𝐹, 𝐵, 𝑅} of 𝑉, one must find a set 𝑆 ⊆ 𝑉 of minimum cardinality satisfying the following three conditions: (i) Every vertex in 𝐵 must either be in 𝑆 or adjacent to (dominated by) a vertex in 𝑆. (ii) Every vertex in 𝑅 must be in 𝑆. (iii) Vertices in 𝐹 need not be in 𝑆 nor adjacent to a vertex in 𝑆, but can be in 𝑆 in order to dominate vertices in 𝐵. Thus, (i) bound vertices must be dominated, either by being in 𝑆 or adjacent to a vertex in 𝑆, (ii) required vertices must be in the set 𝑆, and (iii) free vertices need not be dominated. A solution 𝑆 is called an optional dominating set. Therefore, if you are given an arbitrary graph 𝐺 = (𝑉, 𝐸) and the partition {∅, 𝑉, ∅} of 𝑉, then a solution 𝑆 to the optional domination problem is a minimum dominating set of 𝐺.

60

Chapter 3. Complexity and Algorithms for Domination in Graphs

Let 𝛾opt (𝐺), the optional domination number, equal the minimum cardinality of an optional dominating set for the partition {𝐹, 𝐵, 𝑅} of 𝑉. The correctness of the following algorithm for finding an optional dominating set of any tree is based on the following theorem. Theorem 3.6 ([187]) If 𝑇 = (𝑉, 𝐸) is an arbitrary tree, {𝐹, 𝐵, 𝑅} is a partiton of 𝑉, and 𝑣 is a leaf of 𝑇 whose neighbor is vertex 𝑢, then the following properties hold: (a) If 𝑣 ∈ 𝐹, then 𝛾opt (𝑇) = 𝛾opt (𝑇 − 𝑣). (b) If 𝑣 ∈ 𝐵 and 𝑇 ′ is the tree which results from deleting 𝑣 and placing 𝑢 in 𝑅, then 𝛾opt (𝑇) = 𝛾opt (𝑇 ′ ). (c) If 𝑢, 𝑣 ∈ 𝑅, then 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 − 𝑣). (d) If 𝑣 ∈ 𝑅 and 𝑢 ∉ 𝑅, and if 𝑇 ′ is the tree which results from deleting 𝑣 and placing 𝑢 in 𝐹, then 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 ′ ). Proof (a) Suppose that 𝑣 ∈ 𝐹. Since 𝑣 is free and need not be dominated, any optional dominating set of 𝑇 − 𝑣 is also an optional dominating set of 𝑇. Thus, 𝛾opt (𝑇) ≤ 𝛾opt (𝑇 − 𝑣).
Conversely, let 𝑆 be an optional dominating set of 𝑇. If 𝑣 ∈ 𝑆, then 𝑆 \ {𝑣} ∪ {𝑢} is an optional dominating set of 𝑇 − 𝑣, in which case 𝛾opt (𝑇 − 𝑣) ≤ 𝛾opt (𝑇). If 𝑣 ∉ 𝑆, then 𝑆 is also an optional dominating set of 𝑇 − 𝑣. Thus, 𝛾opt (𝑇 − 𝑣) ≤ 𝛾opt (𝑇). Thus, from both cases we can conclude that 𝛾opt (𝑇) = 𝛾opt (𝑇 − 𝑣). (b) Suppose that 𝑣 ∈ 𝐵. Let 𝑇 ′ be the tree which results from deleting 𝑣 and placing 𝑢 in 𝑅. Then any optional dominating set 𝑆 ′ of 𝑇 ′ is automatically an optional dominating set of 𝑇. Hence, 𝛾opt (𝑇) ≤ 𝛾opt (𝑇 ′ ). Conversely, let 𝑆 be an optional
dominating set of 𝑇. Since 𝑣 ∈ 𝐵, either 𝑣 ∈ 𝑆 or 𝑢 ∈ 𝑆. In either case, 𝑆 − {𝑣} ∪ {𝑢} is an optional dominating set of 𝑇 ′ . Hence, 𝛾opt (𝑇 ′ ) ≤ 𝛾opt (𝑇), and therefore, 𝛾opt (𝑇) = 𝛾opt (𝑇 ′ ). (c) Suppose that 𝑢, 𝑣 ∈ 𝑅. Every optional dominating set 𝑆 of 𝑇 must therefore contain both 𝑢 and 𝑣. It follows that 𝑆 \ {𝑣} is an optional dominating set of 𝑇 − 𝑣. Thus, 𝛾opt (𝑇 − 𝑣) ≤ |𝑆| − 1 = 𝛾opt (𝑇) − 1. Conversely, if 𝑆 ′ is an optional dominating set of 𝑇 − 𝑣, then since it must contain the vertex 𝑢, the set 𝑆 ′ ∪ {𝑣} is an optional dominating set of 𝑇. Thus, 𝛾opt (𝑇) ≤ 1 + |𝑆 ′ | = 1 + 𝛾opt (𝑇 − 𝑣). Consequently, 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 − 𝑣). (d) Suppose that 𝑣 ∈ 𝑅 and 𝑢 ∉ 𝑅. Let 𝑇 ′ be the tree which results from deleting 𝑣 and placing 𝑢 in 𝐹. Let 𝑆 ′ be an optional dominating set of 𝑇 ′ . Then clearly, 𝑆 ′ ∪ {𝑣} is an optional dominating set of 𝑇. Hence, 𝛾opt (𝑇) ≤ 1 + |𝑆 ′ | = 1 + 𝛾opt (𝑇 ′ ). Conversely, let 𝑆 be an optional dominating set of 𝑇, where by assumption, 𝑣 ∈ 𝑆 and 𝑢 ∈ 𝐹. It follows therefore that 𝑆 ′ = 𝑆 \ {𝑣} is an optional dominating set of 𝑇 − 𝑣, and therefore 𝛾opt (𝑇 ′ ) ≤ 𝛾opt (𝑇) − 1. Consequently, 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 ′ ). The four conditions in Theorem 3.6 suggest the following algorithm for computing the optional domination number of any tree 𝑇 with a given vertex partition {𝐹, 𝐵, 𝑅}, and hence for computing the domination number of any tree. The algorithm consists of a series of guarded commands, as originally introduced by Dijkstra, consisting of a guard 𝐺 𝑖 followed by a command 𝐴𝑖 , of the following form:

Section 3.4. A Representative Sample of Domination Algorithms for Trees

61

do 𝐺 1 : 𝐴1 𝐺 2 : 𝐴2 .. . 𝐺 𝑛 : 𝐴𝑛 od Each guard 𝐺 𝑖 is a Boolean expression, and each 𝐴𝑖 consists of a sequence of statements or actions to be executed. An algorithm using this form is executed as follows: find any guard 𝐺 𝑖 whose Boolean expression is TRUE and then execute the statements in 𝐴𝑖 . Repeat this process until none of the guards 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑛 are true. At any given point in the execution of an algorithm more than one guard may be true. In a deterministic algorithm you normally find the first guard 𝐺 𝑖 whose expression evaluates to TRUE and then execute the corresponding statements in 𝐴𝑖 . If several guards evaluate to TRUE and it does not matter which guard’s actions are executed, then the algorithm is called nondeterministic. The algorithm we present is nondeterministic.

Algorithm 1 Minimum Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) with each vertex labeled free, bound, or required Output : A minimum cardinality optional dominating set 𝑆 1 2

3 4

5

6

7

[Initialize] Set 𝑆 ← ∅ while 𝑇 has two or more vertices, execute any one of the following guarded commands do 𝐺 1 : 𝑇 has a free leaf 𝑣: set 𝑇 ← 𝑇 − 𝑣 𝐺 2 : 𝑇 has a bound leaf 𝑣 adjacent to a vertex 𝑢: label 𝑢 required; set 𝑇 ← 𝑇 − 𝑣 𝐺 3 : 𝑇 has a required leaf 𝑣 adjacent to a vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; if vertex 𝑢 is bound, then label 𝑢 free; set 𝑇 ← 𝑇 − 𝑣 od [The last vertex] if the one remaining vertex 𝑣 is bound or required, then set 𝑆 ← 𝑆 ∪ {𝑣}

We again point out, that while Algorithm 1 computes the minimum cardinality of an optional dominating set for a given vertex partition {𝐹, 𝐵, 𝑅}, it will compute the domination number 𝛾(𝑇) of any tree 𝑇 = (𝑉, 𝐸), given the partition {∅, 𝑉, ∅} of 𝑉, in which all vertices are initially labeled bound.

62

Chapter 3. Complexity and Algorithms for Domination in Graphs

3.4.2 Minimum Independent Dominating Set The following algorithm for computing the independent domination number 𝑖(𝑇) of any tree 𝑇 was constructed by Beyer et al. [74] in 1977. Without loss of generality, this algorithm assumes that all trees 𝑇𝑟 are rooted at some vertex 𝑟, and for every vertex 𝑣 ∈ 𝑉 (𝑇𝑟 ), we consider the subtree 𝑇𝑣 of 𝑇𝑟 rooted at 𝑣. Thus, for any vertex 𝑣, consider its parent vertex 𝑢, the edge 𝑢𝑣 in 𝑇𝑟 , and the subtree 𝑇𝑢 − 𝑉 (𝑇𝑣 ) consisting of all descendants of 𝑢 except for the descendants of 𝑢 in 𝑇𝑣 . We then consider what happens to independent dominating sets in both 𝑇𝑣 and 𝑇𝑢 − 𝑉 (𝑇𝑣 ) and how these sets can be merged to form independent dominating sets in the merged subtree 𝑇𝑢 . Recall that an 𝑖-set of a graph 𝐺 is a minimum independent dominating set in 𝐺. Consider any 𝑖-set 𝑆 in a rooted tree 𝑇𝑟 and how 𝑆 intersects the vertices in 𝑇𝑣 for any vertex 𝑣 ∈ 𝑉 (𝑇𝑟 ). Three possibilities exist: 1. 𝑣 ∈ 𝑆, 2. 𝑣 ∉ 𝑆 but 𝑣 is dominated by an immediate descendant (neighbor) of 𝑣 in 𝑇𝑣 , 3. 𝑣 ∉ 𝑆, no neighbor of 𝑣 in 𝑇𝑣 is in 𝑆 but the parent 𝑢 of 𝑣 is in 𝑆. Given this, we can define the following three numbers: IN(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 and for which 𝑣 ∈ 𝑆, OUTC(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 but which does not contain vertex 𝑣, OUTN(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 except vertex 𝑣. For these kinds of sets 𝑆, the parent 𝑢 of 𝑣 will be needed in order to dominate 𝑣. A linear algorithm for computing 𝑖(𝑇) for any tree 𝑇 can be constructed based on the following theorem. Theorem 3.7 ([74]) If 𝑇𝑢 and 𝑇𝑣 are two vertex-disjoint trees, rooted at vertices 𝑢 and 𝑣, respectively, and 𝑇𝑥 is the rooted tree obtained from 𝑇𝑢 and 𝑇𝑣 by adding the edge 𝑢𝑣 and rooting the merged tree at vertex 𝑢 but relabeling this new root 𝑥, then the following hold:  (a) IN(𝑥) = IN(𝑢) + min OUTC(𝑣), OUTN(𝑣) .  (b) OUTC(𝑥) = min OUTN(𝑢) + IN(𝑣), OUTC(𝑢) + OUTC(𝑣), OUTC(𝑢) + IN(𝑣) . (c) OUTN(𝑥) = OUTN(𝑢) + OUTC(𝑣). Proof (a) Let 𝑆 be an 𝑖-set of a rooted tree 𝑇𝑥 by merging two rooted trees 𝑇𝑢 and 𝑇𝑣 , and assume that 𝑢 = 𝑥 ∈ 𝑆. Let 𝑆𝑢 = 𝑆 ∩ 𝑇𝑢 and 𝑆 𝑣 = 𝑆 ∩ 𝑇𝑣 . Clearly, 𝑢 ∈ 𝑆𝑢 and |𝑆𝑢 | = |IN(𝑢)|, else 𝑆 does not have minimum cardinality.  Also, if 𝑢 ∈ 𝑆, then 𝑣 ∉ 𝑆, else 𝑆 is not an independent set. Therefore, |𝑆 𝑣 | = min OUTC(𝑣), OUTN(𝑣) . (b) Let 𝑆𝑢 and 𝑆 𝑣 be as defined above in (a), except that we assume 𝑢 ∉ 𝑆. Vertex 𝑥 = 𝑢 must be dominated in one of three ways, namely: (i) by vertex 𝑣, (ii) by a child of 𝑢 in 𝑇𝑢 , or (iii) by both 𝑣 and a child of 𝑢 in 𝑇𝑢 . In case (i), it follows that |𝑆| = IN(𝑣) + OUTN(𝑢). In case (ii), it follows that |𝑆| = OUTC(𝑢) + OUTC(𝑣). In case (iii), it follows that |𝑆| = OUTC(𝑢) + IN(𝑣).

Section 3.4. A Representative Sample of Domination Algorithms for Trees

63

(c) Let 𝑆 be an 𝑖-set in 𝑇𝑥 that does not contain 𝑥 and such that 𝑥 is the only vertex in 𝑇𝑥 that is not dominated. In this case, it follows that |𝑆| = OUTN(𝑢) + OUTC(𝑣).  Corollary 3.8 If 𝑢 is the root of a tree 𝑇, then 𝑖(𝑇) = min IN(𝑢), OUTC(𝑢) . A simple algorithm for computing 𝑖(𝑇) for any tree 𝑇 can be constructed as follows. Arbitrarily root the tree 𝑇 of order 𝑛 at any vertex 𝑟. Order the vertices according to a breadth-first ordering 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 , where 𝑣 1 = 𝑟. Create an array parent[1, 2, . . . , 𝑛], where for every 2 ≤ 𝑗 ≤ 𝑛, parent[ 𝑗] = 𝑖 for some 𝑖 < 𝑗 and 𝑣 𝑖 is the parent of vertex 𝑣 𝑗 in the breadth-first ordering.

Algorithm 2 Minimum Independent Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) given by an array parent[1, 2, . . . , 𝑛] Output : |𝑆| = 𝑖(𝑇) 1 2 3 4 5

6 7 8 9

10 11

12

[Initialize the values of IN(𝑣), OUTC(𝑣), OUTN(𝑣)] for 1 ≤ 𝑣 ≤ 𝑛 do IN(𝑣) = 1 OUTC(𝑣) = 𝑛 OUTN(𝑣) = 0 od [Propagate values from vertex 𝑣 𝑛 to 𝑣 2 ] for 𝑣 = 𝑛 down to 2 do Let 𝑢 = parent[𝑣]  IN(𝑢) = IN(𝑢) + min OUTC(𝑣), OUTN(𝑣)  OUTC(𝑢) = min OUTN(𝑢) + IN(𝑣), OUTC(𝑢) + OUTC(𝑣), OUTC(𝑢) + IN(𝑣) OUTN(𝑢) = OUTN(𝑢) + OUTC(𝑣) od [Root vertex 𝑟 = 1]  |𝑆| = min IN(1), OUTC(1)

3.4.3

Minimum Total Dominating Set

The following algorithm, due to Laskar et al. [555] in 1984 for computing the total domination number of an arbitrary nontrivial tree, actually solves a more general problem, similar to Algorithm 1 presented earlier in this section. In an optional total dominating set vertices can have one of four different labels: (a) free, meaning that the vertex does not need to be in the TD-set; this happens when the vertex has already been totally dominated by a vertex in 𝑆, (b) bound, meaning that the vertex is neither in 𝑆 nor adjacent to a vertex in 𝑆, but must be totally dominated,

64

Chapter 3. Complexity and Algorithms for Domination in Graphs

(c) required-1, meaning it is in 𝑆 to dominate a vertex in 𝑉, but it is not yet adjacent to a vertex in 𝑆, or (d) required-2, meaning it is a vertex in 𝑆 and is also adjacent to a vertex in 𝑆. Recall that a 𝛾t -set of a graph 𝐺 is a minimum TD-set in 𝐺. In order for the algorithm to find a 𝛾t -set in a tree, all vertices are initially labeled bound, meaning that they must be either in the TD-set 𝑆 being computed or dominated by a vertex in 𝑆. The algorithm proceeds by processing only one leaf 𝑣 in the tree at a time, whose only neighbor is vertex 𝑢. After the vertex 𝑣 has been processed, it is deleted from the tree, resulting in a smaller tree.

Algorithm 3 Minimum Total Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) with all vertices labeled bound Output : A 𝛾t -set 𝑆 1 2

3 4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19

[Initialize] Set 𝑆 ← ∅ while 𝑇 has three or more vertices, execute any one of the following guarded commands do 𝐺 1 : 𝑇 has a free leaf 𝑣: set 𝑇 ← 𝑇 − 𝑣 𝐺 2 : 𝑇 has a bound leaf 𝑣 adjacent to a free or required-2 vertex 𝑢: label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 3 : 𝑇 has a bound leaf 𝑣 adjacent to a bound or required-1 vertex 𝑢: label 𝑢 required-1; set 𝑇 ← 𝑇 − 𝑣 𝐺 4 : 𝑇 has a required-1 leaf 𝑣 adjacent to a vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 5 : 𝑇 has a required-2 leaf 𝑣 adjacent to a required-1 or required-2 vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 6 : 𝑇 has a required-2 leaf 𝑣 adjacent to a free or bound vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 free; set 𝑇 ← 𝑇 − 𝑣 od [Two remaining vertices 𝑢 and leaf 𝑣] case (label(𝑢), label(𝑣)) of :𝑆←𝑆 (free, free) (bound, free) : 𝑆 ← 𝑆 ∪ {𝑣} (bound, bound) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (required-1, . . . ) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (. . . , required-1) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (required-2, free) : 𝑆 ← 𝑆 ∪ {𝑢} (required-2, bound) : 𝑆 ← 𝑆 ∪ {𝑢} (required-2, required-2) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} endcase

Section 3.5. Early Domination Algorithms and NP-Completeness Results

65

The correctness of Algorithm 3 can be seen by the entries in Table 3.2. Consider vertex 𝑣 to be the root of a subtree 𝑇𝑣 positioned to the right of vertex 𝑢, which is the root of its own subtree 𝑇𝑢 , with an edge joining 𝑣 to 𝑢. The label of vertex 𝑣 indicates that in the subtree 𝑇𝑣 we have found a 𝛾t -set 𝑆 𝑣 subject to one of four conditions: (a) vertex 𝑣, labeled free (column 2), is not in 𝑆 𝑣 but is dominated by a vertex in 𝑆 𝑣 , in which case the label of vertex 𝑢 need not change. (b) vertex 𝑣, labeled bound (column 3), is not dominated by any vertex in 𝑆 𝑣 but all other vertices in 𝑇𝑣 are dominated by 𝑆 𝑣 ; in this case vertex 𝑢 must be placed into 𝑆𝑢 in order to dominate vertex 𝑣, the label of 𝑢 then becoming required-2 if it already has a neighbor in 𝑆𝑢 and is currently labeled either free or required-2, or the label of 𝑢 becoming required-1, in case it has no neighbor in 𝑆𝑢 and its label is currently either bound or required-1. (c) vertex 𝑣, labeled required-1 (column 4), is in 𝑆 𝑣 but has no neighbor in 𝑆 𝑣 , which then requires vertex 𝑢 to be placed into 𝑆𝑢 and be labeled required-2, since it will be in 𝑆𝑢 with a neighbor in 𝑆 𝑣 (vertex 𝑣). (d) vertex 𝑣, labeled required-2 (column 5), is in 𝑆 𝑣 and also has a neighbor in 𝑆 𝑣 . In this case, vertex 𝑢 becomes dominated and can be labeled free or can be relabeled required-2 if it is currently labeled required-1. label(𝑣) label(𝑢)

free

bound

required-1

required-2

free bound required-1 required-2

free bound required-1 required-2

required-2 required-1 required-1 required-2

required-2 required-2 required-2 required-2

free free required-2 required-2

Table 3.2 Delete 𝑇𝑣 and label 𝑢 as indicated in the table

3.5

Early Domination Algorithms and NP-Completeness Results

In this section, we present a chronology of early domination algorithms and NPcompleteness results. When a parameter is referred to as weighted, the vertices are labeled with nonnegative integers, and one seeks a set 𝑆 for the associated parameter for which the sum of the weights of the vertices in 𝑆 is a minimum or a maximum, depending on the parameter. 1959

• Maghout [580] presented exponential-time algorithms for the independence number 𝛼(𝐷) and in-domination number 𝛾in (𝐷) in digraphs.

Chapter 3. Complexity and Algorithms for Domination in Graphs

66 1966

• Daykin and Ng [217] presented linear algorithms for unweighted and weighted independence and domination in directed trees; unfortunately, both of the unweighted algorithms for independence and domination contain errors, as do the attempted proofs of correctness of these algorithms. They are noteworthy, nevertheless, for being in some sense ahead of their time, for much was to follow this pioneering paper. 1972

• Gavril [330] presented an O (𝑛3 ) algorithm for independence in chordal graphs, that is, graphs in which every cycle of length greater than three has a chord. • Nieminen [612] presented a greedy algorithm for finding a minimal, but not necessarily minimum, dominating set in an arbitrary graph. 1973

• Gavril [331] presented an O (𝑛3 ) algorithm for independence in circle graphs, that is, graphs whose vertex set is a family of chords of a given circle and in which two vertices are joined by an edge if and only if the corresponding chords intersect. 1974

• Gavril [332] presented an O (𝑛3 ) algorithm for independence in circular arc graphs, that is, graphs whose vertex set is a family of arcs of a given circle and in which two vertices are joined by an edge if and only if the corresponding arcs intersect. 1975

• Cockayne et al. [187] presented the first domination algorithm in graphs, a linear algorithm for domination in trees. • Mitchell et al. [595] presented linear algorithms for trees for independence, vertex covering, and matching. 1977

• Beyer et al. [74] presented a linear algorithm for independent domination in trees. • In her PhD thesis, Mitchell [592, 594] presented a linear algorithm for edge domination 𝛾 ′ (𝑇) in trees, or domination in line graphs of trees, which showed that for trees, 𝛾 ′ (𝑇) = 𝑖 ′ (𝑇), or equivalently that 𝛾(𝐿(𝑇)) = 𝑖(𝐿(𝑇)). She also presented linear algorithms for independence in 𝑘-trees, including trees and maximal outerplanar graphs. 1978

• Natarajan and White [607] presented a linear algorithm for weighted domination in trees. 1979

• In their book [325], Garey and Johnson listed many NP-completeness and algorithmic results for domination parameters. In particular, they stated that INDEPENDENT DOMINATING SET is NP-complete. They also stated that DOMINATING SET is NP-complete for general graphs, planar graphs with Δ(𝐺) ≤ 3, 4-regular

Section 3.5. Early Domination Algorithms and NP-Completeness Results

67

planar graphs, line graphs of planar graphs with Δ(𝐺) ≤ 3, and line graphs of bipartite graphs with Δ(𝐺) ≤ 3. • Mitchell et al. [593] presented linear algorithms on trees for independence, matching, and domination, using recursive representations, or parent arrays, of trees. 1980

• Yannakakis and Gavril [760] constructed an NP-completeness proof for domination and independent domination in line graphs of graphs, or equivalently, an NP-completeness proof for edge domination. 1981

• Dewdney [233] constructed NP-completeness proofs for domination in bipartite graphs and comparability graphs. • Farber [267] presented a linear algorithm for weighted domination in trees. 1982

• Booth and Johnson [87] constructed NP-completeness proofs for domination in chordal graphs and undirected path graphs. They also presented a linear algorithm for domination in directed path graphs, given an appropriate path representation. • Chang [141] in private communication claimed to have constructed an NPcompleteness proof for integer weighted independent domination in chordal graphs. • Chang and Nemhauser [144] constructed an NP-completeness proof for domination in bipartite graphs. • Farber [268] presented a polynomial time algorithm for weighted independent domination in chordal graphs, and thus, in strongly chordal graphs and interval graphs. • Takamizawa et al. [700] presented a ground-breaking, linear algorithm for independence in two-terminal, series-parallel graphs. This would lead to several other series-parallel algorithms. 1983

• Kikuno et al. [523] presented the first domination algorithm in series-parallel graphs, whose running time is linear. • Laskar and Pfaff [554] constructed an NP-completeness proof for total domination in split graphs, and hence, in chordal graphs. • Pfaff et al. [634] constructed an NP-completeness proof for total domination in bipartite graphs. 1984

• Bertossi [71] constructed NP-completeness proofs for domination in split graphs (a subclass of chordal graphs) and in bipartite graphs. • Corneil and Perl [200] constructed NP-completeness proofs for independent domination in bipartite graphs and in comparability graphs. • Farber [269] presented polynomial algorithms for weighted domination and weighted independent domination in strongly chordal graphs.

Chapter 3. Complexity and Algorithms for Domination in Graphs

68

• Pfaff et al. [635] presented linear algorithms for independent domination and total domination in series-parallel graphs. • Laskar et al. [555] presented a linear algorithm for total domination in trees. They also constructed an NP-completeness proof for total domination in undirected path graphs, that is, intersection graphs of families of paths in a graph. 1985


• Chang and Nemhauser [144] presented O |𝑉 ||𝐸 | algorithms for domination and independence in what they call nearly chordal graphs, a family of graphs which properly contain all chordal graphs. These are graphs which are too complex to define in detail here, but roughly they are graphs the maximum length of an induced cycle (or hole) in which is 4, but if there is a 4-hole, then two conditions on each 4-hole must hold. The authors also showed that for what are called odd sun-free chordal graphs, domination and independence can be solved in polynomial time. • Farber and Keil [270] presented O (𝑛3 ) algorithms for weighted domination and weighted independent domination in permutation graphs. They also presented an O (𝑛2 ) algorithm for domination in permutation graphs. • Johnson [511] constructed an NP-completeness proof for domination in arbitrary grid graphs, which include all subgraphs of complete grid graphs, 𝑃𝑚 □ 𝑃𝑛 , for arbitrary 𝑚, 𝑛 ≥ 1. In 2011 Goncalves et al. [362] presented 16 formulas for the domination numbers 𝛾(𝑃𝑚 □ 𝑃𝑛 ), for all 𝑚, 𝑛 ≥ 1. These formulas are given in Chapter 17. 1986

• Bertossi [72] presented an O (𝑛2 ) algorithm for total domination in interval graphs. • Hedetniemi et al. [450] presented a linear algorithm for domination in cacti. • Keil [519] presented an O (𝑛 + 𝑚) algorithm for total domination in interval graphs. 1987

• Brandstädt and Kratsch [100] presented an O (𝑛3 ) algorithm for weighted domination in permutation graphs, an O (𝑛 log2 𝑛) algorithm for independent domination in permutation graphs, and an O (𝑛2 ) algorithm for weighted independent domination in permutation graphs. • Corneil and Keil [199] presented a polynomial algorithm for domination in 𝑘-trees, for fixed 𝑘, but constructed an NP-completeness proof for domination in 𝑘-trees for arbitrary 𝑘. • Hare and Hedetniemi [388] presented a linear algorithm for domination in 𝑘 × 𝑛 knights graphs, for fixed 𝑘. • Hare et al. [390] presented a linear algorithm for upper domination in generalizedseries-parallel graphs, a class of graphs which properly contains two-terminal, series-parallel graphs, trees, outerplanar graphs, unicyclic graphs, 2-trees, and cacti.

Section 3.5. Early Domination Algorithms and NP-Completeness Results

69

• In his PhD thesis, Wimer [752] presented a linear algorithm for upper domination in partial 𝑘-chordal graphs. 1988

• Atallah et al. [40] presented an O (𝑛 log2 𝑛) algorithm for independent domination in permutation graphs. • Bertossi and Gori [73] presented an O (𝑛 log 𝑛) algorithm for total domination in weighted interval graphs. • Ramalingam and Rangan [645] presented linear algorithms for domination, independent domination, and total domination in interval graphs. 1989

• Arnborg and Proskurowski [38] presented linear algorithms for domination and independence in partial 𝑘-trees. 1990

• Cheston et al. [167] constructed the first NP-completeness proof for upper domination. • Corneil and Stewart [201] constructed NP-completeness proofs for domination and total domination in 𝑘-CUBs, for 𝑘 ≥ 2. They also presented polynomial algorithms for domination in 1-CUBs and total domination in permutation graphs. • Elmallah and Stewart [256] presented polynomial algorithms for domination, independent domination, and total domination in 𝑘-polygon graphs, that is, intersection graphs of straight line chords inside a convex 𝑘-gon, for 𝑘 ≥ 3. • Kratochvíl and Nešetřil [543] constructed NP-completeness proofs for independence in what they call 2-DIR and PURE-3-DIR graphs. These refer to intersection graphs of line segments in the plane which have at most 𝑘 different directions, and PURE refers to the condition that parallel segments on the same infinite line do not intersect. 1993

• Kratsch and Stewart [545] presented O (𝑛6 ), O (𝑛6 ), and O (𝑛3 ) algorithms, respectively, for domination, total domination, and independent domination in cocomparability graphs (complements of comparability graphs). • Keil [520] constructed NP-completeness proofs for domination and total domination in circle graphs. 1994

• Fellows et al. [288] constructed NP-completeness proofs for independence and upper domination in graphs with Δ(𝐺) ≤ 5, trestled graphs of index 𝑘 for any 𝑘 ≥ 1, planar graphs, and triangle-free graphs. • Liang and Rhee [563] presented linear O (𝑛 + 𝑚) algorithms for two independent set problems, including weighted independent set, in permutation graphs. 1997

• Chang [146] constructed NP-completeness proofs for weighted domination and total domination in co-bipartite graphs (complements of bipartite graphs). He

Chapter 3. Complexity and Algorithms for Domination in Graphs

70

also presented polynomial algorithms for bounded weighted domination and bounded weighted total domination in cocomparability graphs. 1998

• Fricke et al. [313] constructed NP-completeness proofs for independence in 𝑘-regular graphs, for 𝑘 ≥ 3, and for the upper domination number in 𝑘-regular graphs, for 𝑘 ≥ 6. • Chang [147] presented a unified approach to the design of algorithms for weighted domination, weighted independence, and total domination for both interval graphs (given an interval model with endpoints sorted) and
circular-arc graphs. These algorithms run in either O (𝑛) or O 𝑛 log(log 𝑛) time on interval graphs and O (𝑚 + 𝑛) time on circular-arc graphs. 1999

Since 1999 more than 1100 papers have been published on various kinds of domination algorithms and complexity in a wide variety of classes of graphs. In this section, we have reviewed only what might be considered the early foundational papers on algorithms and complexity of domination in graphs. Indeed, one could publish several books alone on this topic.

3.6

Other Sources for Domination Algorithms and Complexity

Other than the more than 1100 papers on domination algorithms and complexity that have been published since 1999, several books have been published which contain much of this material, which we describe in this section. The book Fundamentals of Domination in Graphs, by Haynes et al. [417], contains the following: • Section 1.11 An Introduction to NP-Completeness. • Section 1.12 NP-Completeness of the Domination Problem, containing an NPcompleteness proof of DOMINATING SET. • Chapter 12 Domination Complexity and Algorithms, 27 pages, contains NPcompleteness proofs of DOMINATING SET when restricted to bipartite graphs and when restricted to chordal graphs. It mentions, with references, published NPcompleteness results for lower irredundance, domination, independent domination, independence, upper domination, upper irredundance, connected domination, and total domination. Interestingly, no mention is made of upper total domination. But in her PhD thesis, McRae [588] settled the NP-completeness of upper total domination. Also presented in this chapter are polynomial algorithms for domination in trees, domination in interval graphs, total domination in interval graphs, independent domination in interval graphs, vertex independence in interval graphs, and minimum weight independent domination in permutation graphs. The book Domination in Graphs: Advanced Topics, edited by Haynes et al. [416], contains the following two chapters:

Section 3.6. Other Sources for Domination Algorithms and Complexity

71

• Chapter 8 Algorithms, by Kratsch, 41 pages and 155 references, contains algorithms for minimum weight dominating set in strongly chordal graphs, connected domination in cocomparability graphs, minimum weight total domination in interval graphs, independent domination in permutation graphs, and dominating clique in dually chordal graphs. • Chapter 9 Complexity Results, by Hedetniemi, McRae, and Parks, 37 pages and 34 references, contains NP-completeness results for bipartite graphs for independent domination, 2-maximal matchings, and minimum maximal strong matchings. In addition, it contains NP-completeness results for both bipartite graphs and chordal graphs for: perfect domination, efficient domination, efficient total domination, minimum maximal strong stable sets or 2-packings, domination, total domination, odd domination, weak vertex-edge domination, lower irredundance, closed open irredundance, open open irredundance, and open irredundance. In addition, in Part III of Structures of Domination in Graphs, edited by Haynes et al. [414], published in 2021, one can find the following chapters on domination algorithms and complexity: • Algorithms and Complexity of Signed, Minus, and Majority Domination, by Hedetniemi, McRae, and Mohan, with 30 pages and 36 references. • Algorithms and Complexity of Power Domination in Graphs, by Hedetniemi, McRae, and Mohan, with 24 pages and 43 references. • Self-Stabilizing Domination Algorithms, by Hedetniemi, with 36 pages and 72 references. • Algorithms and Complexity of Alliances in Graphs, by Hedetniemi, with 15 pages and 21 references. In addition to the above, the book Exact Exponential Algorithms, by Fomin and Kratsch [307], contains results on exact exponential algorithms for independence and domination in graphs. We quote here from Math Reviews MR3234973 about this book: Moderately exponential-time algorithm is the somewhat euphemistic alternative name for an area of algorithms that has taken off and flourished in the past decade. The main thrust is how to cope with problems that have been proven to be NP-hard, yet must be solved in practice, and not only approximately or merely heuristically, but exactly and reliably. Fedor Fomin and Dieter Kratsch have always been in the front line of these developments. They are therefore predestined to write the first textbook in this area. The book covers the main techniques that have been developed section by section: branching, dynamic programming, inclusion-exclusion, treewidth, measure & conquer, subset convolution, local search, split and list, and trade-offs like time versus space. It mostly sticks to only a few problems to illustrate the techniques: independence and domination in graphs, coloring, traveling salesman, and satisfiability. All statements directly concerning the algorithmic techniques come with proofs, only

72

Chapter 3. Complexity and Algorithms for Domination in Graphs some purely combinatorial yet very technical theorems are merely stated, with pointers to the literature in each such case. The coverage of the recent literature on the area is another strength of the book, which can be, overall, recommended as a textbook for a master (or PhD level) course on this still new and hot topic in algorithms.

It is also worth mentioning that the 1994 PhD thesis of McRae [588] contains 81 original NP-completeness results for bipartite graphs, chordal graphs, line graphs, and line graphs of bipartite graphs, for the decision problems relating to: domination, independent domination, total domination, perfect domination, efficient domination, efficient total domination, independence, upper domination, upper total domination, and upper irredundance. We conclude by simply listing by subject area a sampling of more recent papers on domination algorithms and complexity. For domination, see [260, 497, 544, 614, 627]. For independent domination, see [89, 216, 247, 499, 565, 624]. For upper domination, see [2, 62, 91]. For greedy algorithms, see [670, 671, 788]. For self-stabilizing domination algorithms, see [64, 168, 169, 234, 323, 346–348, 367, 441, 442, 449, 503, 690, 718, 756]. For approximation algorithms and complexity, see [170, 212, 213, 374, 611, 623]. For fixed parameter algorithms and complexity, see [13, 90, 244, 292, 371, 515, 584, 706]. For domination algorithms on Cartesian products, see [205, 207, 334]. For online domination algorithms, see [92, 375]. For exact exponential domination algorithms, see [169, 234, 306, 308, 323, 347, 348, 367, 441, 442, 448, 449, 503, 610, 669, 677, 690, 702, 718, 756]. For approximation algorithms and complexity, see [170, 212, 213, 374, 611, 623].

Chapter 4

General Bounds 4.1 Introduction As we have seen in Chapter 3, the decision problems associated with computing the domination, total domination, and independent domination numbers of arbitrary graphs are all NP-complete. Given the difficulty of determining the exact values of these domination numbers, much of the research involves the determination of tight lower and upper bounds for these numbers. In this chapter, we present some of the more basic bounds on the domination, total domination, and independent domination numbers of graphs. Additional bounds on these parameters will be given throughout the text, particularly in Chapter 6 in terms of minimum degree and Chapter 8 in terms of size. Recall that we use the abbreviation “TD-set” for a “total dominating set,” and “ID-set” for an “independent dominating set.” Further, recall that a dominating vertex in a graph 𝐺 of order 𝑛 is a vertex (of degree 𝑛 − 1) adjacent to every other vertex in 𝐺. The corona 𝐺 ◦ 𝐾1 of a graph 𝐺 is the graph obtained from 𝐺 by adding for each vertex 𝑣 ∈ 𝑉 a new vertex 𝑣 ′ and the edge 𝑣𝑣 ′ . A packing in a graph 𝐺 is a set of vertices whose closed neighborhoods are pairwise disjoint. Thus, if 𝑆 is a packing in 𝐺, then N[𝑢] ∩ N[𝑣] = ∅ for all 𝑢, 𝑣 ∈ 𝑆, implying that 𝑑 (𝑢, 𝑣) ≥ 3 for all 𝑢, 𝑣 ∈ 𝑆. The packing number 𝜌(𝐺) is the maximum cardinality of a packing in 𝐺. An open packing in a graph 𝐺 is a set of vertices whose open neighborhoods are pairwise disjoint. Thus, if 𝑆 is an open packing in 𝐺, then N(𝑢) ∩ N(𝑣) = ∅ for all 𝑢, 𝑣 ∈ 𝑆. The open packing number 𝜌 o (𝐺) is the maximum cardinality of an open packing in 𝐺. A perfect packing in 𝐺 is a packing whose closed neighborhoods partition 𝑉 (𝐺), and a perfect open packing in 𝐺 is an open packing whose open neighborhoods partition 𝑉 (𝐺). A perfect packing is also called an efficient dominating set and a perfect open packing is called an efficient total dominating set, to be discussed further in Chapter 9. A graph is claw-free if it contains no induced 𝐾1,3 . For ease of discussion, we say that an edge is between two sets 𝑋 and 𝑌 if it is incident to a vertex in 𝑋 and to a vertex in 𝑌 . © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_4

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Chapter 4. General Bounds

74

4.2 Domination and Maximum Degree In this section, we present some elementary bounds on the domination, total domination, and independent domination numbers of a graph in terms of the maximum degree. We first give some trivial properties of domination in graphs. Adding edges to a graph cannot increase its domination number. This yields the following observation. Observation 4.1 If 𝐻 is a spanning subgraph of a graph 𝐺, then 𝛾(𝐺) ≤ 𝛾(𝐻). Let 𝐻 be a spanning subgraph of a graph 𝐺, where 𝐻 is isolate-free. Thus, 𝑉 (𝐻) = 𝑉 (𝐺) and 𝐸 (𝐻) ⊆ 𝐸 (𝐺), and 𝛿(𝐻) ≥ 1. Let 𝑆 be a 𝛾t -set of 𝐻. Hence, every vertex not in 𝑆 has a neighbor in 𝑆 in the graph 𝐻 and the subgraph 𝐻 [𝑆] of 𝐻 induced by 𝑆 contains no isolated vertex. Adding edges to 𝐻 to rebuild the graph 𝐺 preserves these two properties, that is, every vertex not in 𝑆 has a neighbor in 𝑆 in the graph 𝐺 and the subgraph 𝐻 [𝑆] of 𝐻 induced by 𝑆 contains no isolated vertex. Thus, 𝑆 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆| = 𝛾t (𝐻). We state this trivial observation formally as follows. Observation 4.2 If 𝐻 is an isolate-free spanning subgraph of a graph 𝐺, then 𝛾t (𝐺) ≤ 𝛾t (𝐻).

4.2.1

Domination Number and Maximum Degree

In 1973 Berge [68] observed that if 𝑣 is a vertex of maximum degree in a graph 𝐺 = (𝑉, 𝐸) of order 𝑛, then the vertex 𝑣 together with all its non-neighbors forms a dominating set of 𝐺, and so 𝛾(𝐺) ≤ |𝑉 \ N(𝑣)| = 𝑛 − deg(𝑣) = 𝑛 − Δ(𝐺). If 𝐹 is a graph of order 𝑘 ≥ 1 that contains a dominating vertex, then the corona 𝐺 = 𝐹 ◦ 𝐾1 of 𝐹 is a graph of order 𝑛 = 2𝑘 with maximum degree Δ(𝐺) = 𝑘 and domination number 𝛾(𝐺) = 𝑘 = 𝑛 − Δ(𝐺). Thus, we have the following. Theorem 4.3 ([68]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) ≤ 𝑛 − Δ(𝐺), and this bound is tight. In 1979 Walikar et al. [741] observed that in a 𝛾-set 𝑆 of a graph 𝐺, every vertex of 𝑆 dominates itself and at most Δ(𝐺) other vertices. Thus, at most Δ(𝐺)
+ 1 vertices are dominated by each vertex of 𝑆, implying that at most Δ(𝐺) + 1 |𝑆| distinct every vertex
is dominated by the set 𝑆, vertices are dominated by the set 𝑆. Since
we therefore have that 𝑛 ≤ Δ(𝐺) + 1 |𝑆| = Δ(𝐺) + 1 𝛾(𝐺). Further, if there is equality in this inequality, then the following conditions hold: (i) the set 𝑆 is an independent set, (ii) no two vertices of 𝑆 have a common neighbor, and (iii)
each vertex of 𝑆 has degree Δ(𝐺). In particular, in this case when 𝑛 = Δ(𝐺) + 1 𝛾(𝐺), we have 𝛾(𝐺) = 𝑖(𝐺). These observations yield the following trivial lower bound on the domination number of a graph in terms of its order and maximum degree. Theorem 4.4 ([741]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) ≥

𝑛 , 1 + Δ(𝐺)

Section 4.2. Domination and Maximum Degree

75

with equality if and only if every 𝛾-set in 𝐺 is a perfect packing and every vertex in such a 𝛾-set has degree Δ(𝐺). As an application of Theorem 4.4, we can readily determine the domination and independent domination numberfor a path 𝑃𝑛 and a cycle 𝐶𝑛 of order 𝑛 ≥ 3. By Theorem 4.4, we have 𝛾(𝐶𝑛 ) ≥ 𝑛3 . Choosing every third vertex on the path 𝑃𝑛 , starting with the first vertex if 𝑛 ≡ 1 (mod 3) and starting with the second   vertex otherwise, produces an ID-set of the path 𝑃𝑛 , and so 𝛾(𝑃𝑛 ) ≤ 𝑖(𝑃𝑛 ) ≤ 𝑛3 . Since adding edges cannot increase   the domination number, 𝛾(𝐶𝑛 ) ≤ 𝛾(𝑃𝑛 ). Combining these inequalities, we have 𝑛3 ≤ 𝛾(𝐶𝑛 ) ≤ 𝛾(𝑃𝑛 ) ≤ 𝑖(𝑃𝑛 ) ≤ 𝑛3 , and hence equality     must hold throughout this inequality chain. Moreover, 𝑛3 = 𝛾(𝐶𝑛 ) ≤ 𝑖(𝐶𝑛 ) ≤ 𝑛3 , once again yielding equality throughout the inequality chain. We state this formally as follows. Proposition 4.5 For 𝑛 ≥ 3, 𝛾(𝐶𝑛 ) = 𝛾(𝑃𝑛 ) = 𝑖(𝐶𝑛 ) = 𝑖(𝑃𝑛 ) =

𝑛

.

3

We state next several other upper bounds on the domination number of a graph in terms of its order and both minimum and maximum degrees. The following bound was determined in 1986 by Marcu [583]. Theorem 4.6 ([583]) If 𝐺 is a graph of order 𝑛, then

  𝑛 − Δ(𝐺) − 1 𝑛 − 𝛿(𝐺) − 2 𝛾(𝐺) ≤ + 2. 𝑛−1 Flach and Volkmann [305] in 1990 determined the following bound. Theorem 4.7 ([305]) If 𝐺 is an isolate-free graph of order 𝑛, then  
Δ(𝐺) 1 𝛾(𝐺) ≤ 𝑛 + 1 − 𝛿(𝐺) − 1 . 2 𝛿(𝐺) An immediate corollary, due to Payan [632] in 1975, now follows. Corollary 4.8 ([632]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤
1 𝑛 + 2 − 𝛿(𝐺) . 2 In 1995 Slater [680] gave the following lower bound on the domination number of a graph that involves its nonincreasing degree sequence. The value of the lower bound is known as the Slater number sl(𝐺). Theorem 4.9 ([680]) If 𝐺 is a graph of order 𝑛 with degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾(𝐺) ≥ sl(𝐺) = min 𝑘 : 𝑘 + (𝑑1 + 𝑑2 + · · · + 𝑑𝑘 ) ≥ 𝑛 . Proof Let 𝐷 be 𝛾-set of 𝐺, where |𝐷| = 𝑘. Each vertex 𝑣 ∈ 𝐷 is adjacent to at most deg(𝑣) vertices in 𝑉 \ 𝐷, while each vertex in 𝑉 \ 𝐷 has at least one neighbor in 𝐷. Hence, double counting the edges between 𝐷 and 𝑉 \ 𝐷,

Chapter 4. General Bounds

76 𝑛 − 𝑘 = |𝑉 \ 𝐷 | ≤

∑︁

deg(𝑣) ≤

𝑣 ∈𝐷

𝑘 ∑︁

𝑑𝑖 ,

𝑖=1

and so 𝑘 + (𝑑1 + 𝑑2 + · · · + 𝑑 𝑘 ) ≥ 𝑛. Hence, sl(𝐺) ≤ 𝑘 = |𝐷 | = 𝛾(𝐺).

4.2.2

Total Domination Number and Maximum Degree

An observation similar to Theorem 4.4 can be made for the total domination number. Recall that for sets of vertices 𝑋 and 𝑌 in a graph 𝐺 the set 𝑋 dominates 𝑌 if every vertex of 𝑌 has a neighbor in 𝑋 or belongs to 𝑋, and the set 𝑋 totally dominates 𝑌 if every vertex of 𝑌 has a neighbor in 𝑋. If 𝑋 dominates 𝑌 , 𝑋 = {𝑥}, and 𝑌 = {𝑦}, then we simply write that vertex 𝑥 totally dominates vertex 𝑦. In particular, if 𝑋 dominates the set 𝑉 (𝐺), then 𝑋 is a dominating set of 𝐺, and if 𝑋 totally dominates the set 𝑉 (𝐺), then 𝑋 is a TD-set of 𝐺. In a 𝛾t -set 𝑆 of a graph 𝐺, every vertex 𝑣 ∈ 𝑆 totally dominates at most Δ(𝐺) other vertices, namely all vertices in its open neighborhood N(𝑣). Thus, at most Δ(𝐺) vertices are totally dominated by each vertex of 𝑆, implying that at most Δ(𝐺)|𝑆| distinct vertices are totally dominated by the set 𝑆. Since every vertex is totally dominated by the set 𝑆, we therefore have that 𝑛 ≤ Δ(𝐺)|𝑆| = Δ(𝐺)𝛾t (𝐺). Further, if equality holds in this inequality chain, then the following conditions hold: (i) the set of open neighborhoods N(𝑣) of vertices 𝑣 ∈ 𝑆 is a partition of 𝑉 (𝐺), (ii) the subgraph 𝐺 [𝑆] is the disjoint union of copies of 𝐾2 , and (iii) each vertex of 𝑆 has degree Δ(𝐺). This yields the following elementary lower bound on the total domination number of a graph in terms of its order and maximum degree. Theorem 4.10 ([741]) If 𝐺 is a graph of order 𝑛, then 𝛾t (𝐺) ≥

𝑛 , Δ(𝐺)

with equality if and only if every 𝛾t -set in 𝐺 is a perfect open packing and every vertex in such a 𝛾t -set has degree Δ(𝐺). As an application of Theorem 4.10, we can readily determine the total domination number of a path 𝑃𝑛 and a cycle 𝐶𝑛 .       Proposition 4.11 For 𝑛 ≥ 2, 𝛾t (𝑃𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 , and for 𝑛 ≥ 3, 𝛾t (𝑃𝑛 ) = 𝛾t (𝐶𝑛 ). Proof Let 𝐺 = 𝐶𝑛 , where 𝑛 ≥ 3. The equality is immediate if 𝑛 = 3. Hence, we may assume that 𝐺 is the cycle 𝑣 1 𝑣 2 . . . 𝑣 𝑛 𝑣 1 , where 𝑛 ≥ 4. The elementary lower bound on the total domination number given in Theorem 4.10 yields 𝛾t (𝐺) ≥ 12 𝑛. Let 𝑆1 =

𝑛 ⌊Ø 4 ⌋−1

{𝑣 4𝑖+2 , 𝑣 4𝑖+3 }.

𝑖=0

We consider four cases based on 𝑛.

Section 4.2. Domination and Maximum Degree

77

Case 1. 𝑛 ≡ 0 (mod 4). Thus, 𝑛 = 4𝑘 for some 𝑘 ≥ 1. In this case, the set 𝑆1 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆1 | = 2𝑘 = 21 𝑛. Consequently, 𝛾t (𝐺) = 12 𝑛. Case 2. 𝑛 ≡ 2 (mod 4). Thus, 𝑛 = 4𝑘 + 2 for some 𝑘 ≥ 1. Suppose that 𝛾t (𝐺) = 12 𝑛 = 2𝑘 + 1. Let 𝐷 be a 𝛾t -set of 𝐺, and so |𝐷 | = 2𝑘 + 1 is odd. Thus, at least one component in 𝐺 [𝐷] has odd order at least 3, implying that at least one vertex in 𝐷 has no neighbor in 𝑉 \ 𝐷. Hence, counting edges between 𝐷 and 𝑉 \ 𝐷 yields at most |𝐷| − 1 edges. Since every vertex in 𝑉 \ 𝐷 has a neighbor in 𝐷, this yields an edge count between 𝐷 and 𝑉 \ 𝐷 of at least |𝑉 \ 𝐷| = 𝑛 − |𝐷|. Consequently, 𝑛 − |𝐷| ≤ |𝐷| − 1, and so |𝐷| ≥ 12 (𝑛 + 1), a contradiction. Therefore, 𝛾t (𝐺) > 12 𝑛 = 2𝑘 + 1, that is, 𝛾t (𝐺) ≥ 2𝑘 + 2. The set 𝑆2 = 𝑆1 ∪ {𝑣 4𝑘+1 , 𝑣 4𝑘+2 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆2 | = 2𝑘 + 2. Consequently, 𝛾t (𝐺) ≤ 2𝑘 + 2 = 12 𝑛 + 1. Case 3. 𝑛 ≡ 1 (mod 4). Thus, 𝑛 = 4𝑘 + 1 for some 𝑘 ≥ 1. As observed earlier, 𝛾t (𝐺) ≥ 12 𝑛. Since 𝑛 is odd, we therefore have 𝛾t (𝐺) ≥ 12 (𝑛 + 1) = 2𝑘 + 1. The set 𝑆3 = 𝑆1 ∪ {𝑣 4𝑘 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆3 | = 2𝑘 + 1. Consequently, 𝛾t (𝐺) = 2𝑘 + 1 = 12 (𝑛 + 1). Case 4. 𝑛 ≡ 3 (mod 4). Thus, 𝑛 = 4𝑘 + 3 for some 𝑘 ≥ 1. Since 𝑛 is odd, we have 𝛾t (𝐺) ≥ 12 (𝑛 + 1) = 2𝑘 + 2. The set 𝑆4 = 𝑆1 ∪ {𝑣 4𝑘+1 , 𝑣 4𝑘+2 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆4 | = 2𝑘 + 2. Consequently, 𝛾t (𝐺) = 2𝑘 + 2 = 12 (𝑛 + 1).       In all cases, 𝛾t (𝐶𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 .  𝑛 An 𝑛identical   𝑛  construction of the earlier TD-set in a cycle 𝐶𝑛 yields 𝛾t (𝑃𝑛 ) ≤ + 2 4 − 4 . The desired result for a path now follows by applying Observation 4.2 to the result of a cycle. Suppose that a connected graph 𝐺 = (𝑉, 𝐸) of order 𝑛 ≥ 2 contains a dominating vertex 𝑣, that is, suppose 𝑉 = N[𝑣] and deg(𝑣) = Δ(𝐺) = 𝑛 − 1. In this case, the set consisting of 𝑣 and an arbitrary neighbor of 𝑣 form a TD-set of 𝐺, implying that 𝛾t (𝐺) = 2 = 𝑛 − Δ(𝐺) + 1. We state this formally as follows. Proposition 4.12 If 𝐺 is a connected graph of order 𝑛 ≥ 2 that contains a dominating vertex, then 𝛾t (𝐺) = 2 = 𝑛 − Δ(𝐺) + 1. In 1980 Cockayne et al. [182] established the following upper bound in terms of the order and maximum degree. Theorem 4.13 ([182]) If 𝐺 is a connected graph of order 𝑛 that does not contain a dominating vertex, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺). Proof Suppose that a connected graph 𝐺 of order 𝑛 does not contain a dominating vertex. Necessarily, 𝑛 ≥ 4. Let 𝑣 be a vertex of maximum degree in 𝐺, and so deg𝐺 (𝑣) = Δ(𝐺). Let 𝑇 be a spanning tree of 𝐺 that is distance-preserving from 𝑣. In particular, N𝐺 (𝑣) = N𝑇 (𝑣) and Δ(𝐺) = Δ(𝑇). Let 𝑇 have ℓ(𝑇) leaves. We note that ℓ(𝑇) ≥ Δ(𝑇). Let 𝑆 be the set of vertices that are not leaves of 𝑇. Since the graph 𝐺 does not contain a dominating vertex, removing all leaves from 𝑇 produces a connected graph of order at least 2. Thus, 𝐺 [𝑆] is a connected graph of order at least 2, implying that 𝑆 is a TD-set of 𝐺. Therefore, 𝛾t (𝐺) ≤ |𝑆| = 𝑛 − ℓ(𝑇) ≤ 𝑛 − Δ(𝑇) = 𝑛 − Δ(𝐺). As an application of Proposition 4.12, we have the following upper bound on the total domination number of a disconnected graph.

Chapter 4. General Bounds

78

Theorem 4.14 ([182]) If 𝐺 is a disconnected isolate-free graph of order 𝑛, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1, with equality if and only if every component of 𝐺 is a copy of 𝐾2 , except possibly for one component which contains a dominating vertex. Proof Let 𝐺 be a disconnected graph with components 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 where 𝑘 ≥ 2. We note that Δ(𝐺) = max Δ(𝐺 𝑖 ). 𝑖∈[𝑘]

Since 𝐺 is isolate-free, we note that Δ(𝐺 𝑖 ) ≥ 1 for all 𝑖 ∈ [𝑘]. Renaming components if necessary, we may assume that Δ(𝐺) = Δ(𝐺 1 ). With this assumption, 𝑘 ∑︁

Δ(𝐺 𝑖 ) = Δ(𝐺) +

𝑘 ∑︁

Δ(𝐺 𝑖 ) ≥ Δ(𝐺) + 𝑘 − 1.

(4.1)

𝑖=2

𝑖=1

By linearity and by Inequality (4.1), Proposition 4.12 and Theorem 4.13 yield the desired upper bound on the total domination number of 𝐺 as follows: 𝛾t (𝐺) =

𝑘 ∑︁

𝛾t (𝐺 𝑖 )

𝑖=1

≤

𝑘 ∑︁

𝑛(𝐺 𝑖 ) − Δ(𝐺 𝑖 ) + 1




𝑖=1

=

∑︁ 𝑘

 ∑︁  𝑘 Δ(𝐺 𝑖 ) + 𝑘 𝑛(𝐺 𝑖 ) −

𝑖=1

𝑖=1


≤ 𝑛 − Δ(𝐺) + 𝑘 − 1 + 𝑘 = 𝑛 − Δ(𝐺) + 1. Moreover, suppose that 𝛾t (𝐺) = 𝑛−Δ(𝐺) +1. Thus, we must have equality throughout the above inequality chain. In particular, 𝛾t (𝐺 1 ) = 𝑛(𝐺 1 ) − Δ(𝐺 1 ) + 1, implying by Proposition 4.12 and Theorem 4.13 that the component 𝐺 1 contains a dominating vertex. Furthermore, we must have equality in Inequality (4.1), implying that Δ(𝐺 𝑖 ) = 1 for all 𝑖 ∈ [𝑘] \ {1}, and so all such components 𝐺 𝑖 are 𝐾2 -components. For an example of graphs attaining the upper bound of Theorem 4.13, consider the subdivided star 𝐺 = 𝑆(𝐾1,𝑘 ) obtained from a star 𝐾1,𝑘 by subdividing every edge exactly once. The resulting graph 𝐺 has order 𝑛 = 2𝑘 + 1, Δ(𝐺) = 𝑘, and 𝛾t (𝐺) = 𝑘 + 1 = 𝑛 − Δ(𝐺). In 2001 Haynes and Markus [435] determined a property of the graphs attaining the upper bound of 𝑛 − Δ(𝐺) as follows. For a graph 𝐺 of order  𝑛 and 𝑘 ∈ [𝑛], the generalized maximum degree, denoted Δ 𝑘 (𝐺), of 𝐺 is the max |N(𝑆)| : 𝑆 ⊆ 𝑉 and |𝑆| = 𝑘 . Note that Δ1 (𝐺) = Δ(𝐺). Theorem 4.15 ([435]) If 𝐺 is a connected graph of order 𝑛 ≥ 3 with  Δ(𝐺) ≤ 𝑛 − 2, then 𝛾t (𝐺) = 𝑛−Δ(𝐺) if and only if Δ 𝑘 (𝐺) = Δ(𝐺) + 𝑘 for all 𝑘 ∈ 2, 3, . . . , 𝛾t (𝐺) .

Section 4.3. Domination and Order

79

Henning and Yeo noted in their book [490] that one direction of Theorem 4.15 can be strengthened as follows. of order 𝑛 ≥ 3 with Δ(𝐺) ≤ Theorem 4.16 ([490]) Let 𝐺 be a connected graph
𝑛 − 2. If Δ𝑛−Δ(𝐺) −1 (𝐺) = Δ(𝐺) + 𝑛 − Δ(𝐺) − 1 = 𝑛 − 1, then 𝛾t (𝐺) = 𝑛 − Δ(𝐺). We close this section with the following trivial lower bound on the total domination number that involves the nonincreasing degree sequence of a graph. This result has the same flavor as the Slater lower bound for domination given in Theorem 4.9. Theorem 4.17 If 𝐺 is an isolate-free graph of order 𝑛 with degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ) where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾t (𝐺) ≥ min{𝑘 : 𝑑1 + 𝑑2 + · · · + 𝑑 𝑘 ≥ 𝑛}.

4.2.3 Independent Domination Number and Maximum Degree We establish next a trivial upper bound on the independent domination number of a graph in terms of its order and maximum degree. Let 𝑣 be a vertex of maximum degree Δ(𝐺) in a graph 𝐺, and let 𝐼 𝑣 be a maximal independent set that contains vertex 𝑣. Since 𝐼 𝑣 is an ID-set of 𝐺 that contains no neighbor of 𝑣, we have 𝑖(𝐺) ≤ |𝐼 𝑣 | ≤ |𝑉 \ N(𝑣)| = 𝑛 − deg(𝑣) = 𝑛 − Δ(𝐺). This yields the following upper bound on 𝑖(𝐺), due to Haviland [398] in 1991. Theorem 4.18 ([398]) If 𝐺 is a graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺).

4.3

Domination and Order

An obvious upper bound on the (independent/total) domination number of any graph is its order. And since every vertex must have a neighbor in a TD-set, an immediate lower bound on the total domination number is 2. We state this formally as follows. Observation 4.19 If 𝐺 is a graph of order 𝑛, then the following hold: (a) 1 ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝑛. (b) If 𝐺 is isolate-free, then 2 ≤ 𝛾t (𝐺) ≤ 𝑛. We note that the bounds of Observation 4.19 are tight. Clearly, for 𝑛 ≥ 2, if 𝐺 is a complete graph 𝐾𝑛 or a star 𝐾1,𝑛−1 , then 𝛾(𝐺) = 𝑖(𝐺) = 1 and 𝛾t (𝐺) = 2. On the other hand, for the empty graph 𝐺 = 𝐾 𝑛 , we have 𝛾(𝐺) = 𝑖(𝐺) = 𝑛. Similarly, for the graph 𝑘𝐾2 of order 𝑛 = 2𝑘 consisting of 𝑘 ≥ 1 vertex-disjoint copies of 𝐾2 , we have 𝛾t (𝑘𝐾2 ) = 𝑛. Observation 4.20 If 𝐺 is a graph of order 𝑛, then the following hold: (a) 𝛾(𝐺) = 1 (respectively, 𝑖(𝐺) = 1) if and only if 𝐺 has a dominating vertex. (b) 𝛾(𝐺) = 𝑛 (respectively, 𝑖(𝐺) = 𝑛) if and only if 𝐺 is the empty graph 𝐾 𝑛 .

Chapter 4. General Bounds

80

4.3.1 Domination Number and Order Recall that in Section 2.6, we presented Ore’s Lemmas [622] from 1962. In particular, Lemma 2.73 stated that the complement 𝑉 \ 𝐷 of any minimal dominating set 𝐷 in an isolate-free graph 𝐺 = (𝑉, 𝐸) is a dominating set. As animmediate consequence of this lemma, we have if 𝐺 has order 𝑛, then 𝛾(𝐺) ≤ min |𝐷|, 𝑛 − |𝐷| ≤ 21 𝑛. We state this formally as follows. Theorem 4.21 (Ore’s Theorem [622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 12 𝑛. In 1979 Bollobás and Cockayne [84] established the following property of minimum dominating sets in graphs. Recall that for a set 𝑆 and a vertex 𝑣 ∈ 𝑆, the 𝑆-external private neighborhood of 𝑣 is abbreviated epn[𝑣, 𝑆] or epn(𝑣, 𝑆). Thus, epn[𝑣, 𝑆] = epn(𝑣, 𝑆). Lemma 4.22 ([84]) Every isolate-free graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. Proof Among all 𝛾-sets of 𝐺, let 𝐷 be chosen so that the number of edges in 𝐺 [𝐷] is a maximum. Let 𝑣 be an arbitrary vertex of 𝐷 and suppose, to the contrary, that epn[𝑣, 𝐷] = ∅. By Ore’s Lemma 2.72, ipn[𝑣, 𝐷] ≠ ∅, and so 𝑣 is an isolate in 𝐺 [𝐷] and therefore all neighbors of 𝑣 belong outside the set 𝐷. Since epn[𝑣, 𝐷] = ∅, every neighbor of 𝑣 is dominated by some vertex in 𝐷 \ {𝑣}. Replacing the vertex 𝑣 in 𝐷 by an arbitrary neighbor of 𝑣 outside 𝐷 produces a new 𝛾-set of 𝐺 that induces a subgraph containing more edges than does the subgraph induced by 𝐷, contradicting our choice of 𝐷. Hence, epn[𝑣, 𝐷] ≠ ∅. We note that if 𝐷 is a 𝛾-set of an isolate-free graph 𝐺 of order 𝑛 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷, then ∑︁ 𝑛 − |𝐷| = |𝑉 \ 𝐷| ≥ |epn[𝑣, 𝐷] | ≥ |𝐷 |, (4.2) 𝑣 ∈𝐷

and so |𝐷 | ≤ 12 𝑛. Hence, the bound of Theorem 4.21 also follows as an immediate consequence of Lemma 4.22. A vertex of degree 1 is called a leaf and the neighbor of a leaf is called a support vertex. We note that the set 𝑉 (𝐺) is a dominating set of the corona 𝐺 ◦ 𝐾1 , and so 𝛾(𝐺 ◦ 𝐾1 ) ≤ |𝑉 (𝐺)|. Moreover, every vertex of 𝐺 is a support vertex of 𝐺 ◦ 𝐾1 and either it or its leaf neighbor is in every 𝛾-set of 𝐺 ◦ 𝐾1 , implying that 𝛾(𝐺 ◦ 𝐾1 ) ≥ |𝑉 (𝐺)|. Consequently, 𝛾(𝐺 ◦ 𝐾1 ) = |𝑉 (𝐺)|. For example, the corona 𝐶5 ◦𝐾1 of the cycle 𝐶5 shown in Figure 4.1 has 𝛾(𝐶5 ◦𝐾1 ) = 5 and the five highlighted vertices form a 𝛾-set of 𝐶5 ◦ 𝐾1 . This yields the following observation. Observation 4.23 If 𝐺 is a corona 𝐻 ◦ 𝐾1 of order 𝑛, then 𝛾(𝐺) = 12 𝑛, and if 𝐻 is isolate-free, then 𝛾t (𝐺) = 12 𝑛. In 1982 Payan and Xuong [633] characterized the graphs achieving equality in Theorem 4.21, and showed that with the exception of a 4-cycle, coronas are the only

Section 4.3. Domination and Order

81

Figure 4.1 The corona 𝐶5 ◦ 𝐾1

graphs achieving equality in Ore’s Theorem 4.21. We remark that this characterization due to Payan and Xuong came 20 years after Ore’s Theorem. Theorem 4.24 ([633]) If 𝐺 is an isolate-free graph of even order 𝑛, then 𝛾(𝐺) = 12 𝑛 if and only if every component of 𝐺 is a 4-cycle or a corona 𝐻 ◦ 𝐾1 for some graph 𝐻. Proof Let 𝐺 be an isolate-free graph of even order 𝑛. Since the domination number of a graph is the sum of the domination numbers of its components and by Theorem 4.21 each component has domination number at most half its order, it suffices to prove the result when 𝐺 is connected. If 𝐺 = 𝐶4 , then 𝛾(𝐺) = 2 = 12 𝑛, and if 𝐺 = 𝐻 ◦ 𝐾1 for some graph 𝐻, then by Observation 4.23, we have 𝛾(𝐺) = 12 𝑛. Hence, the sufficiency is immediate. To prove the necessity, assume that 𝐺 is a connected graph of order 𝑛 satisfying 𝛾(𝐺) = 12 𝑛. Let 𝑘 = 12 𝑛 and let 𝐷 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a 𝛾-set of 𝐺 satisfying the statement of Lemma 4.22; that is, epn[𝑣 𝑖 , 𝐷] ≠ ∅ for every 𝑖 ∈ [𝑘]. Let 𝐷 = 𝑉 \ 𝐷. Since |𝐷 | = 12 𝑛, we must have equality throughout Inequality (4.2), implying that |epn[𝑣 𝑖 , 𝐷] | = 1 for every 𝑖 ∈ [𝑘] and 𝐷=

𝑘 Ø

epn[𝑣 𝑖 , 𝐷].

𝑖=1

Let epn[𝑣 𝑖 , 𝐷] = {𝑢 𝑖 } for 𝑖 ∈ [𝑘]. Thus, 𝐷 = {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } and N𝐺 (𝑢 𝑖 )∩𝐷 = {𝑣 𝑖 } for 𝑖 ∈ [𝑘], implying that the set of edges between 𝐷 and 𝐷 is the set [𝐷, 𝐷] =

𝑘 Ø

{𝑢 𝑖 𝑣 𝑖 }.

𝑖=1

If 𝑛 = 2, then 𝐺 = 𝐾2 = 𝐾1 ◦ 𝐾1 . If 𝑛 = 4, then either 𝐺 = 𝐶4 or 𝐺 = 𝑃4 = 𝐾2 ◦ 𝐾1 . Hence, we may assume that 𝑛 ≥ 6, for otherwise the desired result follows. Thus, 𝑘 = 12 𝑛 ≥ 3. We show first that exactly one of 𝑢 𝑖 and 𝑣 𝑖 has degree 1 in 𝐺 for all 𝑖 ∈ [𝑘]. Renaming vertices if necessary, suppose, to the contrary, that both 𝑢 1 and 𝑣 1 have degree at least 2 in 𝐺. In this case, 𝑣 1 𝑣 𝑖 ∈ 𝐸 (𝐺) and 𝑢 1 𝑢 𝑗 ∈ 𝐸 (𝐺) for some 𝑖 ∈ [𝑘] \ {1} and some 𝑗 ∈ [𝑘] \ {1}. If 𝑖 ≠ 𝑗, then 𝑣 𝑗 𝑢 𝑗 𝑢 1 𝑣 1 𝑣 𝑖 𝑢 𝑖 is a path 𝑃6 in 𝐺. If 𝑖 = 𝑗, then 𝑢 1 𝑣 1 𝑣 𝑖 𝑢 𝑖 𝑢 1 is an (induced) 4-cycle 𝐶 in 𝐺. Since 𝐺 is a connected graph of order 𝑛 ≥ 6, there is an edge in 𝐺 joining a vertex in 𝐶 to a vertex outside 𝐶. Renaming vertices if necessary, we may assume that 𝑣 1 𝑣 𝑘 is such an edge, and so

82

Chapter 4. General Bounds

𝑢 𝑘 𝑣 𝑘 𝑣 1 𝑢 1 𝑢 𝑖 𝑣 𝑖 is a path
𝑃6 in 𝐺. In both cases, 𝐺 contains a spanning subgraph 𝐻 isomorphic to 𝑃6 ∪ 𝑛−6 2 𝐾2 . Since adding edges to a graph does not increase its domination number, 𝛾(𝐺) ≤ 𝛾(𝐻) = 𝛾(𝑃6 ) + 𝑛−6 2 𝛾(𝐾2 ) = 2 + (𝑛 − 6)/2 < 𝑛/2, a contradiction. Hence, exactly one of 𝑢 𝑖 and 𝑣 𝑖 has degree 1 in 𝐺 for all 𝑖 ∈ [𝑘]. Let 𝑋 be the set of vertices of degree at least 2 in 𝐺, and let 𝑌 = 𝑉 \ 𝑋. We note that |𝑋 | = |𝑌 | = 12 𝑛. Further, 𝑌 is an independent set and each vertex in 𝑌 has degree 1 in 𝐺. Since 𝐺 is a connected graph, 𝐺 = 𝐻 ◦ 𝐾1 , where 𝐻 = 𝐺 [𝑋] is the connected graph induced by the set 𝑋. In 1998 Randerath and Volkmann [648], and independently in 2000 Baogen et al. [57], characterized the isolate-free graphs 𝐺 of odd order 𝑛 satisfying 𝛾(𝐺) = 1 2 (𝑛 − 1). A proof of this result can be found in [417]. In Chapter 6, we present improved upper bounds on the domination number of a connected graph 𝐺 in terms of its order 𝑛 when the minimum degree is at least 2. For example, in Chapter 6 we present the 1989 result of McCuaig and Shepherd [586] that if 𝛿(𝐺) ≥ 2 and 𝑛 ≥ 8, then 𝛾(𝐺) ≤ 25 𝑛, and the 1996 result of Reed [655] that if 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 38 𝑛.

4.3.2

Total Domination Number and Order

A fundamental property of minimal TD-sets was established by Cockayne et al. [182] in 1980. Lemma 4.25 ([182]) A TD-set 𝑆 in a graph 𝐺 is a minimal TD-set if and only if every vertex 𝑣 ∈ 𝑆 has an open 𝑆-external private neighbor or an open 𝑆-internal private neighbor, that is, if and only if |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1. Proof Let 𝑆 be a minimal TD-set in 𝐺 and let 𝑣 ∈ 𝑆. If |epn(𝑣, 𝑆)| = 0 and |ipn(𝑣, 𝑆)| = 0, then every vertex in 𝐺 has a neighbor in 𝑆 \ {𝑣}, implying that 𝑆 \ {𝑣} is a TD-set of 𝐺, contradicting the minimality of 𝑆. Therefore, |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1 for every vertex 𝑣 ∈ 𝑆. Conversely, if |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1 for each 𝑣 ∈ 𝑆, then 𝑆 \ {𝑣} is not a TD-set, implying that 𝑆 is a minimal TD-set in 𝐺. The following stronger property of a minimum TD-set in a graph was established in 2000 by Henning [453]. Lemma 4.26 ([453]) If 𝐺 ≠ 𝐾𝑛 is a connected graph of order 𝑛 ≥ 3, then 𝐺 has a 𝛾t -set 𝑆 such that every vertex 𝑣 ∈ 𝑆 has an open 𝑆-external private neighbor or has an open 𝑆-internal private neighbor which in turn has an open 𝑆-external private neighbor, that is, for every vertex 𝑣 ∈ 𝑆, we have |epn(𝑣, 𝑆)| ≥ 1 or there exists a vertex 𝑣 ′ ∈ ipn(𝑣, 𝑆) with |epn(𝑣 ′ , 𝑆)| ≥ 1. Proof Let 𝐺 ≠ 𝐾𝑛 be a connected graph of order 𝑛 ≥ 3. Among all 𝛾t -sets of 𝐺, let 𝑆 be chosen so that (i) the number of edges in 𝐺 [𝑆] is a maximum and (ii) subject ′ to (i), the number of vertices 𝑣 ∈ 𝑆 having |epn(𝑣, 𝑆)|  ≥ 1 or |epn(𝑣 , 𝑆)| ≥ 1 for ′ some neighbor 𝑣 ∈ 𝑆 of 𝑣 is a maximum. Let 𝐴 = 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 , and

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let 𝐵 be a minimum set of vertices of 𝑆 \ 𝐴 such that 𝐺 [ 𝐴 ∪ 𝐵] has no isolated vertices. Necessarily, |𝐵| ≤ | 𝐴|. Further, let 𝐶 = 𝑆 \ ( 𝐴 ∪ 𝐵). We show that 𝐶 = ∅. Suppose, to the contrary, that 𝐶 ≠ ∅. Let 𝐹 = 𝐺 [𝐶]. Since epn(𝑣, 𝑆) = ∅ for every vertex 𝑣 ∈ 𝐶, by Lemma 4.25, each such vertex 𝑣 satisfies |ipn(𝑣, 𝑆)| ≥ 1. Further, since 𝐺 [ 𝐴 ∪ 𝐵] contains no isolated vertex, we deduce that 𝐹 =𝑘𝐾2 for some 𝑘 ≥ 1 and that each vertex of 𝐶 has degree 1 in 𝐹. Let 𝐸 (𝐹) = 𝑢 𝑖 𝑣 𝑖 : 𝑖 ∈ [𝑘] . Further, let 𝑁𝑖 = N𝐺 ({𝑢 𝑖 , 𝑣 𝑖 }) \ {𝑢 𝑖 , 𝑣 𝑖 } for all 𝑖 ∈ [𝑘], and let 𝑘 Ø 𝑁= 𝑁𝑖 . 𝑖=1

Since 𝐺 is connected and 𝑛 ≥ 3, we note that the set 𝑁𝑖 ≠ ∅ for 𝑖 ∈ [𝑘]. Let 𝑤 𝑖 ∈ 𝑁𝑖 for 𝑖 ∈ [𝑘]. Since each vertex of 𝐶 has degree 1 in 𝐺 [𝑆], we note that 𝑁𝑖 ⊆ 𝑉 \ 𝑆. Renaming 𝑢 𝑖 and 𝑣 𝑖 if necessary, we may assume that 𝑣
𝑖 𝑤 𝑖 ∈ 𝐸 (𝐺). If 𝑤 𝑖 is adjacent to a vertex of 𝑆 different from 𝑢 𝑖 and 𝑣 𝑖 , then 𝑆 \ {𝑢 𝑖 } ∪ {𝑤 𝑖 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, contradicting our choice of 𝑆. Hence, no vertex in 𝑁𝑖 has a neighbor in 𝑆 \ {𝑢 𝑖 , 𝑣 𝑖 }. Thus, the sets 𝑁1 , 𝑁2 , . . . , 𝑁 𝑘 are pairwise disjoint subsets. Since epn(𝑢 𝑖 , 𝑆) = epn(𝑣 𝑖 , 𝑆) = ∅, each vertex in 𝑁𝑖 is adjacent to both 𝑢 𝑖 and 𝑣 𝑖 for all 𝑖 ∈ [𝑘]. Suppose that 𝑆 = {𝑢 1 , 𝑣 1 }. In this case, since 𝐺 ≠ 𝐾𝑛 and 𝑉 = 𝑁1 ∪ {𝑢 1 , 𝑣 1 }, the set 𝑁1 contains two nonadjacent vertices. We may assume that 𝑤 1 ∈ 𝑁1 is not adjacent to some vertex of 𝑁1 . The set 𝑆 ′ = {𝑣 1 , 𝑤 1 } is a 𝛾t -set of 𝐺 such that |epn(𝑣 1 , 𝑆 ′ )| ≥ 1, contradicting our choice of the set 𝑆. Hence 𝑆 ≠ {𝑢 1 , 𝑣 1 }. Thus, since 𝐺 is connected, each set 𝑁𝑖 contains a vertex that is adjacent to a vertex of 𝑉 \ (𝑆 ∪ 𝑁𝑖 ) for all 𝑖 ∈ [𝑘]. We may assume 𝑤 𝑖 is such a vertex of 𝑁𝑖 . We show next that there is no edge joining a vertex of 𝑁𝑖 and a vertex of 𝑁 𝑗 , vertices if necessary, where 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗. If this is not the case, then renaming
we may assume 𝑤 1 𝑤 2 ∈ 𝐸 (𝐺). Thus, 𝑆 \ {𝑢 1 , 𝑢 2 } ∪ {𝑤 1 , 𝑤 2 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, there is no edge joining a vertex of 𝑁𝑖 and a vertex of 𝑁 𝑗 where 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗, implying that each vertex 𝑤 𝑖 is adjacent to a vertex, say 𝑦 𝑖 , of 𝑉 \ (𝑆 ∪ 𝑁) for all 𝑖 ∈ [𝑘]. Further, we note that such a vertex 𝑦 𝑖 has no neighbor in 𝐶 and therefore has at least one neighbor in 𝐴 ∪ 𝐵.
If 𝑤 𝑖 is adjacent to every other vertex of 𝑁𝑖 for some 𝑖 ∈ [𝑘], then 𝑆 \ {𝑢 𝑖 , 𝑣 𝑖 } ∪ {𝑤 𝑖 , 𝑦 𝑖 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, 𝑤 𝑖 is not adjacent to at least one vertex, say 𝑥𝑖 , of 𝑁𝑖 for all 𝑖 ∈ [𝑘].
We now consider the set 𝑆 ′ = 𝑆 \ {𝑢 1 } ∪ {𝑤 1 }. Necessarily, 𝑆 ′ is a 𝛾t -set of 𝐺. Since 𝑥 1 ∈ epn(𝑣 1 , 𝑆 ′ ), we note that |epn(𝑣 1 , 𝑆 ′ )| ≥ 1. If every  vertex 𝑎 ∈ 𝐴 satisfies ′ )| ≥ 1, then 𝐺 [𝑆 ′ ] has the same size as 𝐺 [𝑆] but 𝑣 ∈ 𝑆 ′ : |epn(𝑣, 𝑆 ′ )| ≥ |epn(𝑎, 𝑆  1 > 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 , contradicting our choice of the set 𝑆. Hence, there exists a vertex 𝑎 ∈ 𝐴 such that epn(𝑎, 𝑆 ′ ) = ∅, implying that vertex 𝑤 1 dominates the set epn(𝑎, 𝑆). By Lemma 4.25, |ipn(𝑎, 𝑆 ′ )| ≥ 1. Let 𝑎 ′ ∈ epn(𝑎, 𝑆) and let 𝑎★ ∈ ipn(𝑎, 𝑆 ′ ). We note that vertex 𝑎 is the only vertex in 𝑆 ′ that is adjacent to 𝑎★, and so the vertex 𝑎★ has degree 1 in 𝐺 [𝑆 ′ ]. If 𝑎★ ∈ 𝐵 or if 𝑎★ ∈ 𝐴 and

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epn(𝑎★, 𝑆 ′ ) = ∅, then 𝑆 ′ \ {𝑎 ∗ } ∪ {𝑎 ′ } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, 𝑎★ ∈ 𝐴. Thus, vertex 𝑎 is adjacent to a vertex 𝑎★ ∈ 𝑆 ′ such that |epn(𝑎★, 𝑆 ′ )| ≥ 1. This implies that 𝐺 [𝑆 ′ ] has the same size as 𝐺 [𝑆], but there are more vertices 𝑣 ∈ 𝑆 ′ such that |epn(𝑣, 𝑆 ′ )| ≥ 1 or |epn(𝑣 ′ , 𝑆 ′ )| ≥ 1 for some neighbor 𝑣 ′ ∈ 𝑆 ′ of 𝑣 than there are vertices in 𝑆, once again contradicting our choice of the set 𝑆. Therefore, 𝐶 = ∅. By our choice of the set 𝐵, every vertex in 𝐵 has a neighbor in 𝐴. Let 𝑤 ∈ 𝐵, and so epn(𝑤, 𝑆) = ∅. By Lemma 4.25, we note that |ipn(𝑤, 𝑆)| ≥ 1. Let 𝑣 ∈ ipn(𝑤, 𝑆). Since every vertex in 𝐵 has a neighbor in 𝐴, we note that 𝑣 ∉ 𝐵, and so 𝑣 ∈ 𝐴, implying that the vertex 𝑣 has degree 1 in 𝐺 [𝑆]. Since 𝑤 is an arbitrary vertex in 𝐵, this shows that every vertex in 𝐵 has a neighbor 𝑣 ∈ 𝑆 of degree 1 in 𝐺 [𝑆] such that |epn(𝑣, 𝑆)| ≥ 1. Cockayne et al. [182] proved that the total domination of a connected graph of order at least 3 is at most two-thirds its order. The proof follows from a simple application of Lemma 4.26. Theorem 4.27 ([182]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛. Proof Let 𝐺 be a connected graph of order 𝑛 ≥ 3. If 𝐺 = 𝐾𝑛 , then 𝛾t (𝐺) = 2 ≤ 23 𝑛. Hence, we may assume that 𝐺 ≠ 𝐾𝑛 . Let 𝑆 be a 𝛾t -set of 𝐺 satisfying the statement of Lemma 4.26. Let 𝐴 = 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 and let 𝐵 = 𝑆 \ 𝐴. Thus,  𝐵 = 𝑣 ∈ 𝑆 : epn(𝑣, 𝑆) = ∅ , and so, by Lemma 4.26, each vertex 𝑢 ∈ 𝐵 has a neighbor 𝑣 ∈ 𝐴 of degree 1 in 𝐺 [𝑆], and so |epn(𝑣, 𝑆)| ≥ 1 and the vertex 𝑢 is the unique neighbor of 𝑣 in 𝐺 [𝑆]. Thus, |𝐵| ≤ | 𝐴|, and so |𝑆| = | 𝐴| + |𝐵| ≤ 2| 𝐴|. Let 𝐶 be the set of all external 𝑆-private neighbors, and so 𝐶 ⊆ 𝑉 \ 𝑆. Thus, Ø ∑︁ 𝐶= epn(𝑣, 𝑆) and |𝐶 | = |epn(𝑣, 𝑆)| ≥ | 𝐴|. 𝑣∈ 𝐴

𝑣∈ 𝐴

Hence, 𝑛 − |𝑆| = |𝑉 \ 𝑆| ≥ |𝐶 | ≥ | 𝐴| ≥ 12 |𝑆|,

(4.3)

and so 𝛾t (𝐺) = |𝑆| ≤ 23 𝑛. The 2-corona 𝐹 ◦ 𝑃2 of a connected graph 𝐹 is the graph of order 3|𝑉 (𝐹)| obtained from 𝐹 by attaching a path of length 2 to each vertex of 𝐹 so that the resulting paths are vertex-disjoint. For example, the 2-corona 𝐶4 ◦ 𝑃2 of a 4-cycle is illustrated in Figure 4.2. We note that every TD-set of the 2-corona 𝐹 ◦ 𝑃2 of a graph 𝐹 contains all support vertices of 𝐹 ◦ 𝑃2 . Moreover in order to totally dominate the support vertices, every TD-set also contains a neighbor of each support vertex. Thus, if 𝐹 has order 𝑘, then 𝛾t (𝐹 ◦ 𝑃2 ) ≥ 2𝑘, noting that 𝐹 has 𝑘 support vertices. The set 𝑉 (𝐹), together with the set of all support vertices of 𝐹 ◦ 𝑃2 , is a TD-set of 𝐹 ◦ 𝑃2 , and so 𝛾t (𝐹 ◦ 𝑃2 ) ≤ 2𝑘. Consequently, 𝛾t (𝐹 ◦ 𝑃2 ) = 2𝑘. In Figure 4.2, 𝛾t (𝐶4 ◦ 𝑃2 ) = 8 and the eight highlighted vertices, for example, form a 𝛾t -set of 𝐶4 ◦ 𝑃2 . This yields the following observation. Observation 4.28 If 𝐺 is a 2-corona 𝐹 ◦ 𝑃2 of order 𝑛, then 𝛾t (𝐺) = 23 𝑛.

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85

Figure 4.2 The 2-corona 𝐶4 ◦ 𝑃2

In 2000 Brigham et al. [117] characterized the connected graphs that achieve equality in the upper bound of Theorem 4.27, and showed that, with the exception of two small cycles, the extremal graphs are precisely the family of 2-coronas of connected graphs. The proof we present of their characterization follows readily from Lemma 4.26. Theorem 4.29 ([117]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) = 23 𝑛 if and only if 𝐺 is 𝐶3 , 𝐶6 , or 𝐹 ◦ 𝑃2 for some connected graph 𝐹. Proof Let 𝐺 be a connected graph of order 𝑛 ≥ 3. If 𝐺 = 𝐾𝑛 and 𝛾t (𝐺) = 23 𝑛, then 𝛾t (𝐺) = 2 = 23 𝑛, implying that 𝑛 = 3 and 𝐺 is the cycle 𝐶3 . Hence, we may assume that 𝐺 ≠ 𝐾𝑛 . Adopting the notation in the proof of Theorem 4.27, if 𝛾t (𝐺) = 23 𝑛, then there must be equality throughout Inequality (4.3). In particular, | 𝐴| = |𝐵| = |𝐶 | = 13 𝑛 and 𝑉 \ 𝑆 = 𝐶, implying that each vertex of 𝐴 has degree 2 in 𝐺 and has one neighbor in 𝐵 and its other neighbor in 𝐶. Thus, the graph 𝐺 contains a spanning subgraph 𝐻 = 𝑘 𝑃3 , where 𝑘 = 13 𝑛. Suppose that both sets 𝐵 and 𝐶 contain a vertex of degree 2 or more in 𝐺. In this case, by the connectivity of 𝐺, the graph 𝐺 contains a spanning subgraph 𝐻, where 𝐻 = 𝐶6 or 𝐻 = 𝑃9 ∪ (𝑘 − 3)𝑃3 . If 𝐻 = 𝑃9 ∪ (𝑘 − 3)𝑃3 , then 𝛾t (𝐻) = 5 + 2(𝑘 − 3) = 2𝑘 − 1 < 23 𝑛. Since adding edges to a graph does not increase the total domination number, this would imply that 𝛾t (𝐺) ≤ 𝛾t (𝐻) < 23 𝑛, a contradiction. Hence, 𝐻 = 𝐶6 and 𝐺 contains no spanning subgraph isomorphic to 𝑃9 ∪ (𝑘 − 3)𝑃3 . This implies that 𝐺 = 𝐻 = 𝐶6 . Hence, we may assume that one of the sets 𝐵 or 𝐶 contains only vertices of degree 1 in 𝐺. If every vertex of 𝐵 has degree 1 in 𝐺, then by the connectivity of 𝐺, the subgraph 𝐺 [𝐶] is connected and 𝐺 = 𝐹 ◦ 𝑃2 where 𝐹 = 𝐺 [𝐶], while if every vertex of 𝐶 has degree 1 in 𝐺, then the subgraph 𝐺 [𝐵] is connected and 𝐺 = 𝐹 ◦ 𝑃2 where 𝐹 = 𝐺 [𝐵]. In both cases, 𝐺 = 𝐹 ◦ 𝑃2 for some connected graph 𝐹. The total domination number of a graph is equal to the sum of the total domination numbers of its components. Hence, as a consequence of Theorem 4.27, we have the following trivial characterization of graphs having total domination number equal to their order. Corollary 4.30 If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾t (𝐺) = 𝑛 if and only if 𝐺 = 𝑘𝐾2 for some 𝑘 ≥ 1.

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In Chapter 6, we present improved upper bounds on the total domination number of a connected graph 𝐺 in terms of its order 𝑛, when the minimum degree is at least 2. For example, we show in this chapter that if 𝛿(𝐺) ≥ 2 and 𝑛 ≥ 11, then 𝛾t (𝐺) ≤ 74 𝑛. We will also prove among other bounds the result that if 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛, and if 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛.

4.3.3

Independent Domination Number and Order

In this section, we present two upper bounds on the independent domination number of a graph in terms of its order. Using the property of a minimum dominating set of a graph established in Lemma 4.22, Bollobás and Cockayne [84] in 1979 proved the following upper bound on the independent domination number. Theorem 4.31 ([84]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 𝛾(𝐺) −

𝑛 . 𝛾(𝐺)

Proof By Lemma 4.22, there exists a 𝛾-set 𝐷 of any isolate-free graph 𝐺 such that epn(𝑣, 𝐷) ≠ ∅ for every vertex 𝑣 ∈ 𝐷. For each vertex 𝑣 ∈ 𝐷, choose an arbitrary vertex 𝑣 ′ ∈ epn(𝑣, 𝐷). Let 𝛾 = 𝛾(𝐺). By
the Pigeonhole Principle, there is a vertex 𝑣 ∈ 𝐷 that is adjacent to at least 𝑛 − |𝐷| / |𝐷| = (𝑛 − 𝛾)/𝛾 vertices  of 𝑉 \ 𝐷. Let 𝑁 𝑣 be the set of neighbors of 𝑣 that belong to 𝑉 \ 𝐷, and so |𝑁 𝑣 | ≥ (𝑛 − 𝛾)/𝛾 . Let 𝐼 be a maximal independent set in 𝐺 that contains the vertex 𝑣. We note that 𝐼 ∩ 𝑁 𝑣 = ∅. Further, the set 𝐼 contains at most one of 𝑢 and 𝑢 ′ for every vertex 𝑢 ∈ 𝐷 \ {𝑣}, and so there are at least |𝐷| − 1 vertices not in 𝑁 𝑣 that do not belong to the set 𝐼. Therefore,
|𝐼 | ≤ 𝑛 − |𝐷| − 1 − |𝑁 𝑣 |   𝑛−𝛾 ≤ 𝑛 − (𝛾 − 1) − 𝛾   𝑛 =𝑛+2−𝛾− 𝛾 𝑛 ≤ 𝑛+2−𝛾− . 𝛾 Since 𝐼 is a maximal ID-set of 𝐺, it follows that 𝑖(𝐺) ≤ |𝐼 |. √ Treating 𝑛 as fixed, the function 𝑓 (𝛾) = 𝑛 + 2 − 𝛾 − 𝛾𝑛 is maximized at 𝛾 = 𝑛. √
√ Thus since 𝑓 𝑛 = 𝑛 + 2 − 2 𝑛, as an immediate consequence of Theorem 4.31, we obtain the following bound, first observed in 1988 by Favaron [274] (and also proved in 1995 by Gimbel and Vestergaard [336]). Theorem 4.32 ([274]) If 𝐺 is an isolate-free graph of order 𝑛, then √ 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛.

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That the bound of Theorem 4.32 is tight may be seen as follows. For any integer 𝑟 ≥ 1, the generalized corona cor(𝐺, 𝑟) of a graph 𝐺 is the graph obtained from 𝐺 by adding 𝑟 pendant edges to each vertex of 𝐺, that is, for each vertex 𝑣 of 𝐺, we add 𝑟 new vertices and add an edge from each new vertex to the vertex 𝑣. In particular, if 𝑟 = 1, then cor(𝐺, 𝑟) is the corona 𝐺 ◦ 𝐾1 (also denoted cor(𝐺)). For example, if 𝐺 is the cycle 𝐶5 and 𝑟 = 3, then the generalized corona cor(𝐺, 𝑟) of 𝐺 is illustrated in Figure 4.3.

Figure 4.3 The generalized corona cor(𝐶5 , 3) For 𝑟 ≥ 2, if we take 𝐺 √= cor(𝐾𝑟 , 𝑟 − 1), then 𝐺 has order 𝑛 = 𝑟 2 and 𝑖(𝐺) = (𝑟 − 1) 2 + 1 = 𝑛 + 2 − 2 𝑛, showing that the upper bound of Theorem 4.32 is tight. This implies that, unlike the domination and total domination numbers, there is no constant 𝑘 < 1 such that for every isolate-free graph 𝐺 of order 𝑛, 𝑖(𝐺) ≤ 𝑘𝑛. Brigham et al. [117] investigated the graphs that attain (the floor of) the bound in Theorem 4.32. In particular, they showed that if 𝑛 is a square, then the generalized coronas cor(𝐾𝑟 , 𝑟 − 1) given above are the only extremal graphs. Thus, in this case all extremal graphs achieving the bound of Theorem 4.32 have minimum degree 1. In Chapter 6, we present improved upper bounds on the independent domination number of a connected graph 𝐺 in terms of its order 𝑛 when the minimum degree is at least 2. In particular, we will prove a much stronger result than Theorem 4.32, namely √ that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿, then 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛.

4.4

Basic Relationships Among Core Parameters

As mentioned in Chapter 2, Cockayne et al. [196] were the first to observe the important Domination Chain relating the core parameters 𝛾, 𝑖, 𝛼, and Γ. Theorem 4.33 ([196]) For every graph 𝐺, 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺). In 1979 Bollobás and Cockayne [84] were the first to observe the following relationship between the domination and total domination numbers of an isolate-free graph. Theorem 4.34 ([84]) If 𝐺 is an isolate-free graph, then 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ 2𝛾(𝐺). Theorems 4.33 and 4.34 establish the fundamental relationships among the core domination parameters. In Chapter 15, we revisit these results, present a proof of Theorem 4.34, and explore other relationships among these parameters.

88

Chapter 4. General Bounds

4.5 Domination and Distance Our next result gives a lower bound on the domination number in terms of diameter. Theorem 4.35 If 𝐺 is a connected graph, then 1
 3 diam(𝐺) + 1 ≤ 𝛾(𝐺). Proof Let 𝐺 be a connected graph with diam(𝐺) = 𝑑, and let 𝑆 be a 𝛾-set of 𝐺. Let 𝑃 : 𝑣 1 𝑣 2 . . . 𝑣 𝑑+1 be a path such that 𝑑 (𝑣 1 , 𝑣 𝑑+1 ) = diam(𝐺) = 𝑑. We note that any vertex on 𝑃 dominates at most three vertices on 𝑃. Suppose that there is a vertex 𝑢 ∈ 𝑆, such that 𝑢 is not on 𝑃 and 𝑢 dominates at least four vertices on 𝑃. Thus, N(𝑢) contains two vertices 𝑣 𝑖 and 𝑣 𝑗 on 𝑃 such that 𝑑 (𝑣 𝑖 , 𝑣 𝑗 ) ≥ 3 and 𝑖 < 𝑗. However, replacing the subpath on 𝑃 from 𝑣 𝑖 to 𝑣 𝑗 with the path 𝑣 𝑖 𝑢 𝑣 𝑗 creates a path 𝑃′ : 𝑣 1 𝑣 2 . . . 𝑣 𝑖 𝑢 𝑣 𝑗 𝑣 𝑗+1 . . . 𝑣 𝑑+1 from 𝑣 1 to 𝑣 𝑑+1 that is shorter than 𝑃, contradicting the fact that 𝑃 is a shortest path from 𝑣 1 to 𝑣 𝑑+1 . Hence, every  vertex  in 𝑆 dominates
 at most three vertices of 𝑃, implying that 𝛾(𝐺) = |𝑆| ≥ 13 |𝑃| = 13 diam(𝐺) + 1 . Theorem 4.35 is tight as can be seen by the following result.   
 Proposition 4.36 For 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝑛3 = 13 diam(𝑃𝑛 ) + 1 . A graph 𝐺 for which diam(𝐺) = 2 is called a diameter-2 graph. For a diameter-2 graph 𝐺 and any vertex 𝑣 of 𝐺, the open neighborhood N(𝑣) of 𝑣 dominates 𝐺, while the closed neighborhood N[𝑣] of 𝑣 totally dominates 𝐺. Choosing such a vertex 𝑣 of minimum degree immediately yields the following observation. Observation 4.37 If 𝐺 is a diameter-2 graph, then 𝛾(𝐺) ≤ 𝛿(𝐺) and 𝛾t (𝐺) ≤ 𝛿(𝐺) + 1. We remark that if 𝐺 is a star 𝐾1,𝑘 where 𝑘 ≥ 2, then 𝐺 is a diameter-2 graph satisfying 𝛾(𝐺) = 𝛿(𝐺) = 1 and 𝛾t (𝐺) = 𝛿(𝐺) + 1 = 2. If 𝐺 is a diameter-2 graph with 𝛿(𝐺) = 2 that does not contain a dominating vertex, then 𝛾(𝐺) = 𝛿(𝐺) = 2. In Chapter 7, we study the (total) domination number of a diameter-2 graph in more depth. For example, we will show that given any 𝜀 > 0, if 𝐺 is
√︁a diameter-2 graph of sufficiently large order 𝑛, then 𝛾(𝐺) ≤ 𝛾t (𝐺) < √1 + 𝜀 𝑛 ln(𝑛). We 2 will also show that if we choose the probability 𝑝 carefully, then any random graph in G(𝑛, 𝑝) (which we will define formally in Section 7.3) is a diameter-2 graph with√︁its domination √︁and total domination numbers concentrated between roughly 1 √ 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). 2 2 2 Let diam(𝐺) = ∞ for a disconnected graph. Since any pair of vertices at distance 3 or more apart in 𝐺 form a dominating set of its complement 𝐺, we have the following observation made by Brigham et al. [118] in 1988. Observation 4.38 ([118]) If 𝐺 is a graph with diam(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 2. In 2010 DeLaViña et al. [222] proved the following bound on the domination number in terms of the radius.

Section 4.5. Domination and Distance Theorem 4.39 ([222]) If 𝐺 is a connected graph, then 𝛾(𝐺) ≥

89 2 3

rad(𝐺).

The bound of Theorem 4.39 is tight. For example, if 𝐺 = 𝐶6𝑘 for 𝑘 ≥ 1, then rad(𝐺) = 3𝑘 and 𝛾(𝐺) = 2𝑘, that is, 𝛾(𝐺) = 23 rad(𝐺). On the other hand, for 𝑘 ≥ 1 if 𝐺 = 𝑃2𝑘+1 ◦ 𝐾1 , then rad(𝐺) = 𝑘 + 1 and 𝛾(𝐺) = 2𝑘 + 1. In this case, 𝛾(𝐺) = 2 rad(𝐺) − 1, showing that the difference between these two values can be arbitrarily large. In 2007 DeLaViña et al. [221] proved that the radius of a connected graph is a lower bound on its total domination number. In order to prove this result, they first proved the following three preliminary lemmas. Lemma 4.40 ([221]) If 𝐷 is a dominating set in a tree 𝑇, then the subgraph 𝑇 − 𝐷 has at most 𝑘 − 1 edges, where 𝑘 is the number of components of the subgraph 𝑇 [𝐷] induced by 𝐷. Proof Let 𝑇 = (𝑉, 𝐸) be a tree of order 𝑛 and size 𝑚. Let 𝐷 be a dominating set in 𝑇 and let 𝑘 be the number of components of the subgraph 𝑇 [𝐷]. We note that the subgraph 𝑇 − 𝐷 is the forest 𝑇 [𝑉 \ 𝐷] induced by 𝑉 \ 𝐷. Let 𝑚 1 denote the number of edges in 𝑇 [𝐷], let 𝑚 2 be the number of edges between 𝐷 and 𝑉 \ 𝐷, and let 𝑚 3 denote the number of edges in 𝑇 [𝑉 \ 𝐷]. Since 𝑇 is a tree, 𝑛 − 1 = 𝑚 = 𝑚 1 + 𝑚 2 + 𝑚 3 . We wish to show that 𝑚 3 ≤ 𝑘 − 1. Suppose, to the contrary, that 𝑚 3 ≥ 𝑘. Since 𝑇 [𝐷] is a forest with 𝑘 (tree) components, 𝑚 1 = |𝐷| − 𝑘. Since 𝐷 is a dominating set, every vertex in 𝑉 \ 𝐷 is adjacent to at least one vertex
in 𝐷, and so
𝑚 2 ≥ |𝑉 \ 𝐷| = 𝑛 − |𝐷|. Therefore, 𝑛 − 1 = 𝑚 1 + 𝑚 2 + 𝑚 3 ≥ |𝐷 | − 𝑘 + 𝑛 − |𝐷 | + 𝑘 = 𝑛, a contradiction. Consequently, 𝑚 3 ≤ 𝑘 − 1. Lemma 4.41 ([221]) If 𝐷 is a 𝛾t -set in a nontrivial connected graph 𝐺, then there exists a spanning tree 𝑇 of 𝐺 such that the following properties hold: (a) 𝐷 is a 𝛾t -set in 𝑇, and (b) the number of components in 𝑇 [𝐷] equals the number of components in 𝐺 [𝐷]. Proof Let 𝐺 be a nontrivial connected graph. Let 𝐷 be a 𝛾t -set of 𝐺, and let 𝐷 = 𝑉 \ 𝐷. If 𝐺 is a tree, then we take 𝑇 = 𝐺, and the desired result is immediate. Otherwise, 𝐺 contains at least one cycle 𝐶. We delete an edge 𝑒 from 𝐶 forming 𝐺 𝑒 = 𝐺 − 𝑒 as follows. If 𝐶 has two consecutive vertices 𝑢 and 𝑣 in 𝐷, then let 𝑒 = 𝑢𝑣 and delete the edge 𝑒. We note that in this case, the set 𝐷 remains a TD-set of the (connected) graph 𝐺 𝑒 . Further, the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷]. If 𝐶 has no two consecutive vertices in 𝐷, but 𝐶 has two consecutive vertices 𝑢 and 𝑣 such that 𝑢 ∈ 𝐷 and 𝑣 ∈ 𝐷, then let 𝑒 = 𝑢𝑣 and delete the edge 𝑒. We note that in this case, the second neighbor of 𝑣 on 𝐶 belongs to 𝐷, and therefore the set 𝐷 is a TD-set of the (connected) graph 𝐺 𝑒 . Further, the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷]. If neither of the above two cases applies, then the cycle 𝐶 is entirely contained in 𝐺 [𝐷]. In this case, we choose 𝑒 as an arbitrary edge of 𝐶 and delete the edge 𝑒. Once again, the set 𝐷 is still a TD-set of the (connected) graph 𝐺 𝑒 , and the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷].

Chapter 4. General Bounds

90

We now repeat the above process in the connected graph 𝐺 𝑒 until all cycles are removed. Let 𝑇 denote the resulting subgraph. Since we are careful always to preserve connectivity, 𝑇 is a spanning tree of 𝐺. By construction, 𝐷 is a TD-set of 𝑇, and so 𝛾t (𝑇) ≤ |𝐷| = 𝛾t (𝐺). By Observation 4.2, 𝛾t (𝐺) ≤ 𝛾t (𝑇). Consequently, 𝛾t (𝑇) = 𝛾t (𝐺) and 𝐷 is a 𝛾t -set in 𝑇. Lemma 4.42 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≥

1 2


diam(𝐺) + 1 .

Proof Let 𝐷 be a 𝛾t -set in 𝐺, and let 𝑘 be the number of components in 𝐺 [𝐷]. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇 and the number of components in 𝑇 [𝐷] equals 𝑘. Since 𝐷 is a TD-set of 𝑇, every component in 𝑇 [𝐷] contains at least two vertices. Thus, since 𝑇 [𝐷] is a forest with 𝑘 (tree) components, 𝛾t (𝑇) = |𝐷 | ≥ 2𝑘. Let 𝑃 be a longest path in 𝑇, and so |𝐸 (𝑃)| = diam(𝑇). Let 𝑝 1 denote the number of edges of 𝑃 in 𝑇 [𝐷], let 𝑝 2 be the number of edges of 𝑃 between 𝐷 and 𝑉 \ 𝐷, and let 𝑝 3 denote the number of edges of 𝑃 in 𝑇 [𝑉 \ 𝐷]. Since 𝑇 [𝐷] is a forest with 𝑘 components, there are |𝐷| − 𝑘 = 𝛾t (𝑇) − 𝑘 edges in 𝑇 [𝐷], and so 𝑝 1 ≤ 𝛾t (𝑇) − 𝑘. In traversing a longest path in 𝑇, we can enter and leave each component of 𝑇 [𝐷] at most once, implying that 𝑝 2 ≤ 2𝑘. By Lemma 4.40, the subgraph 𝑇 − 𝐷 has at most 𝑘
− 1 edges, and so 𝑝 3 ≤ 𝑘 − 1. Thus, diam(𝑇) = |𝐸 (𝑃)| = 𝑝 1 + 𝑝 2 + 𝑝 3 ≤ 𝛾t (𝑇) − 𝑘 + 2𝑘 + (𝑘 − 1) = 𝛾t (𝑇) + 2𝑘 − 1 ≤ 2𝛾t (𝑇) − 1. However, since the diameter of a graph is at most the diameter of any of its spanning trees, diam(𝐺) ≤ diam(𝑇) ≤ 2𝛾t (𝑇) − 1, or
equivalently, 𝛾t (𝐺) ≥ 12 diam(𝐺) + 1 . With the above three lemmas, DeLaViña et al. [221] proved that the radius of a connected graph is a lower bound on its total domination number. Theorem 4.43 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≥ rad(𝐺). Proof Let 𝐷 be a 𝛾t -set in 𝐺. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇. Since 𝑇 is a tree, 2 rad(𝑇) − 1 ≤ diam(𝑇). By Lemma 4.42, diam(𝑇) ≤ 2𝛾t (𝑇) − 1 = 2|𝐷 | − 1 = 2𝛾t (𝐺) − 1. Consequently, 2 rad(𝑇) − 1 ≤ 2𝛾t (𝐺) − 1, or equivalently, rad(𝑇) ≤ 𝛾t (𝐺). Since the radius of a graph is at most the radius of any of its spanning trees, rad(𝐺) ≤ rad(𝑇). Therefore, rad(𝐺) ≤ 𝛾t (𝐺). We note that if 𝐺 = 𝑃4𝑘 where 𝑘 ≥ 1, then 𝛾t (𝐺) = 2𝑘 = rad(𝐺). Furthermore, in this case, the graph 𝐺 has a unique 𝛾t -set 𝐷 and the subgraph 𝐺 [𝐷] = 𝑘𝐾2 , where 𝑘 = 12 rad(𝐺). DeLaViña et al. [221] characterized the graphs attaining the bound of Theorem 4.43. Theorem 4.44 ([221]) If 𝐷 is a 𝛾t -set of a nontrivial
connected graph 𝐺, then 𝛾t (𝐺) = rad(𝐺) if and only if 𝐺 [𝐷] has size 𝑚 𝐺 [𝐷] = 12 rad(𝐺).

Section 4.6. Domination and Packing

91

Proof Suppose that 𝛾t (𝐺) = rad(𝐺). Let 𝐷 be a 𝛾t -set in 𝐺, and let 𝑘 be the number of components in 𝐺 [𝐷]. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇 and the number of components in 𝑇 [𝐷] equals 𝑘. As shown in the proofs of Lemma 4.42 and Theorem 4.43, and by our supposition that 𝛾t (𝐺) = rad(𝐺), the following inequality chain holds: =rad(𝐺) z}|{ (4.4) 2 rad(𝐺) − 1 ≤ diam(𝑇) ≤ 𝛾t (𝑇) + 2𝑘 − 1 ≤ 2𝛾t (𝑇) − 1 = 2𝛾t (𝐺) − 1. We must have equality throughout Inequality Chain (4.4), implying that rad(𝐺) = 2𝑘 = 𝛾(𝐺). Since 𝐷 is a TD-set of 𝐺, every component in 𝐺 [𝐷] contains at least two vertices, and so 2𝑘 = 𝛾t (𝐺) = |𝐷 | ≥ 2𝑘. Consequently, |𝐷| = 2𝑘, implying that each of the 𝑘 components of 𝐺 [𝐷] is a
𝐾2 -component. Therefore, 𝐺 [𝐷] has size equal to 𝑘 = 21 rad(𝐺), that is, 𝑚 𝐺 [𝐷] = 12 rad(𝐺).
Conversely, suppose that 𝑚 𝐺 [𝐷] = 12 rad(𝐺). Since 𝐷 is a TD-set of 𝐺, every vertex in 𝐷 has degree at least 1 in 𝐺 [𝐷], and so, by Theorem 4.43, the following inequality chain holds:
∑︁ rad(𝐺) = 2𝑚 𝐺 [𝐷] = deg𝐺 [𝐷 ] (𝑣) ≥ |𝐷| = 𝛾t (𝐺) ≥ rad(𝐺) (4.5) 𝑣 ∈𝐷

We must have equality throughout Inequality (4.5), implying that 𝛾t (𝐺) = rad(𝐺).

4.6

Domination and Packing

Recall that 𝜌(𝐺) and 𝜌 o (𝐺) denote the packing number and open packing number, respectively, of 𝐺. The following result shows that the domination number of a graph is at least its packing number. Theorem 4.45 If 𝐺 is a graph, then 𝛾(𝐺) ≥ 𝜌(𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a maximum packing in 𝐺, and so 𝑘 = 𝜌(𝐺). Let 𝐷 be a 𝛾-set in 𝐺. In order to dominate the vertex 𝑣 𝑖 , we have |𝐷 ∩ N[𝑣 𝑖 ] | ≥ 1 for all 𝑖 ∈ [𝑘]. Since N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] are vertex-disjoint sets, 𝛾(𝐺) = |𝐷| ≥

𝑘 ∑︁

|𝐷 ∩ N[𝑣 𝑖 ] | ≥ 𝑘 = 𝜌(𝐺).

𝑖=1

A result similar to that of Theorem 4.45 holds for total domination and open packing numbers. Theorem 4.46 If 𝐺 is an isolate-free graph, then 𝛾t (𝐺) ≥ 𝜌 o (𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a maximum open packing in 𝐺, and so 𝑘 = 𝜌 o (𝐺). Let 𝐷 be a 𝛾t -set in 𝐺. In order to totally dominate the vertex 𝑣 𝑖 , we have

Chapter 4. General Bounds

92

|𝐷∩N(𝑣 𝑖 )| ≥ 1 for all 𝑖 ∈ [𝑘]. Since N(𝑣 1 ), N(𝑣 2 ), . . . , N(𝑣 𝑘 ) are vertex-disjoint sets, 𝛾t (𝐺) = |𝐷 | ≥

𝑘 ∑︁

|𝐷 ∩ N(𝑣 𝑖 )| ≥ 𝑘 = 𝜌 o (𝐺).

𝑖=1

In 1975 Meir and Moon [589] proved that the domination number of a tree equals its packing number. Similarly, in 2005 Rall [644] proved that the total domination number of a tree equals its open packing number. Proofs of these results will be given in Chapter 5.

4.7

Gallai Type Theorems

As we mentioned in Chapter 1, after Gallai [324] published his well-known theorem, that for any connected graph 𝐺 of order 𝑛, the sum of the independence number of 𝐺 plus the vertex covering number of 𝐺 always equals the order 𝑛 of 𝐺, a number of other theorems of this form, A (𝐺) + B (𝐺) = 𝑛, began to be published. Results like this are often called Gallai theorems. For all six of the core domination parameters in this book, there are corresponding Gallai theorems. Theorem 4.47 For any connected graph 𝐺 of order 𝑛 ≥ 2, the following hold: (a) 𝛼(𝐺) + 𝛽(𝐺) = 𝑛 (1959 Gallai [324]). (b) 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛 (1974 Nieminen [613]). (c) 𝛾(𝐺) + Ψ(𝐺) = 𝑛 (1977 Slater [679]). (d) Γ(𝐺) + 𝜓(𝐺) = 𝑛 (1977 Slater [679]). (e) 𝑖(𝐺) + 𝛽+ (𝐺) = 𝑛 (1980 McFall and Nowakowski [587]). (f) 𝛾t (𝐺) + Ψo (𝐺) = 𝑛 (1988 Cockayne et al. [195]). (g) Γt (𝐺) + 𝜓 o (𝐺) = 𝑛 (1988 Cockayne et al. [195]). As a reminder, other than the six core parameters which appear on the left in these equalities, the corresponding parameters in these sums are the following: (a) 𝛽(𝐺), the minimum cardinality of a vertex cover of 𝐺. (b) 𝜀 𝑓 (𝐺), the maximum number of pendant edges in a spanning forest of 𝐺. (c) Ψ(𝐺), the maximum cardinality of an enclaveless set in 𝐺. (d) 𝜓(𝐺), the minimum cardinality of a maximal enclaveless set in 𝐺. (e) 𝛽+ (𝐺), the maximum cardinality of a minimal vertex cover of 𝐺. (f) Ψo (𝐺), the maximum cardinality of an open enclaveless set in 𝐺. (g) 𝜓 o (𝐺), the minimum cardinality of an open enclaveless set in 𝐺. From these equalities above and other established inequalities, a number of inequalities can be established where the sum of two parameters is bounded above by 𝑛. For example, since 𝑖(𝐺) ≤ 𝛼(𝐺) and 𝛾(𝐺) ≤ 𝛽(𝐺) if 𝐺 is isolate-free, we can conclude from Gallai’s Theorem that 𝑖(𝐺) + 𝛾(𝐺) ≤ 𝑛. But this inequality can be improved, as was shown by Allan et al. [16] in 1984, as follows. Theorem 4.48 ([16]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝑖(𝐺) + 𝛾t (𝐺) ≤ 𝑛. Proof Among all 𝛾-sets of 𝐺, let 𝑆 be chosen so that

Section 4.7. Gallai Type Theorems

93

(a) the number of isolated vertices in 𝐺 [𝑆] is a minimum, and (b) subject to (a), the sum of degrees of vertices that belong to 𝑆 is a maximum. Let 𝑆 ′ ⊆ 𝑆 be the set of isolated vertices in 𝐺 [𝑆] and assume that |𝑆 ′ | = 𝑘. If 𝑘 = 0, then 𝑆 is a TD-set and therefore 𝛾(𝐺) = 𝛾t (𝐺), and since 𝑖(𝐺) + 𝛾(𝐺) ≤ 𝑛 it follows that 𝑖(𝐺) + 𝛾t (𝐺) ≤ 𝑛, as required. Assume therefore that 𝑘 > 0. Suppose that deg(𝑣) = 1 for some 𝑣 ∈ 𝑆 ′ . Let 𝑢 be the only neighbor of 𝑣. By supposition, we
have 𝑛 ≥ 3, implying that deg(𝑢) ≥ 2. We now consider the 𝛾-set 𝑆 ′′ = 𝑆 \ {𝑣} ∪ {𝑢}. If 𝑢 is not isolated in 𝐺 [𝑆 ′′ ], then 𝑆 ′′ is a 𝛾-set having fewer isolated vertices in 𝐺 [𝑆 ′′ ] than in 𝐺 [𝑆], contradicting our choice of 𝑆. Hence, 𝑢 is isolated in 𝐺 [𝑆 ′′ ]. However, the sum of degrees of vertices that belong to 𝑆 ′′ exceeds that in 𝑆, once again contradicting our choice of 𝑆. Hence, deg(𝑣) ≥ 2 for all 𝑣 ∈ 𝑆 ′ . We claim that each 𝑣 ∈ 𝑆 has an 𝑆-external private neighbor, that is, |epn[𝑣, 𝑆] | ≥ 1 for all 𝑣 ∈ 𝑆. By Ore’s Lemma 2.72, we have ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. If 𝑣 ∈ 𝑆 \ 𝑆 ′ , then ipn[𝑣, 𝑆] = ∅, and therefore epn[𝑣, 𝑆] ≠ ∅, as desired. Let 𝑣 ∈ 𝑆 ′ and suppose, to the contrary, that epn[𝑣, 𝑆] = ∅. Thus, every neighbor of 𝑣 (which necessarily belongs to the set 𝑉 \ 𝑆) has at least two neighbors in 𝑆. In this case, replacing 𝑣 in the set 𝑆 with an arbitrary neighbor of 𝑣 produces a new 𝛾-set of 𝐺 having fewer isolated vertices in its induced subgraph than does 𝑆, contradicting our choice of 𝑆. Hence, epn[𝑣, 𝑆] ≠ ∅. Thus, |epn[𝑣, 𝑆] | ≥ 1 for all 𝑣 ∈ 𝑆. For each vertex 𝑣 ∈ 𝑆, let 𝑤 𝑣 be an arbitrary vertex in epn[𝑣, 𝑆]. By our earlier observations, deg(𝑣) ≥ 2 for all 𝑣 ∈ 𝑆 ′ . For each vertex 𝑣 ∈ 𝑆 ′ , let 𝑢 𝑣 be a neighbor of 𝑣 different from 𝑤 𝑣 . Let Ø Ø 𝑈= {𝑤 𝑣 } and 𝑊= {𝑢 𝑣 }. 𝑣 ∈𝑆 ′

𝑣 ∈𝑆

We note that |𝑊 | = |𝑆| = 𝛾(𝐺). We also note that |𝑈| ≤ |𝑆 ′ | and 𝑈 ⊆ 𝑉 \ (𝑆 ∪𝑊). Moreover, the set 𝑆 ∪ 𝑈 is a TD-set of 𝐺. Therefore, 𝛾t (𝐺) ≤ |𝑆| + |𝑈| = 𝛾(𝐺) + |𝑈|.

(4.6)

𝑆′

We can extend the independent set to a maximal independent set of 𝐺, call it 𝑋. Therefore, 𝑖(𝐺) ≤ |𝑋 |. For each vertex 𝑣 ∈ 𝑆, at most one of two adjacent vertices 𝑣 and 𝑤 𝑣 belongs to the set 𝑋. Therefore, there exists a subset 𝑊 ′ ⊂ 𝑆 ∪ 𝑊 such that |𝑊 ′ | = 𝛾(𝐺) and 𝑊 ′ ∩ 𝑋 = ∅. In addition, since 𝑆 ′ ⊆ 𝑋 and each vertex in 𝑆 ′ has a neighbor in 𝑈, we note that 𝑈 ∩ 𝑋 = ∅. Hence, by our earlier observations, the sets 𝑈, 𝑋, and 𝑊 ′ are pairwise disjoint, and so |𝑈| + |𝑋 | + |𝑊 ′ | = |𝑈 ∪ 𝑋 ∪ 𝑊 ′ | ≤ |𝑉 | = 𝑛. Since 𝑖(𝐺) ≤ |𝑋 |, then by Inequality (4.6), 𝑖(𝐺) + 𝛾t (𝐺) ≤ |𝑋 | + 𝛾(𝐺) + |𝑈| = |𝑋 | + |𝑊 ′ | + |𝑈| = |𝑈 ∪ 𝑋 ∪ 𝑊 ′ | ≤ 𝑛. If 𝐺 = 𝑘𝐾3 for some 𝑘 ≥ 1, then 𝑖(𝐺) = 𝑘 and 𝛾t (𝐺) = 2𝑘, while 𝑛 = 3𝑘. Thus, in this case, 𝑖(𝐺) + 𝛾t (𝐺) = 𝑛. Hence, the bound in Theorem 4.48 is best possible. We remark that if 𝐻 = 𝑘𝐾2 for some 𝑘 ≥ 1, then 𝑖(𝐺) = 𝑘 and 𝛾t (𝐺) = 2𝑘, while 𝑛 = 2𝑘. Thus, in this case, 𝑖(𝐺) + 𝛾t (𝐺) = 32 𝑛. Hence, the requirement in this theorem that every component of 𝐺 has order at least 3 is necessary.

94

Chapter 4. General Bounds

4.8 Domination and Matching Recall that the matching number 𝛼′ (𝐺) of 𝐺 is the maximum cardinality of a matching in 𝐺. As an immediate consequence of Lemma 4.22 by Bollobás and Cockayne, the domination number of an isolate-free graph is at most its matching number. Corollary 4.49 ([84]) If 𝐺 is an isolate-free graph, then 𝛾(𝐺) ≤ 𝛼′ (𝐺). Proof By Lemma 4.22, the graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. For each vertex 𝑣 ∈ 𝐷, let 𝑣 ′ be an arbitrary vertex in epn[𝑣, 𝐷]. The set 𝑀 = {𝑣𝑣 ′ : 𝑣 ∈ 𝐷} is a matching in 𝐺, implying that 𝛼′ (𝐺) ≥ |𝑀 | = |𝐷| = 𝛾(𝐺). We note that for a corona 𝐺 = 𝐻 ◦ 𝐾1 of any graph 𝐻, 𝛾(𝐺) = 𝛼′ (𝐺). Since the vertices incident to the edges in a maximum matching in a graph form a TD-set in the graph, the total domination number of an isolate-free graph is trivially at most twice the matching number. Observation 4.50 If 𝐺 is an isolate-free graph, then 𝛾t (𝐺) ≤ 2𝛼′ (𝐺). Since 𝛾(𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺, it is natural to ask whether the total domination number lies between the domination number and the matching number. However, it is not necessarily true that 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for all isolate-free graphs 𝐺. For example, if 𝐺 = 𝑃2𝑘 ◦ 𝑃2 is the 2-corona of a path 𝑃2𝑘 for 𝑘 ≥ 1, then 𝐺 has order 𝑛 = 6𝑘, 𝛼′ (𝐺) = 12 𝑛 = 3𝑘, and 𝛾t (𝐺) = 23 𝑛 = 4𝑘, showing that the difference 𝛾t (𝐺) − 𝛼′ (𝐺) = 𝑘 can be arbitrarily large. It is also not necessarily true that 𝛼′ (𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺. As a simple example, if 𝐺 is the graph obtained from a complete graph 𝐾2𝑘+5 for 𝑘 ≥ 1 by adding a new vertex (of degree 1) and joining it to one vertex of the complete graph, then 𝐺 has order 𝑛 = 2(𝑘 + 3), 𝛼′ (𝐺) = 12 𝑛 = 𝑘 + 3, and 𝛾t (𝐺) = 2, showing that the difference 𝛼′ (𝐺) − 𝛾t (𝐺) can be arbitrarily large. More generally, Henning et al. [465] in 2008 showed that even for graphs 𝐺 with arbitrarily large minimum degree (but fixed with respect to the order of the graph), it is not necessarily true that the inequality 𝛾t (𝐺) ≤ 𝛼′ (𝐺) must hold, or that the inequality 𝛼′ (𝐺) ≤ 𝛾t (𝐺) must hold. Theorem 4.51 ([465]) For every integer 𝛿 ≥ 1, there exist graphs 𝐺 and 𝐻 with 𝛿(𝐺) = 𝛿(𝐻) = 𝛿 satisfying 𝛾t (𝐺) > 𝛼′ (𝐺) and 𝛾t (𝐻) < 𝛼′ (𝐻). A path covering of a graph 𝐺 is a collection of vertex-disjoint paths of 𝐺 that partition 𝑉 (𝐺). The minimum cardinality of a path covering of 𝐺 is the path covering number of 𝐺, denoted pc(𝐺). DeLaViña et al. [221] established a relationship between the total domination number, the matching number, and the path covering number of an isolate-free graph. Theorem 4.52 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺) + pc(𝐺). Proof Let Q be a path covering of 𝐺 of minimum cardinality, that is, |Q| = pc(𝐺) and every vertex of 𝐺 belongs to exactly one path in Q. Let Q2 ⊆ Q be the paths

Section 4.8. Domination and Matching

95

in Q of order at least 2. Let  𝑃 be an arbitrary path in Q2 , and let 𝑃 have order 𝑝 ≥ 2. We note that 𝛼′ (𝑃) ≥ 𝑝2 and, by Proposition 4.11, 𝛾t (𝑃) =

 𝑝 2

+

 𝑝 4

−

 𝑝 4

≤

 𝑝 2

+ 1 ≤ 𝛼′ (𝑃) + 1.

Taking the union of a maximum matching over all paths 𝑃 ∈ Q2 produces a matching in 𝐺. Therefore, by our earlier observations,  ∑︁  ∑︁ ∑︁
𝛾t (𝑃) ≤ 𝛼′ (𝑃) + 1 = 𝛼′ (𝑃) + |Q2 | ≤ 𝛼′ (𝐺) + |Q2 |. (4.7) 𝑃 ∈ Q2

𝑃∈ Q2

𝑃∈ Q2

Let Q1 be the paths in Q of order 1. Possibly, Q1 = ∅. However, if Q1 ≠ ∅, then by the minimality of the path covering Q, no two vertices in different paths in Q1 are adjacent. For each path 𝑃 ∈ Q2 , let 𝑆 𝑃 be a 𝛾t -set of the path 𝑃. For each path 𝑃 ∈ Q1 , let 𝑣 𝑃 be the one vertex on the path 𝑃, and let 𝑥 𝑃 be an arbitrary neighbor of 𝑣 𝑃 . By our earlier observations, the vertex 𝑥 𝑃 lies on a path 𝑄 that belongs to Q2 and therefore the vertex 𝑥 𝑃 is totally dominated by the set 𝑆 𝑄 . Let Ø  Ø  𝑆= 𝑆𝑃 ∪ {𝑥 𝑃 } . 𝑃∈ Q2

𝑃∈ Q1

The set 𝑆 so constructed is a TD-set of 𝐺. Hence, by Inequality (4.7), ∑︁ ∑︁ 𝛾t (𝐺) ≤ |𝑆| ≤ |𝑆 𝑃 | + |{𝑥 𝑃 }| 𝑃∈ Q2

=

∑︁

𝑃∈ Q1

𝛾t (𝑃) + |Q1 |

𝑃∈ Q2 ′


≤ 𝛼 (𝐺) + |Q2 | + |Q1 | = 𝛼′ (𝐺) + |Q| = 𝛼′ (𝐺) + pc(𝐺). That the bound of Theorem 4.52 is tight may be seen as follows. For 𝑘 ≥ 1, let 𝐺 𝑘 be the graph of order 7𝑘 obtained from a cycle 𝐶 𝑘 , if 𝑘 ≥ 3, or a path 𝑃 𝑘 , if 𝑘 = 1 or 𝑘 = 2, by identifying each vertex of the cycle or path with the center of a path 𝑃7 . The resulting graph 𝐺 𝑘 satisfies 𝛾t (𝐺 𝑘 ) = 4𝑘, 𝛼′ (𝐺 𝑘 ) = 3𝑘, and pc(𝐺 𝑘 ) = 𝑘, that is, 𝛾t (𝐺 𝑘 ) = 𝛼′ (𝐺 𝑘 ) + pc(𝐺 𝑘 ). When 𝑘 = 4, the graph 𝐺 𝑘 is illustrated in Figure 4.4, where the highlighted vertices form a 𝛾t -set of 𝐺 𝑘 . In view of Theorem 4.51, a natural problem is to determine for which graph classes G ′ it is true that 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for all graphs 𝐺 ∈ G ′ . Several such classes of graphs are given in the literature. We list only a few of these given by Henning et al. [465] and Henning and Yeo [480, 489].

Chapter 4. General Bounds

96

Figure 4.4 The graph 𝐺 4

Theorem 4.53 If 𝐺 is a connected graph of order 𝑛, then the following hold: (a) ([465]) If 𝐺 is claw-free with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). (b) ([480]) If 𝐺 is 𝑟-regular where 𝑟 ≥ 3, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). (c) ([489]) If all vertices of 𝐺 are contained in a triangle and 𝑛 ≥ 4, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). We omit proofs of the results in Theorem 4.53(a) and (c). However, we do present a proof of the 3-regular case in Theorem 4.53(b) since as remarked by O and West [617], the “3-regular case is the only case where the inequality between 𝛾t and 𝛼′ is delicate. When more edges are added, 𝛼′ tends to increase and 𝛾t tends to decrease, so the separation increases.” For this purpose, we recall a classic result about the matching number of a graph due to Berge [66], which is sometimes referred to as the Tutte-Berge formulation for the matching number. For a graph 𝐺 and a subset 𝑋 ⊂ 𝑉 (𝐺), let oc(𝐺 − 𝑋) denote the number of components of 𝐺 − 𝑋 that have odd order. Theorem 4.54 (Tutte-Berge Formula [66]) For every graph 𝐺 and subset 𝑋 ⊆ 𝑉 (𝐺),
𝛼′ (𝐺) = min 12 |𝑉 (𝐺)| + |𝑋 | − oc(𝐺 − 𝑋) . 𝑋⊆𝑉 (𝐺)

We shall also need the following result that is proved in Chapter 8. Theorem 4.55 ([458, 672]) If 𝐺 is a connected graph of order
𝑛 ≥ 3 and size 𝑚 with Δ(𝐺) ≤ Δ for some integer Δ ≥ 3, then 𝑚 ≤ Δ 𝑛 − 𝛾t (𝐺) . Theorem 4.56 ([480]) If 𝐺 is a 3-regular connected graph, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). Proof Let 𝐺 be a 3-regular connected graph of order 𝑛. By the Tutte-Berge Formula in Theorem 4.54, there is a subset 𝑋 of vertices in 𝐺 satisfying
𝛼′ (𝐺) = 12 |𝑉 (𝐺)| + |𝑋 | − oc(𝐺 − 𝑋) . (4.8) If 𝑋 = ∅, then oc(𝐺 − 𝑋) = oc(𝐺) = 0, and so by Equation (4.8), we have 𝛼′ (𝐺) = 12 𝑛. Applying Theorem 4.55 with Δ = 3 to the 3-regular graph 𝐺, we have

Section 4.8. Domination and Matching

97


= 𝑚 ≤ 3 𝑛 − 𝛾t (𝐺) , or equivalently, 𝛾t (𝐺) ≤ 12 𝑛 = 𝛼′ (𝐺). Hence, we may assume that 𝑋 ≠ ∅. Let 𝐺 ★ be an arbitrary odd component of 𝐺 − 𝑋, and let 𝐺 ★ have (odd) order 𝑛★ and size 𝑚★. If there are 𝑠 edges between 𝑋 and 𝐺 ★, then 2𝑚★ = 3𝑛★ − 𝑠, which implies that 𝑠 is odd since 𝑛★ is odd. Thus, every odd component of 𝐺 − 𝑋 is joined to 𝑋 by either exactly one edge or by at least three edges. Hence, if 𝑐 1 is the number of odd components of 𝐺 − 𝑋 having only one edge to 𝑋, then
counting the edges joining 𝑋 to odd components yields at least 𝑐 1 + 3 oc(𝐺 − 𝑋) − 𝑐 1 . However, since 𝐺 is 3-regular, the number of edges joining 𝑋 to odd components is at most 3|𝑋 |. Consequently, 3oc(𝐺 − 𝑋) − 2𝑐 1 ≤ 3|𝑋 |, or equivalently, oc(𝐺 − 𝑋) − |𝑋 | ≤ 23 𝑐 1 .

Hence, by Equation (4.8), we have 𝛼′ (𝐺) = 12 𝑛 + |𝑋 | − oc(𝐺 − 𝑋) ≥ 12 𝑛 − 23 𝑐 1 , yielding the inequality 𝛼′ (𝐺) ≥ 12 𝑛 − 13 𝑐 1 . (4.9) 3 2𝑛

We define a 1-odd-component of 𝐺 − 𝑋 as an odd component of 𝐺 − 𝑋 joined to 𝑋 by exactly one edge. Let 𝐺 ★ be a 1-odd-component of 𝐺 − 𝑋, and let 𝐺 ★ have (odd) order 𝑛★ and size 𝑚★. We note that one vertex of 𝐺 ★ has degree 2 in 𝐺 ★ and the other vertices of 𝐺 ★ have degree 3 in 𝐺 ★, and so 𝑚★ = 12 (3𝑛★ − 1). By Theorem 4.55, 𝛾t (𝐺 ★) ≤ 𝑛★ − 13 𝑚★ = 𝑛★ − 16 (3𝑛★ − 1) = 12 𝑛★ + 16 . Since 𝑛★ is odd and 𝛾t (𝐺 ★) is an integer, 𝛾t (𝐺 ★) ≤ 12 (𝑛★ − 1). Let 𝐺 ′ be the graph obtained from 𝐺 by removing all vertices that belong to a 1-odd-component of 𝐺 − 𝑋. Suppose that there is an isolated vertex 𝑥 in 𝐺 ′ . Since 𝐺 is connected, the graph 𝐺 is then determined and consists of the vertex 𝑥 and three 1-odd-components of 𝐺 − 𝑋, with the vertex 𝑥 joined to one vertex from each such component. In particular, 𝑐 1 = 3. Let 𝐺 1 , 𝐺 2 , and 𝐺 3 denote these three 1-odd-components, and let 𝑣 𝑖 be the vertex in 𝐺 𝑖 adjacent to 𝑥 for 𝑖 ∈ [3]. Let 𝐺 𝑖 have order 𝑛𝑖 and 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [3]. We note that 𝑛 = 𝑛1 + 𝑛2 + 𝑛3 + 1. By our earlier observations, |𝐷 𝑖 | ≤ 12 (𝑛𝑖 − 1). The set 𝐷 1 ∪ 𝐷 2 ∪ 𝐷 3 ∪ {𝑣 1 }, for example, is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 12 (𝑛1 + 𝑛2 + 𝑛3 − 3) + 1 = 12 (𝑛 + 1 − 3) = 𝛼′ (𝐺) (noting here that |𝑋 | = 1 and oc(𝐺 − 𝑋) = 3). Hence, we may assume that 𝐺 ′ is isolate-free. Suppose that 𝐺 ′ has a component of order 2. Let {𝑥, 𝑦} be the vertex set of this component. Since 𝐺 is connected, the graph 𝐺 is then determined and consists of the (adjacent) vertices 𝑥 and 𝑦 and four 1-odd-components of 𝐺 − 𝑋 (where 𝑥 is joined to two such components and 𝑦 to the other two components). In particular, 𝑐 1 = 4. Let 𝐺 1 , 𝐺 2 , 𝐺 3 , and 𝐺 4 denote these four 1-odd-components. Adopting our earlier notation, let 𝐺 𝑖 have order 𝑛𝑖 , and let 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [4]. The set 𝐷 1 ∪𝐷 2 ∪𝐷 3 ∪𝐷 4 ∪{𝑥, 𝑦} is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 12 (𝑛1 +𝑛2 +𝑛3 +𝑛4 −4)+2 = 1 ′ 2 (𝑛 + 2 − 4) = 𝛼 (𝐺) (noting here that |𝑋 | = 2 and oc(𝐺 − 𝑋) = 4). Hence, we may assume that every component in 𝐺 ′ has order at least 3. Let 𝐺 ′ have order 𝑛′ and size 𝑚 ′ . By assumption, 𝑛′ ≥ 3. Applying Theorem 4.55 with Δ = 3 to each component of 𝐺 ′ yields a TD-set 𝐷 ′ of 𝐺 ′ such that |𝐷 ′ | ≤ 𝑛′ − 𝑚 ′ /3. Since there are exactly 𝑐 1 edges joining 𝑉 (𝐺 ′ ) and 𝑉 (𝐺) \ 𝑉 (𝐺 ′ ), ∑︁ 2𝑚 ′ = deg𝐺 ′ (𝑣) = 3𝑛′ − 𝑐 1 . (4.10) 𝑣 ∈𝑉 (𝐺 ′ )

Chapter 4. General Bounds

98

Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑐1 denote the 1-odd-components in 𝐺 − 𝑋. Adopting our earlier notation, let 𝐺 𝑖 have order 𝑛𝑖 and let 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [𝑐 1 ]. By our earlier observations, |𝐷 𝑖 | ≤ 21 (𝑛𝑖 − 1) for 𝑖 ∈ [𝑐 1 ]. We note that 𝑛 = 𝑛′ + 𝑛1 + · · · + 𝑛𝑐1 . Consider now the set 𝐷 = 𝐷 ′ ∪ 𝐷 1 ∪ · · · ∪ 𝐷 𝑐1 . The set 𝐷 is a TD-set of 𝐺. Thus, by Inequality (4.9) and Equation (4.10), and our earlier observations, 𝛾t (𝐺) ≤ |𝐷| = |𝐷 ′ | +

𝑐1 ∑︁

|𝐷 𝑖 |

𝑖=1

≤ 𝑛′ − 13 𝑚 ′ +

𝑐1 ∑︁

1 2 (𝑛𝑖

− 1)

𝑖=1

≤ 𝑛′ − 16 (3𝑛′ − 𝑐 1 ) − 12 𝑐 1 +

𝑐1 ∑︁

1 2 𝑛𝑖

𝑖=1

= 12 𝑛 − 13 𝑐 1 ≤ 𝛼′ (𝐺). O and West [617] defined a balloon in a graph 𝐺 as a maximal 2-edge-connected subgraph incident to exactly one cut-edge of 𝐺. As remarked by O and West, the “term arises from viewing the cut-edge as a string tying the balloon to the rest of the graph.” Further, they let 𝑏(𝐺) be the number of balloons in 𝐺. We remark that using our notation in the proof of Theorem 4.56, we have that 𝑏(𝐺) ≥ 𝑐 1 , and so Inequality (4.9) can be written as 𝛼′ (𝐺) ≥ 12 𝑛 − 13 𝑏(𝐺).

(4.11)

However, O and West [617] showed that going deeper into the proof of Theorem 4.56, one can improve the upper bound on the total domination number slightly. Theorem 4.57 ([617]) If 𝐺 is a 3-regular connected graph of order 𝑛, then 𝛾t (𝐺) ≤ 12 𝑛 − 12 𝑏(𝐺),

(4.12)

(except that 𝛾t (𝐺) ≤ 12 𝑛 − 1 when 𝑏(𝐺) = 3 and the three balloons have a common neighbor), and this is tight for all even values of 𝑏(𝐺). Inequalities (4.11) and (4.12) together improve the bound 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for 3-regular connected graphs. Corollary 4.58 ([617]) If 𝐺 is a 3-regular connected graph of order 𝑛, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺) − 16 𝑏(𝐺), except when 𝑏(𝐺) = 3 and there is exactly one vertex outside the balloons, in which case 𝛾t (𝐺) ≤ 𝛼′ (𝐺).

Chapter 5

Domination in Trees 5.1

Introduction

Trees are perhaps the simplest of all graph families, so it is not surprising that much research has been done involving domination in trees. Some of the most significant results on domination in trees are presented in this chapter. Many problems that are NP-complete for general graphs can be solved in polynomial time for trees. This is the case for the decision problem DOMINATING SET. In fact, linear algorithms exist for computing 𝛾(𝑇), 𝑖(𝑇), 𝛼(𝑇), Γ(𝑇), 𝛾t (𝑇), and Γt (𝑇) for arbitrary trees 𝑇, a discussion of which is found in Chapter 3. Interest in trees also arises because they are minimally connected, that is, among all connected graphs on 𝑛 vertices, trees have the minimum possible number, namely, 𝑛 − 1, of edges. This inherent skeleton-like structure of trees gives rise to many natural applications, ranging from representing data structures in theoretical computer science to modeling molecules in chemistry. Domination in trees aids in the study of some of these applications. For example, chemical indices of trees having a given domination number were investigated in [742] and in [755]. For a different type of example, domination numbers of trees were used in [434] to predict which trees were likely (or unlikely) to represent native RNA structures. In this chapter, we present theoretical results involving domination in trees. We introduce some additional terminology for this presentation. Recall that a vertex of degree 1 is a leaf, its neighbor is a support vertex, and a support vertex adjacent to two or more leaves is a strong support vertex. For an integer 𝑖 ≥ 1 and a tree 𝑇, the number of vertices of degree 𝑖 in 𝑇 is denoted by 𝑛𝑖 (𝑇), or simply by 𝑛𝑖 . Thus, 𝑛1 is the number of leaves of 𝑇. A caterpillar is a tree with the property that removing all of its leaves forms a path called its spine. The code of the caterpillar having spine 𝑃 𝑘 : 𝑣 1 𝑣 2 . . . 𝑣 𝑘 is the ordered 𝑘-tuple (ℓ1 , ℓ2 , . . . , ℓ𝑘 ), where ℓ𝑖 is the number of leaves in the caterpillar adjacent to 𝑣 𝑖 for 𝑖 ∈ [𝑘]. A rooted tree 𝑇 = 𝑇𝑟 distinguishes one vertex 𝑟 called the root and the tree 𝑇𝑟 is said to be rooted at vertex 𝑟. For each vertex 𝑣 ≠ 𝑟 of 𝑇𝑟 , the parent of 𝑣 is the neighbor of 𝑣 on the unique (𝑟, 𝑣)-path, while a child of 𝑣 is any other neighbor of 𝑣. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_5

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100

The root 𝑟 does not have a parent in 𝑇𝑟 and all of its neighbors are its children. A descendant of 𝑣 is a vertex 𝑥 such that the unique (𝑟, 𝑥)-path contains 𝑣. Thus, every child of 𝑣 is a descendant of 𝑣. Let 𝐶 (𝑣) and 𝐷 (𝑣) denote the set of children and descendants, respectively, of 𝑣, and let 𝐷 [𝑣] = 𝐷 (𝑣) ∪ {𝑣}. The maximal subtree 𝑇𝑣 rooted at 𝑣 is the subtree induced by 𝐷 [𝑣].

5.2 Domination in Trees We begin with a simple observation about dominating sets in trees. If a 𝛾-set 𝑆 in a tree 𝑇 of order 𝑛 ≥ 3 contains a leaf, then we can simply replace this leaf in 𝑆 with its support vertex to produce another 𝛾-set of 𝑇. Thus, we have the following well-known result. Observation 5.1 If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, then the following properties hold: (a) There is a 𝛾-set of 𝑇 that contains no leaf. (b) 𝛾(𝑇) ≤ 𝑛 − ℓ. As observed in Chapter 2 and proven in Chapter 15, the three upper parameters of the Domination Chain are equal for bipartite graphs. Hence, the Domination Chain for domination and independence numbers of trees can be stated as follows. Theorem 5.2 If 𝑇 is a tree, then 𝛾(𝑇) ≤ 𝑖(𝑇) ≤ 𝛼(𝑇) = Γ(𝑇). For a simple example, the double star 𝑇 = 𝑆(𝑟, 𝑠), where 1 ≤ 𝑟 ≤ 𝑠, has 𝛾(𝑇) = 2, 𝑖(𝑇) = 𝑟 + 1, and 𝛼(𝑇) = Γ(𝑇) = 𝑟 + 𝑠. For instance, the double star 𝑇 = 𝑆(3, 4) shown in Figure 5.1 has 𝛾(𝑇) = 2, 𝑖(𝑇) = 4, and 𝛼(𝑇) = Γ(𝑇) = 7.

Figure 5.1 A double star 𝑆(3, 4)

Much research has focused on when equality is achieved between two parameters in the Domination Chain. As we shall see later in this chapter, the trees for which 𝛾(𝑇) = 𝑖(𝑇) have been characterized.

5.2.1

Domination Bounds in Trees

Recall that in Section 4.3.1, we presented Ore’s Theorem [622] from 1962. We also presented Theorem 4.24 from 1982 by Payan and Xuong [633] providing a characterization of the graphs achieving equality in Ore’s Theorem. Here we restate the characterization in the special case of trees.

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101

Theorem 5.3 ([622, 633]) If 𝑇 is tree of order 𝑛 ≥ 2, then 𝛾(𝑇) ≤ 21 𝑛, with equality if and only if 𝑇 = 𝑇 ′ ◦ 𝐾1 is the corona of some tree 𝑇 ′ . We note that Hansberg and Volkmann [377] characterized the trees of even order having domination number one less than half their order. Theorem 4.3 states Berge’s simple and tight upper bound on the domination number that for any graph 𝐺 of order 𝑛, 𝛾(𝐺) ≤ 𝑛 − Δ(𝐺). It is easy to see that this upper bound also holds for 𝑖(𝐺). In 1997 Domke et al. [235] characterized the connected bipartite graphs 𝐺 achieving equality in this bound for the domination parameters 𝛾(𝐺) and 𝑖(𝐺). Later in 1997, Favaron and Mynhardt [283] extended this work. Next we state the characterization from [235] of the trees attaining the bound for 𝛾(𝐺). A tree is a wounded spider if it can be obtained from a nontrivial star 𝐾1,𝑘 , where 𝑘 ≥ 1, by subdividing at most 𝑘 − 1 of its edges exactly once. Thus, a nontrivial star is a wounded spider. As a simple example, the wounded spider obtained from 𝐾1,6 by subdividing three of its edges is shown Figure 5.2. Let 𝑇 be a subdivided star obtained from a nontrivial star 𝐾1,𝑘 with center 𝑥 by subdividing 𝑗 edges, where 𝑗 ∈ [𝑘 − 1] 0 . Then 𝑥 along with the 𝑗 vertices of degree 2 created by the subdivision form a 𝛾-set (𝛾t -set) of 𝑇. Furthermore, the set consisting of 𝑥 together with the leaves adjacent to the vertices of degree 2 is an 𝑖-set. Thus, for this tree and all wounded spiders, 𝛾(𝑇) = 𝑖(𝑇) = 𝛾t (𝑇).

Figure 5.2 A wounded spider

Theorem 5.4 ([235]) If 𝑇 is tree of order 𝑛 ≥ 2, then 𝛾(𝑇) ≤ 𝑛 − Δ(𝑇), with equality if and only if 𝑇 is a wounded spider. In 2006 Blidia et al. [81] generalized Berge’s bound given in Theorem 4.3 as follows. For a vertex 𝑣 of 𝐺, let 𝛼𝑣′ (𝐺) denote the size of a largest matching in the graph 𝐺 − N[𝑣], that is, 𝛼𝑣′ (𝐺) = 𝛼′ (𝐺 − N[𝑣]). Let Δ′ (𝐺) denote the maximum of deg(𝑣) + 𝛼𝑣′ (𝐺) among all vertices 𝑣 of 𝐺. Theorem 5.5 ([81]) If 𝐺 is a graph, then 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝑛 − Δ′ (𝐺). The authors of [81] characterized the trees 𝑇 for which 𝛾(𝑇) = 𝑛 − Δ′ (𝑇) and those for which 𝑖(𝑇) = 𝑛 − Δ′ (𝑇). We refer the reader to their paper for these characterizations, but we will give examples of trees in these families here. For an example of a tree 𝑇 with 𝛾(𝑇) = 𝑛 − Δ′ (𝑇), see Figure 5.3 where the highlighted

Chapter 5. Domination in Trees

102 𝑣

Figure 5.3 A tree 𝑇 satisfying 𝛾(𝑇) = 11 = 𝑛 − Δ′ (𝑇)

vertices form a 𝛾-set of 𝑇. We note that 𝑇 has order 𝑛 = 26, 𝛾(𝑇) = 𝑖(𝑇) = 11, and Δ′ (𝑇) = 15, where for example, deg(𝑣) = 7 and 𝛼𝑣′ (𝑇) = 8. For an example of a tree 𝑇 ★ with 𝛾(𝑇 ★) < 𝑖(𝑇 ★) = 𝑛 − Δ′ (𝑇 ★), see Figure 5.4 where the highlighted vertices form an 𝑖-set of 𝑇 ★. We note that 𝑇 ★ has order 𝑛 = 27, 𝛼𝑣′ (𝑇 ★) = 8, 𝛾(𝑇 ★) = 11 < 12 = 𝑖(𝑇 ★), and Δ′ (𝑇 ★) = 15. 𝑣

Figure 5.4 A tree 𝑇 ★ with 𝑖(𝑇 ★) = 12 = 𝑛 − Δ′ (𝑇 ★) Next we present a lower bound on the domination number of a tree. Vertices of maximum eccentricity are called peripheral. Let ecc★ (𝐺) denote the maximum distance from the set of peripheral vertices of 𝐺 to a vertex not in the set, where the distance from a vertex to a set is the smallest distance from the vertex to any of the vertices in the set. DeLaViña et al. [222] proved a lower bound on the domination number of a tree 𝑇 in terms ecc★ (𝑇).
Theorem 5.6 ([222]) If 𝑇 is a nontrivial tree, then 𝛾(𝑇) ≥ 12 1 + ecc★ (𝑇) . Theorem 5.6 is tight as can be seen with paths of odd order. Although they only proved it for trees, the authors of [222] believe that the bound of Theorem 5.6 holds for general graphs. Let 𝛼L′ (𝐺) denote the minimum cardinality of a maximal matching of 𝐺, also called the lower matching number in the literature. We note that 𝛼L′ (𝐺) is not as well-studied as the cardinality of a maximum matching 𝛼′ (𝐺). DeLaViña et al. [222] also determined upper and lower bounds on the domination number of a tree 𝑇 in terms of 𝛼L′ (𝑇). Theorem 5.7 ([222]) If 𝑇 is a tree, then 𝛼L′ (𝑇) ≤ 𝛾(𝑇) ≤ 2𝛼L′ (𝑇).

Section 5.2. Domination in Trees

103

The bounds stated in Theorem 5.7 are tight. For example, if 𝑇 is the path 𝑃𝑛 for 𝑛 ≡ 0 (mod 3), then 𝛾(𝑇) = 𝛼L′ (𝑇) = 𝑛3 . For the upper bound, if 𝑇 is the corona of an even path 𝑃2𝑘 for 𝑘 ≥ 1, then 𝛾(𝑇) = 2𝑘 = 2𝛼L′ (𝑇). We note that the lower bound of Theorem 5.7 is not true for graphs in general. In fact, the difference 𝛼L′ (𝐺) − 𝛾(𝐺) can be made arbitrarily large. Consider the graph 𝐺 𝑚 formed from 𝑚𝐾2 labeled 𝑢 𝑖 𝑣 𝑖 for 𝑖 ∈ [𝑚] by adding two new vertices 𝑢 and 𝑣 and edges 𝑢𝑢 𝑖 , 𝑣𝑣 𝑖 for 𝑖 ∈ [𝑚]. The resulting graph 𝐺 𝑚 satisfies 𝛾(𝐺 𝑚 ) = 2, while 𝛼L′ (𝐺 𝑚 ) = 𝑚.

5.2.2 Domination Lower Bounds Involving the Number of Leaves Let R be the family of trees in which the distance between any two distinct leaves is congruent to 2 modulo 3. As an illustration, the caterpillar 𝑇 with code (2, 0, 0, 3, 0, 0, 3, 0, 0, 2) shown in Figure 5.5 is an example of a tree that belongs to the family R. In this example, 𝑇 has order 𝑛 = 20, ℓ = 10 leaves, and 𝛾(𝑇) = 4 = 13 (𝑛 − ℓ + 2), where the four highlighted vertices in Figure 5.5 form a 𝛾-set of 𝑇.

Figure 5.5 A tree 𝑇 in the family R

Lemańska [557] established the following lower bound on the domination number of a tree in terms of its order and number of leaves, and characterized the family of trees achieving this bound. Theorem 5.8 ([557]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) ≥ 1 3 (𝑛 − ℓ + 2), with equality if and only if 𝑇 ∈ R. Proof Let 𝑇 be a tree of order 𝑛 ≥ 3. We proceed by induction on 𝑛. If 𝑛 = 3, then 𝑇 is the path 𝑃3 ∈ R and the result is immediate, establishing the base case. Let 𝑇 be a tree of order 𝑛 ≥ 4 and assume that if 𝑇 ′ is a tree with ℓ ′ leaves and order 𝑛′ , where 3 ≤ 𝑛′ < 𝑛, then 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2), with equality if and only if 𝑇 ′ ∈ R. By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇. Among all paths in 𝑇 with length diam(𝑇) = 𝑑, let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be one such that deg𝑇 (𝑣 1 ) is maximized. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Suppose that deg𝑇 (𝑣 1 ) > 2. In this case, we let 𝑇 ′ = 𝑇 − 𝑣 0 with order 𝑛′ = 𝑛 − 1. Since 𝑣 1 is a support vertex in 𝑇 ′ , it follows that ℓ ′ = ℓ − 1 and 𝛾(𝑇 ′ ) = 𝛾(𝑇). By our inductive hypothesis, 𝛾(𝑇) = 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 2).

(5.1)

Moreover, if 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), then we have equality throughout Inequality (5.1) and so 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2), implying that 𝑇 ′ ∈ R. Let 𝑢 0 be a child of 𝑣 1 different from 𝑣 0 . Now 𝑑𝑇 (𝑢 0 , 𝑣 0 ) = 2. Since 𝑢 0 and 𝑣 0 are at the same distance from every

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other leaf in 𝑇, and 𝑇 ′ ∈ R, the tree 𝑇 therefore also belongs to R. Hence, we may assume that deg𝑇 (𝑣 1 ) = 2, for otherwise the desired result follows. Suppose that deg𝑇 (𝑣 2 ) > 2. In this case, we let 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 } with order ′ 𝑛 = 𝑛−2. We note that ℓ ′ = ℓ−1. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 . If 𝑢 1 is a leaf, then 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then it is a support vertex of degree 2 and 𝑢 1 ∈ 𝑆. In both cases, the set 𝑆\{𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆|−1 = 𝛾(𝑇)−1. By our inductive hypothesis, 𝛾(𝑇) ≥ 𝛾(𝑇 ′ )+1 ≥ 31 (𝑛′ −ℓ ′ +2)+1 = 13 (𝑛−ℓ+4) > 13 (𝑛−ℓ+2). Hence, we may assume that deg𝑇 (𝑣 2 ) = 2, for otherwise the lower bound holds with strict inequality. With our earlier assumptions, if 𝑛 ∈ {4, 5}, then 𝑇 is a path 𝑃𝑛 , and 𝛾(𝑇) = 2 > 13 𝑛 = 13 (𝑛 − ℓ + 2). Hence, we may assume that 𝑛 ≥ 6. Therefore, let 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 } with order 𝑛′ = 𝑛 − 3 ≥ 3. If deg𝑇 (𝑣 3 ) > 2, then ℓ ′ = ℓ − 1. If deg𝑇 (𝑣 3 ) = 2, then 𝑣 3 is a leaf in 𝑇 ′ and ℓ ′ = ℓ. In both cases, ℓ ′ ≤ ℓ. Recall that 𝑣 1 ∈ 𝑆. If 𝑣 2 ∈ 𝑆, then we can replace 𝑣 2 in 𝑆 with the vertex 𝑣 3 . Hence, we may assume that 𝑣 2 ∉ 𝑆, implying that the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , whence 𝛾(𝑇 ′ ) ≤ |𝑆| −1 = 𝛾(𝑇) −1. By our inductive hypothesis, 𝛾(𝑇) ≥ 𝛾(𝑇 ′ ) + 1 ≥ 13 (𝑛′ − ℓ ′ + 2) + 1 ≥ 13 (𝑛 − ℓ + 2), noting that 𝑛′ = 𝑛 − 3 and ℓ ′ ≤ ℓ. Moreover, if 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), then we have equality throughout the previous inequality chain, implying that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) and ℓ ′ = ℓ. This in turn implies that 𝑇 ′ ∈ R and the vertex 𝑣 3 is a leaf in 𝑇 ′ . Since 𝑇 ′ ∈ R, the distance between any two distinct leaves in 𝑇 ′ is congruent to 2 modulo 3. Let 𝑢 and 𝑣 be two leaves in 𝑇. If both 𝑢 and 𝑣 belong to 𝑇 ′ , then 𝑑𝑇 (𝑢, 𝑣) = 𝑑𝑇 ′ (𝑢, 𝑣) ≡ 2 (mod 3). If one of 𝑢 and 𝑣 does not belong to 𝑇 ′ , then we may assume that 𝑢 = 𝑣 0 . In this case, 𝑑𝑇 (𝑢, 𝑣) = 𝑑𝑇 (𝑢, 𝑣 3 ) + 𝑑𝑇 (𝑣 3 , 𝑣) = 3 + 𝑑𝑇 ′ (𝑣 3 , 𝑣) ≡ 2 (mod 3) noting that the distance between 𝑣 3 and 𝑣 in 𝑇 ′ is congruent to 2 modulo 3. Thus, the distance between any two distinct leaves in 𝑇 is congruent to 2 modulo 3, that is, 𝑇 ∈ R. In 2019 Hajian et al. [372] defined a family G00 of trees as follows. Definition 5.9 Let G00 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1, such that 𝑇1 is a star with at least three vertices, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O defined below, for all 𝑖 ∈ [𝑘 − 1]. Operation O. Add a star 𝑄 𝑖 with at least three vertices to the tree 𝑇𝑖 and add an edge joining a leaf of 𝑄 𝑖 and a leaf of 𝑇𝑖 . Hajian et al. [372] showed that G00 ⊆ R and R ⊆ G00 , that is, the family R is precisely the family G00 . Theorem 5.8 due to Lemańska [557] can therefore be restated as follows. Theorem 5.10 ([557]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) ≥ 1 0 3 (𝑛 − ℓ + 2), with equality if and only if 𝑇 ∈ G0 . We shall need the following elementary property of graphs in the family G00 . Proposition 5.11 If 𝑇 ∈ G00 is obtained from a star with at least three vertices by applying Operation O exactly 𝑘 − 1 times for some 𝑘 ≥ 1, then 𝛾(𝑇) = 𝜌(𝑇) = 𝑘. Further, 𝛾(𝑇 − 𝑣) ≥ 𝛾(𝑇) for every vertex 𝑣 in 𝑇.

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105

In order to generalize the result of Lemańska [557], the authors in [372] defined two additional families G01 and G02 of trees, as follows. Definition 5.12 Let T01,1 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star with at least three vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . Let G01 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees where 𝑘 ≥ 1 such that 𝑇1 ∈ T01,1 ∪ {𝑃2 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O (see Definition 5.9) for all 𝑖 ∈ [𝑘 − 1]. The following lemma determines the domination number of a graph that belongs to the family G01 . The proofs we present of Lemma 5.13 and Theorem 5.14 are from [372]. Lemma 5.13 ([372]) If 𝑇 ∈ G01 has order 𝑛 ≥ 2 and ℓ leaves, then 𝛾(𝑇) = 1 3 (𝑛 − ℓ + 3). Proof Let 𝑇 ∈ G01 be a tree of order 𝑛 ≥ 2 with ℓ leaves. The tree 𝑇 can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees where 𝑘 ≥ 1 such that 𝑇1 ∈ T01,1 ∪ {𝑃2 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O for all 𝑖 ∈ [𝑘 − 1]. We prove by induction on 𝑘 ≥ 1 that 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). Suppose that 𝑘 = 1. Thus, either 𝑇 = 𝑃2 or 𝑇 ∈ T01,1 . If 𝑇 = 𝑃2 , then 𝑛 = ℓ = 2 and 𝛾(𝑇) = 1 = 13 (𝑛 − ℓ + 3). Hence, we may assume that 𝑇 ∈ T01,1 . Thus, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star 𝑄 with 𝑡 ≥ 3 vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . We note that 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ and ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 2. By Theorem 5.10, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 3). This establishes the base case. Suppose that 𝑘 ≥ 2 and let 𝑇 ★ = 𝑇𝑘−1 . Let 𝑇 ★ have order 𝑛★ and ℓ★ leaves. Applying the induction hypothesis to the tree 𝑇 ★ ∈ G01 , we have 𝛾(𝑇 ★) = 13 (𝑛★ −ℓ★ +3). The tree 𝑇 can be obtained from 𝑇 ★ by adding a star 𝑄★ with 𝑞 ≥ 3 vertices and adding an edge from a leaf of 𝑄★ to a leaf of 𝑇 ★. By construction of the tree 𝑇, we note that 𝛾(𝑇) = 𝛾(𝑇 ★) + 1. Further, 𝑛 = 𝑛★ + 𝑞 and ℓ = (ℓ★ − 1) + (𝑞 − 2). Thus, 𝛾(𝑇) = 𝛾(𝑇 ★) + 1 = 13 (𝑛★ − ℓ★ + 3) + 1 = 13 (𝑛 − ℓ) + 1 = 13 (𝑛 − ℓ + 3). Theorem 5.14 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) = 1 1 3 (𝑛 − ℓ + 3) if and only if 𝑇 ∈ G0 . Proof The sufficiency follows from Lemma 5.13. To prove the necessity, let 𝑇 be a tree of order 𝑛 ≥ 2 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). We proceed by induction on 𝑛 ≥ 2 to show that 𝑇 ∈ G01 . If 𝑛 = 2, then 𝑇 = 𝑃2 ∈ G01 . This establishes the base case. Let 𝑛 ≥ 3 and assume that if 𝑇 ′ is a tree of order 𝑛′ ≥ 2, where 𝑛′ < 𝑛 with ℓ ′ leaves such that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3), then 𝑇 ′ ∈ G01 . Let 𝑇 be a tree of order 𝑛 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇.

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If 𝑇 is a star, then 𝑇 ∈ G00 and by Theorem 5.10, 𝛾(𝑇) = 31 (𝑛 − ℓ + 2), a contradiction. If 𝑇 is a double star, then ℓ = 𝑛 − 2 and 𝛾(𝑇) = 2 > 13 (𝑛 − ℓ + 3), a contradiction. Hence, diam(𝑇) ≥ 4 and 𝑛 ≥ 5. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the 𝛾-set 𝑆 contains no leaf of 𝑇, we note that 𝑣 1 ∈ 𝑆 and no child of 𝑣 1 belongs to 𝑆. Let deg𝑇 (𝑣 1 ) = 𝑡. Since {𝑣 0 , 𝑣 2 } ⊆ N𝑇 (𝑣 1 ), we note that 𝑡 ≥ 2. We show that deg𝑇 (𝑣 2 ) = 2. Suppose, to the contrary, that deg𝑇 (𝑣 2 ) ≥ 3. We now consider the tree 𝑇 ′ obtained from 𝑇 by removing the maximal subtree at 𝑇𝑣1 , that is, 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1). Let 𝑢 1 be a child of 𝑣 2 in 𝑇 different from 𝑣 1 . If 𝑢 1 is a leaf, then by our choice of the set 𝑆, we note that 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then every child of 𝑢 1 is a leaf, and so 𝑢 1 ∈ 𝑆. In both cases, the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , implying that 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 3) − 1 = 13 (𝑛 − ℓ), contradicting our earlier observation that 𝛾(𝑇 ′ ) ≥ 13 (𝑛 − ℓ + 1). Hence, deg𝑇 (𝑣 2 ) = 2. Suppose next that deg𝑇 (𝑣 3 ) ≥ 3. We now consider the tree 𝑇 ′ obtained from 𝑇 by removing the maximal subtree at 𝑇𝑣2 , that is, 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. Recall that 𝑣 1 ∈ 𝑆. If 𝑣 2 ∈ 𝑆, then we can simply replace 𝑣 2 in 𝑆 by its parent 𝑣 3 . Hence, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ). This implies that the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 3) − 1 = 13 (𝑛 − ℓ). By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, we have 1 1 ′ 1 ′ 1 ′ ′ ′ ′ 3 (𝑛 − ℓ) ≥ 𝛾(𝑇 ) ≥ 3 (𝑛 − ℓ + 2) = 3 (𝑛 − ℓ), implying that 𝛾(𝑇 ) = 3 (𝑛 − ℓ + 2). ′ 0 ′ By Theorem 5.10, the tree 𝑇 ∈ G0 . Thus, the tree 𝑇 is obtained from 𝑇 ∈ G00 by attaching the star 𝑇𝑣2 (centered at 𝑣 1 ) and adding an edge from the leaf 𝑣 2 of the added star 𝑇𝑣2 to the non-leaf 𝑣 3 of 𝑇 ′ , implying that 𝑇 ∈ T01,1 ⊂ G01 . We may therefore assume that deg𝑇 (𝑣 3 ) = 2, for otherwise 𝑇 ∈ G01 as desired. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = (ℓ ′ −1) + (𝑡 −1) = ℓ ′ +𝑡 −2. By Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ −ℓ ′ +2) = 13 (𝑛−ℓ−1). If 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2), then 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ − 1) + 1 = 13 (𝑛 − ℓ + 2), a contradiction. Thus, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ). Hence, 13 (𝑛 − ℓ + 3) = 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 ≥ 31 (𝑛 − ℓ + 3), implying that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3). Applying the inductive hypothesis to the tree 𝑇 ′ , we have 𝑇 ′ ∈ G01 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G01 by applying Operation O, implying that 𝑇 ∈ G01 . We consider next the family G02 defined in [372]. Definition 5.15 Let T02,1 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star (with at least two vertices) and adding an edge from the center of the added star to a non-leaf in 𝑇 ′ . Let T02,2 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G01 by adding a star with at least three vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . Now, let G02 be the

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family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1, such that 𝑇1 ∈ T02,1 ∪ T02,2 ∪ {𝑃4 }, 𝑇 = 𝑇𝑘 , and if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O (see Definition 5.9) for all 𝑖 ∈ [𝑘 − 1]. The following lemma determines the domination number of a graph that belongs to the family G02 . The proofs we present of Lemma 5.16 and Theorem 5.17 are from [372]. Lemma 5.16 ([372]) If 𝑇 ∈ G02 has order 𝑛 ≥ 2 and ℓ leaves, then 𝛾(𝑇) = 1 3 (𝑛 − ℓ + 4). Proof Let 𝑇 ∈ G02 be a tree of order 𝑛 ≥ 2 with ℓ leaves. The tree 𝑇 can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1 such that 𝑇1 ∈ T02,1 ∪T02,2 ∪{𝑃4 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O for all 𝑖 ∈ [𝑘 −1]. We prove by induction on 𝑘 ≥ 1 that 𝛾(𝑇) = 13 (𝑛−ℓ+4). If 𝑘 = 1, then either 𝑇 ∈ T02,1 ∪ T02,2 ∪ {𝑃4 }. If 𝑇 = 𝑃4 , then 𝑛 = 4, ℓ = 2, and 𝛾(𝑇) = 2 = 13 (𝑛 − ℓ + 4). Hence, we may assume that 𝑇 ∈ T02,1 ∪ T02,2 . Assume that 𝑇 ∈ T02,1 . In this case, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star with 𝑡 ≥ 3 vertices and adding an edge from the center of the added star to a non-leaf in 𝑇 ′ . By Proposition 5.11, we have 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ and ℓ ′ leaves. By construction of the tree 𝑇, 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 4). Assume that 𝑇 ∈ T02,2 . In this case, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G01 by adding a star with 𝑡 ≥ 3 vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . We note that 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 2. By Lemma 5.13, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ + 1). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 4). This establishes the base case. Suppose that 𝑘 ≥ 2 and let 𝑇 ★ = 𝑇𝑘−1 . Let 𝑇 ★ have order 𝑛★ and ℓ★ leaves. Applying the induction hypothesis to the tree 𝑇 ★ ∈ G02 , we have 𝛾(𝑇 ★) = 13 (𝑛★ −ℓ★ +4). The tree 𝑇 can be obtained from 𝑇 ★ by adding a star with 𝑞 ≥ 3 vertices and adding an edge from a leaf of the added star to a leaf of 𝑇 ★. By construction of the tree 𝑇, we have 𝛾(𝑇) = 𝛾(𝑇 ★) + 1. Further, 𝑛 = 𝑛★ + 𝑞 and ℓ = (ℓ★ − 1) + (𝑞 − 2). Thus, 𝛾(𝑇) = 𝛾(𝑇 ★) + 1 = 13 (𝑛★ − ℓ★ + 4) + 1 = 13 (𝑛 − ℓ + 1) + 1 = 13 (𝑛 − ℓ + 4). Theorem 5.17 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) = 1 2 3 (𝑛 − ℓ + 4) if and only if 𝑇 ∈ G0 . Proof The sufficiency follows from Lemma 5.16. To prove the necessity, let 𝑇 be a tree of order 𝑛 ≥ 2 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 4). We proceed by induction on 𝑛 ≥ 2 to show that 𝑇 ∈ G02 . If 𝑛 = 2, then 𝑇 = 𝑃2 and 𝛾(𝑇) = 1 = 13 (𝑛 − ℓ + 3), a contradiction. If 𝑇 is a star, then 𝑇 ∈ G00 and by Theorem 5.10, 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), a contradiction. If 𝑇 = 𝑃4 , then 𝑇 ∈ G02 . This establishes the base case. Let 𝑛 ≥ 5 and assume that if 𝑇 ′ is a tree of order 𝑛′ ≥ 2, where 𝑛′ < 𝑛

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with ℓ ′ leaves such that 𝛾(𝑇 ′ ) = 31 (𝑛′ − ℓ ′ + 4), then 𝑇 ′ ∈ G02 . Let 𝑇 be a tree of order 𝑛 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 4). By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇. If 𝑇 is a double star, then 𝑇 ∈ T02,1 ⊆ G02 . Hence, we may assume that diam(𝑇) ≥ 4. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the 𝛾-set 𝑆 contains no leaf of 𝑇, we note that 𝑣 1 ∈ 𝑆 and no child of 𝑣 1 belongs to 𝑆. Let deg𝑇 (𝑣 1 ) = 𝑡, where 𝑡 ≥ 2. Suppose that deg𝑇 (𝑣 2 ) ≥ 3. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 . If 𝑢 1 is a leaf, then 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then 𝑢 1 is a support vertex and 𝑢 1 ∈ 𝑆. In both cases, the vertex 𝑣 2 is dominated by 𝑆 \ {𝑣 1 }, implying that 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 4) − 1 = 13 (𝑛 − ℓ + 1). By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1).

(5.2)

Since we must have equality throughout Inequality (5.2), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +2). By Theorem 5.10, the tree 𝑇 ′ is in G00 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G00 by attaching the star 𝑇𝑣1 centered at 𝑣 1 (possibly, 𝑇𝑣1 = 𝑃2 ) and adding an edge from the center 𝑣 1 of the added star 𝑇𝑣1 to the non-leaf 𝑣 2 of 𝑇 ′ , implying that 𝑇 ∈ T02,1 ⊂ G02 . We may assume therefore that deg𝑇 (𝑣 2 ) = 2, for otherwise 𝑇 ∈ G02 as desired. Suppose next that deg𝑇 (𝑣 3 ) ≥ 3, and consider the tree 𝑇 ′ = 𝑇 −𝑉 (𝑇𝑣2 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. If 𝑣 2 ∈ 𝑆, then we can simply replace 𝑣 2 in 𝑆 by its parent 𝑣 3 . Hence, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ), implying that the set 𝑆\{𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| −1 = 13 (𝑛−ℓ+4) −1 = 1 ′ ′ 3 (𝑛 − ℓ + 1). By construction of the tree 𝑇, we have 𝑛 = 𝑛 + 𝑡 + 1 and ℓ = ℓ + 𝑡 − 1. If 1 ′ ′ ′ ′ 𝛾(𝑇 ) = 3 (𝑛 − ℓ + 2), then since every 𝛾-set of 𝑇 can be extended to a dominating set of 𝑇 by adding 𝑣 1 to it, 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 = 13 (𝑛′ − ℓ ′ + 2) + 1 = 13 (𝑛 − ℓ + 3), a contradiction. Hence, by Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3). Thus, by our earlier observations, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ + 1).

(5.3)

Since we must have equality throughout Inequality (5.3), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +3). By Theorem 5.14, the tree 𝑇 ′ is in G01 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G01 by attaching the star 𝑇𝑣2 , centered at 𝑣 1 , to 𝑇 ′ by adding an edge between the leaf 𝑣 2 of the star 𝑇𝑣2 to the non-leaf vertex 𝑣 3 of 𝑇 ′ , implying that 𝑇 ∈ T02,2 ⊂ G02 . We may assume therefore that deg𝑇 (𝑣 3 ) = 2, for otherwise 𝑇 ∈ G02 as desired. We again consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). As before, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ), implying that 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 4) − 1 = 13 (𝑛 − ℓ + 1). Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it the vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = (ℓ ′ − 1) + (𝑡 − 1) = ℓ ′ + 𝑡 − 2. If 𝛾(𝑇 ′ ) ≤ 13 (𝑛′ − ℓ ′ + 3),

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then 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 ≤ 31 (𝑛′ − ℓ ′ + 3) + 1 = 13 (𝑛 − ℓ + 3), a contradiction. Hence, by Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 4). Thus, by our earlier observations, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 4) = 13 (𝑛 − ℓ + 1).

(5.4)

Since we must have equality throughout Inequality (5.4), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +4). Applying the inductive hypothesis to the tree 𝑇 ′ , we have 𝑇 ′ is in G02 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G02 by applying Operation O, implying that 𝑇 ∈ G02 . The results of Theorems 5.10, 5.14, and 5.17 are summarized in the following theorem. Theorem 5.18 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾(𝑇) ≥ , 3 with equality if and only if 𝑇 ∈ G00 ∪ G01 ∪ G02 . We remark that Theorem 5.18 was generalized in [372] to connected graphs of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves. In order to state this stronger result, the authors in [372] defined the families G𝑘0 , G𝑘1 , and G𝑘2 of graphs for all integers 𝑘 ≥ 0, where in the special case when 𝑘 = 0, the families G00 , G01 , and G02 are defined earlier in Definitions 5.9, 5.12, and 5.15, respectively. However, we omit the definitions of the families G𝑘0 , G𝑘1 , and G𝑘2 for 𝑘 ≥ 1, since this chapter focuses only on domination in trees. We are now in a position to state the main result in [372] that generalizes Theorem 5.18. In the special case when 𝑘 = 0, Theorem 5.19 is a restatement of the result of Theorem 5.18. Theorem 5.19 ([372]) If 𝐺 is a connected graph of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves, then   𝑛 − ℓ + 2(1 − 𝑘) 𝛾(𝐺) ≥ , 3 with equality if and only if 𝐺 ∈ G𝑘0 ∪ G𝑘1 ∪ G𝑘2 .

5.2.3

Slater Lower Bound on the Domination Number

As we have seen in Theorem 4.9 in Chapter 4, Slater [680] determined a lower bound on the domination number of a graph 𝐺 in terms of a parameter, which is now known as the Slater number, based on the non-increasing degree sequence of 𝐺. Recall that the Slater number sl(𝐺) of a graph 𝐺 is the smallest integer 𝑡 such that 𝑡 added to the sum of the first 𝑡 terms of the non-increasing degree sequence of 𝐺 is at least as large as the number of vertices of 𝐺. We repeat Theorem 4.9. Theorem 5.20 ([680]) For any graph 𝐺, 𝛾(𝐺) ≥ sl(𝐺).

110

Chapter 5. Domination in Trees

Desormeaux et al. [227] in 2014 continued this work and showed that the lower bound on the domination number for trees given in Theorem 5.10 is also a lower bound on the Slater number for trees. Theorem 5.21 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves and degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then sl(𝑇) ≥ 31 (𝑛 − ℓ + 2), with equality if and only if 𝑛 − ℓ ≡ 1 (mod 3) and 𝑑𝑡+1 ≤ 2, where 𝑡 = 13 (𝑛 − ℓ + 2). Corollary 5.22 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, where 𝑛 − ℓ ≡ 1 (mod 3) and ℓ ≤ 14 𝑛 + 2, then sl(𝑇) = 13 (𝑛 − ℓ + 2). Theorem 5.10 now follows as an immediate consequence of Theorems 5.20 and 5.21, and we can restate it as follows. Corollary 5.23 If 𝑇 is a tree of order 𝑛 with ℓ leaves and degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾(𝑇) ≥ sl(𝑇) ≥ 13 (𝑛 − ℓ + 2). Further, if 𝑛 − ℓ . 1 (mod 3) or if 𝑛 − ℓ ≡ 1 (mod 3) and 𝑑𝑡+1 ≥ 3, where 𝑡 = 13 (𝑛 − ℓ + 2), then 𝛾(𝑇) > 13 (𝑛 − ℓ + 2). There exist trees for which the difference between the Slater number and 13 (𝑛−ℓ+2) is arbitrarily large. For example, let 𝑘 ≥ 3 be any odd integer, and let 𝑇 be a caterpillar with spine 𝑃 𝑘 and code (ℓ1 , ℓ2 , . . . , ℓ𝑘 ), where ℓ1 = ℓ𝑘 = 2 and ℓ𝑖 = 1 for 2 ≤ 𝑖 ≤ 𝑘 −1. Then, 𝑇 has order 𝑛 = 2𝑘 + 2, sl(𝑇) = 12 (𝑘 + 1), and 13 (𝑛 − ℓ + 2) = 13 (𝑘 + 2). Recall that R denotes the family of trees 𝑇 for which 𝑑 (𝑥, 𝑦) ≡ 2 (mod 3) between any pair of distinct leaves 𝑥 and 𝑦 of 𝑇. As mentioned earlier, Hajian et al. [372] showed that R = G00 , where G00 is the family defined in Definition 5.9. As an immediate consequence of Theorem 5.20, Corollary 5.22, and Theorem 5.10, we have the following result. Corollary 5.24 If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, where 𝑛 − ℓ ≡ 1 (mod 3) and ℓ ≤ 2 + 𝑛4 , then 𝛾(𝑇) = sl(𝑇) if and only if 𝑇 ∈ G00 . Next we present an upper bound on the domination number of a tree in terms of the Slater number. Let 𝑉≥2 (𝑇) be the set of 𝑛 − ℓ non-leaf vertices of a tree 𝑇. Theorem 5.25 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3, then 𝛾(𝑇) ≤ 3 sl(𝑇) − 2, with equality if and only if sl(𝑇) = 13 (𝑛 − ℓ + 2) and every vertex of 𝑇 is a leaf or a support vertex. Proof By Observation 5.1, 𝛾(𝑇) ≤ 𝑛 − ℓ. By Theorem 5.21, we have sl(𝑇) ≥ 1 3 (𝑛 − ℓ + 2). Hence, 𝛾(𝑇) ≤ 𝑛 − ℓ ≤ 3 sl(𝑇) − 2, (5.5)

Section 5.2. Domination in Trees

111

which establishes the desired upper bound. To characterize the trees achieving the upper bound, first assume that sl(𝑇) = 31 (𝑛 − ℓ + 2) and every vertex of 𝑇 is a leaf or a support vertex. Then every vertex in 𝑉≥2 (𝑇) is a support vertex of 𝑇. By Observation 5.1(a), there exists a 𝛾-set 𝐷 of 𝑇 that contains no leaf, and so 𝐷 ⊆ 𝑉≥2 (𝑇). However, in order to dominate the leaves of 𝑇, the set 𝐷 contains every support vertex of 𝑇, and so 𝑉≥2 (𝑇) ⊆ 𝐷. Consequently, 𝐷 = 𝑉≥2 (𝑇), and so 𝛾(𝑇) = |𝐷 | = 𝑛 − ℓ = 3 sl(𝑇) − 2. This proves the sufficiency. To prove the necessity, assume that 𝛾(𝑇) = 3 sl(𝑇) − 2. Then we must have equality throughout Inequality (5.5). Thus, sl(𝑇) = 13 (𝑛 − ℓ + 2) and 𝛾(𝑇) = 𝑛 − ℓ. If there is a vertex 𝑣 in 𝑉≥2 (𝑇) that is not a support vertex of 𝑇, then 𝑉≥2 (𝑇) \ {𝑣} is a dominating set of 𝑇, and so 𝛾(𝑇) ≤ |𝑉≥2 (𝑇)| − 1 = 𝑛 − ℓ − 1, a contradiction. Therefore, every vertex in 𝑉≥2 (𝑇) is a support vertex of 𝑇. For an example of a tree attaining the upper bound of Theorem 5.25, let 𝑡 ≥ 2 and let 𝑇 ′ be a caterpillar with spine 𝑣 1 𝑣 2 . . . 𝑣 𝑡 and code (ℓ1 , ℓ2 , . . . , ℓ𝑡 ), where ℓ𝑖 = 5 for 𝑖 ∈ [𝑡]. Let 𝑇 be the tree obtained from 𝑇 ′ by subdividing exactly two pendant edges incident to 𝑣 𝑖 for each 𝑖 ∈ [𝑡 − 1]. For 𝑡 = 3, the tree 𝑇 is illustrated in Figure 5.6. The resulting tree 𝑇 has 𝑛 = 8𝑡 − 2 = 22 vertices, sl(𝑇) = 𝑡 = 3, and 𝛾(𝑇) = 3𝑡 − 2 = 3 sl(𝑇) − 2 = 7. 𝑣1

𝑣2

𝑣3

Figure 5.6 A tree 𝑇 with 𝛾(𝑇) = 3 sl(𝑇) − 2 By Theorem 5.20 and Theorem 5.25, if 𝑇 is a tree of order 𝑛 ≥ 3, then sl(𝑇) ≤ 𝛾(𝑇) ≤ 3 sl(𝑇) − 2.

(5.6)

We conclude this section by showing that all values of the lower and upper bounds on 𝛾(𝑇) in Inequality (5.6) are attainable. Theorem 5.26 ([227]) For integers 𝑎 and 𝑏, such that 𝑎 ≥ 1 and 0 ≤ 𝑏 ≤ 2𝑎 − 2, there exists a tree 𝑇 for which sl(𝑇) = 𝑎 and 𝛾(𝑇) = 𝑎 + 𝑏. Proof Let 𝑎 and 𝑏 be integers where 𝑎 ≥ 1 and 0 ≤ 𝑏 ≤ 2𝑎 − 2. In the case when 𝑎 = 1, we have that 𝑏 = 0, and we take 𝑇 to be a star on at least three vertices. Then, sl(𝑇) = 1 = 𝑎 and 𝛾(𝑇) = 1 = 𝑎 + 𝑏. Assume that 𝑎 ≥ 2. Begin with the caterpillar 𝑇 ′ having spine 𝑣 1 𝑣 2 . . . 𝑣 𝑎 and code (ℓ1 , ℓ2 , . . . , ℓ𝑎 ), where ℓ𝑖 = 2𝑎 for 1 ≤ 𝑖 ≤ 𝑎. Let ′ 𝑅 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑎 }. If 𝑏 is even, let 𝑇 be the tree obtained  𝑏  from 𝑇 by subdividing exactly two pendant edges incident to 𝑣 𝑖 for each 𝑖 ∈ 2 . If 𝑏 is odd, let 𝑇 be the tree obtained from 𝑇 ′ by subdividing exactly two pendant edges incident to 𝑣 𝑖 for

112

Chapter 5. Domination in Trees

  each 𝑖 ∈ 𝑏−1 2 , and subdividing exactly one pendant edge incident to 𝑣 (𝑏+1)/2 . Thus, the resulting tree 𝑇 has order 𝑛 = 2𝑎 2 + 𝑎 + 𝑏. It follows that ∑︁ |𝑅| + deg(𝑣) = |𝑅| + (2𝑎 + 2)|𝑅| − 2 𝑣∈𝑅 = (2𝑎 + 3)|𝑅| − 2 = (2𝑎 + 3)𝑎 − 2 = 2𝑎 2 + 3𝑎 − 2 ≥ 2𝑎 2 + 𝑎 + 𝑏 = 𝑛, implying that sl(𝑇) ≤ |𝑅| = 𝑎. However, if 𝑅 ′ = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑎−1 }, then 𝑅 ′ is a set of 𝑎 − 1 vertices of largest possible degree and ∑︁ |𝑅 ′ | + deg(𝑣) = |𝑅 ′ | + (2𝑎 + 2)|𝑅 ′ | − 1 ′ 𝑣 ∈𝑅 = (2𝑎 + 3)|𝑅 ′ | − 1 = (2𝑎 + 3) (𝑎 − 1) − 1 = 2𝑎 2 + 𝑎 − 4 < 𝑛, implying that sl(𝑇) > |𝑅 ′ | = 𝑎 − 1. Consequently, sl(𝑇) = 𝑎. We note that every vertex of 𝑇 is a leaf or a support vertex and since 𝑇 has 𝑎 + 𝑏 support vertices, every one of which must be in every 𝛾-set of 𝑇, it follows that 𝛾(𝑇) = 𝑎 + 𝑏.

5.2.4

Vertices in All or No Minimum Dominating Sets

A vertex can appear in every 𝛾-set, in no 𝛾-set, or in some but not all 𝛾-sets of a graph. For example, the double star 𝑇 = 𝑆(1, 4) with centers 𝑥 and 𝑦, where 𝑥 is adjacent to one leaf 𝑥 ′ and 𝑦 is adjacent to four leaves, has exactly two 𝛾-sets. The vertex 𝑦 is in both 𝛾-sets of 𝑇, each of the four leaves adjacent to 𝑦 is in no 𝛾-set of 𝑇, and each of 𝑥 and 𝑥 ′ appears in exactly one of the two 𝛾-sets of 𝑇. In this section, we are concerned with the vertices in all or no minimum dominating sets of a tree 𝑇. Let these sets be defined as follows: • A (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in all 𝛾-sets of 𝐺 , and  • N (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in no 𝛾-set of 𝐺 . For example,  the path 𝑃𝑛 given by 𝑣 1 𝑣 2 . . . 𝑣 𝑛−1 𝑣 𝑛 , where 𝑛 ≡ 0 (mod 3), has A (𝑃𝑛 ) = 𝑣 𝑖 : 𝑖 ≡ 2 (mod 3) and 𝑖 ∈ [𝑛] and N (𝑃𝑛 ) = 𝑣 𝑖 : 𝑖 ≡ 0, 1 (mod 3) and 𝑖 ∈ [𝑛] . Mynhardt [604] in 1999 introduced a novel technique called tree pruning to help characterize the sets A (𝑇) and N (𝑇) for an arbitrary tree 𝑇. Variations of this technique have been used to characterize vertices contained in all, or in no, 𝜇-sets of trees for other domination parameters 𝜇. For examples, see [80, 197, 470, 474]. We give a slightly modified description of Mynhardt’s pruning technique here. A branch vertex is a vertex of degree at least 3 in 𝑇. Let 𝐿 (𝑇) denote the set of leaves

Section 5.2. Domination in Trees

113

of 𝑇. We denote the set of leaves in the rooted tree 𝑇 = 𝑇𝑣 distinct from 𝑣 by 𝐿(𝑣), that is, 𝐿(𝑣) = 𝐷 (𝑣) ∩ 𝐿 (𝑇). For 𝑗 ∈ [2], we define  𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿 (𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 3) . The pruning of a tree is performed with respect to the root. Hence, suppose 𝑇 is rooted at 𝑣, that is, 𝑇 = 𝑇𝑣 . If deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇𝑣 ) \ {𝑣}, then let 𝑇 𝑣 = 𝑇. Otherwise, let 𝑢 be a branch vertex at maximum distance from 𝑣. Note that |𝐶 (𝑢)| ≥ 2 and deg(𝑥) ≤ 2 for each 𝑥 ∈ 𝐷 (𝑢). The pruning process is applied as follows: • If |𝐿 1 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a new leaf adjacent to 𝑢. • If 𝐿 1 (𝑢) = ∅ and |𝐿 2 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 3 to 𝑢. This step of the pruning process, where all the descendants of 𝑢 are deleted and a path of length 1, 2, or 3 is attached to 𝑢 to give a tree in which 𝑢 has degree 2, is called a pruning of 𝑇𝑣 at 𝑢. Repeat the above process until a tree 𝑇 𝑣 is obtained with deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇 𝑣 ) \ {𝑣}. Then, 𝑇 𝑣 is called a pruning of 𝑇𝑣 . The tree 𝑇 𝑣 is unique. We note that if 𝑇 is a star 𝐾1,𝑛−1 for 𝑛 ≥ 3 rooted at its center 𝑣, then 𝑇 = 𝑇 𝑣 . To illustrate the pruning process, consider the tree 𝑇𝑣 in Figure 5.7(a). The vertices 𝑢, 𝑤, and 𝑥 are branch vertices at maximum distance 2 from 𝑣. Since 𝐿 1 (𝑢) = ∅ and |𝐿 2 (𝑢)| = 1, we delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. Since |𝐿 1 (𝑤)| = 2, we delete 𝐷 (𝑤) and attach a path of length 1 to 𝑤. Similarly, since |𝐿 1 (𝑥)| = 2, we delete 𝐷 (𝑥) and attach a path of length 1 to 𝑥. This pruning of 𝑇𝑣 at 𝑢, 𝑤, and 𝑥 produces the intermediate tree 𝑇𝑣′ shown in Figure 5.7(b). In tree 𝑇𝑣′ , the vertices 𝑦 and 𝑧 are branch vertices at maximum distance 1 from 𝑣. Since 𝐿 1 (𝑦) = 𝐿 2 (𝑦) = ∅, we delete 𝐷 (𝑦) and attach a path of length 3 to 𝑦. Since |𝐿 1 (𝑧)| = 1, we delete 𝐷 (𝑧) and attach a path of length 1 to 𝑧. This pruning of 𝑇𝑣′ at 𝑦 and 𝑧 produces the pruning 𝑇 𝑣 of 𝑇𝑣 . The following characterization of the sets A (𝑇) and N (𝑇) for an arbitrary tree 𝑇 is a slightly modified version of the one presented in [604]. Theorem 5.27 ([604]) If 𝑣 is a vertex of a tree 𝑇 and 𝐿 𝑖 (𝑣) equals 𝐿 𝑖 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 , then the following hold: (a) 𝑣 ∈ A (𝑇) if and only if |𝐿 1 (𝑣)| ≥ 2. (b) 𝑣 ∈ N (𝑇) if and only if 𝐿 1 (𝑣) = ∅ and 𝐿 2 (𝑣) ≠ ∅. To illustrate Theorem 5.27, note that in the pruning 𝑇 𝑣 of the tree 𝑇𝑣 in Figure 5.7(a), |𝐿 0 (𝑣)| = 1, |𝐿 1 (𝑣)| = 3, and |𝐿 2 (𝑣)| = 2. Since |𝐿 1 (𝑣)| ≥ 2, 𝑣 ∈ A (𝑇) by Theorem 5.27. We conclude this section by briefly mentioning a couple of related topics. Note that if a graph has a unique minimum dominating set, then each vertex either belongs to the unique 𝛾-set or to no 𝛾-set. Gunther et al. [369] in 1994 characterized the trees having a unique 𝛾-set as follows. We note that Fischermann [300] generalized this result to block graphs in 2001.

Chapter 5. Domination in Trees

114 𝑣

𝑣

𝑦

𝑧

𝑢

𝑤

𝑦 𝑥

𝑧

𝑢

𝑤

𝑥

(b) 𝑇𝑣′

(a) 𝑇𝑣

𝑣

𝑦

𝑧

𝑢

𝑤

(c) 𝑇 𝑣

Figure 5.7 The pruning 𝑇 𝑣 of the tree 𝑇𝑣

Theorem 5.28 ([369]) If 𝑇 is a tree of order 𝑛 ≥ 3, then the following conditions are equivalent: (a) 𝑇 has a unique 𝛾-set. (b) 𝑇 has a 𝛾-set 𝑆 for which every vertex in 𝑆 has at least two 𝑆-external private neighbors. (c) 𝑇 has a 𝛾-set 𝑆 for which every vertex 𝑥 ∈ 𝑆 has the property 𝛾(𝑇 − 𝑥) > 𝛾(𝑇). Considering the vertices of a graph that appear in at least one 𝛾-set, in 2002 Fricke et al. [311] defined a partition of the vertices of a graph 𝐺 into two sets as follows. A vertex is called 𝛾-good if it is contained in some 𝛾-set of 𝐺 and 𝛾-bad, otherwise. Further, a graph 𝐺 is called 𝛾-excellent if every vertex of 𝐺 is 𝛾-good. Terminology for good, bad, and excellent is defined similarly for other domination parameters. The families of 𝛾-excellent trees, 𝛾t -excellent trees, and 𝑖-excellent trees were also studied in [127], [420], and [456], respectively.

5.2.5

Domination and Packing in Trees

Recall that a packing in a graph 𝐺 is a set of vertices whose closed neighborhoods are pairwise disjoint, and the packing number 𝜌(𝐺) is the maximum cardinality of a packing in 𝐺. In Theorem 4.45 of Chapter 4, we showed that if 𝐺 is a graph, then 𝛾(𝐺) ≥ 𝜌(𝐺). That is, the packing number is a lower bound on the domination number for general graphs. In 1975 Meir and Moon [589] proved that these numbers are equal for trees.

Section 5.3. Total Domination in Trees

115

Theorem 5.29 (Meir-Moon Theorem [589]) For every tree 𝑇, 𝜌(𝑇) = 𝛾(𝑇). Proof We proceed by induction on the order 𝑛 of a tree 𝑇 to show that 𝜌(𝑇) = 𝛾(𝑇). If 𝑛 ≤ 3, then 𝜌(𝑇) = 𝛾(𝑇) = 1. This establishes the base cases. Let 𝑛 ≥ 4 and assume that if 𝑇 ′ is a tree of order 𝑛′ , where 1 ≤ 𝑛′ < 𝑛, then 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Let 𝑇 be a tree of order 𝑛. By Theorem 4.45, we have 𝛾(𝑇) ≥ 𝜌(𝑇). If diam(𝑇) = 2, then 𝑇 is a star and 𝜌(𝑇) = 𝛾(𝑇) = 1. If diam(𝑇) = 3, then 𝑇 is a double star and 𝜌(𝑇) = 𝛾(𝑇) = 2. Hence, we may assume that diam(𝑇) ≥ 4, and so 𝑛 ≥ 5, for otherwise the desired result follows. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 4. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Suppose that deg𝑇 (𝑣 2 ) = 2. In this case, we let 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ), where 𝑇𝑣2 is the maximal subtree of 𝑇 rooted at 𝑣 2 . The resulting tree 𝑇 ′ has order 𝑛′ , where 2 ≤ 𝑛′ ≤ 𝑛 − 3. Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it the vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Every maximum packing in 𝑇 ′ can be extended to a packing in 𝑇 by adding to it the vertex 𝑣 0 , and so 𝜌(𝑇) ≥ 𝜌(𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have that 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Therefore, 𝜌(𝑇) ≥ 𝜌(𝑇 ′ ) + 1 = 𝛾(𝑇 ′ ) + 1 ≥ 𝛾(𝑇) ≥ 𝜌(𝑇).

(5.7)

Consequently, we must have equality throughout Inequality (5.7). In particular, 𝛾(𝑇) = 𝜌(𝑇). Hence, we may assume that deg𝑇 (𝑣 2 ) ≥ 3, for otherwise the desired result holds. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ), where 𝑇𝑣1 is the maximal subtree of 𝑇 rooted at 𝑣 1 . The resulting tree 𝑇 ′ has order 𝑛′ , where 4 ≤ 𝑛′ ≤ 𝑛 − 2. Let 𝑆 be a maximum packing in 𝑇 ′ , and so 𝜌(𝑇 ′ ) = |𝑆|. If 𝑣 2 ∈ 𝑆, then since every descendant of 𝑣 2 is at distance 1 or 2 from 𝑣 2 , we note that no descendant of 𝑣 2 belongs to the set 𝑆. In this case, if 𝑢 1 is an arbitrary child of 𝑣 2 different from 𝑣 1 , then we can replace the vertex 𝑣 2 in 𝑆 with the vertex 𝑢 1 to yield a new maximum packing in 𝑇 ′ . Hence, we can choose the maximum packing 𝑆 in 𝑇 ′ so that 𝑣 2 ∉ 𝑆. With this choice of 𝑆, the set 𝑆 ∪ {𝑣 0 } is a packing in 𝑇, implying that 𝜌(𝑇) ≥ |𝑆| + 1 = 𝜌(𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have that 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Furthermore, every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Therefore, once again Inequality (5.7) holds, implying that 𝛾(𝑇) = 𝜌(𝑇).

5.3

Total Domination in Trees

In this section, we present several upper and lower bounds on the total domination number of a tree, we present a discussion of vertices in all or no 𝛾t -sets in trees, and we discuss trees having unique 𝛾t -sets. We also present Rall’s theorem that for all trees, the total domination number equals the open packing number.

5.3.1

Total Domination Bounds in Trees

Recall that in Section 4.3.2, we presented the result due to Cockayne et al. [182] showing that the total domination of a connected graph of order at least 3 is at most

Chapter 5. Domination in Trees

116

two-thirds its order. We also presented the characterization due to Brigham et al. [117] of the graphs that achieve equality in this upper bound. Here we restate these results in the special case of trees. Theorem 5.30 ([117, 182]) If 𝑇 is tree of order 𝑛 ≥ 3, then 𝛾(𝑇) ≤ equality if and only if 𝑇 = 𝑇 ′ ◦ 𝑃2 is the 2-corona of some tree 𝑇 ′ .

2 3 𝑛,

with

5.3.2 Total Domination Bounds Involving the Number of Leaves In 2006 Chellali and Haynes [153] were the first to establish a lower bound on the total domination number of a tree in terms of the order, number of leaves, and number of support vertices in the tree. In order to state their result, we define the family F0 of trees as follows. Definition 5.31 Let F0 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 , for 𝑘 ≥ 1, of trees, where the tree 𝑇1 is the path 𝑃4 with support vertices 𝑥 and 𝑦, and where the tree 𝑇 = 𝑇𝑘 . Further, if 𝑘 ≥ 2, then for each 𝑖, where 2 ≤ 𝑖 ≤ 𝑘, the tree 𝑇𝑖 can be obtained from the tree 𝑇𝑖−1 by applying one of the Operations O1 , O2 , and O3 defined as follows. Let 𝐴(𝑇1 ) = {𝑥, 𝑦} and 𝐻 be a path 𝑃4 with non-leaf vertices labeled 𝑢 and 𝑣. Operation O1 . Add a new vertex to 𝑇𝑖−1 and join it to a vertex of 𝐴(𝑇𝑖−1 ). Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ). Operation O2 . Add a copy of 𝐻 to 𝑇𝑖−1 and add an edge from a leaf of 𝐻 to a leaf of 𝑇𝑖−1 . Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ) ∪ {𝑢, 𝑣}. Operation O3 . Add a copy of 𝐻 to 𝑇𝑖−1 and add a new vertex 𝑤, an edge from 𝑤 to a support vertex 𝑢 of 𝐻, and an edge from 𝑤 to a leaf of 𝑇𝑖−1 . Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ) ∪ {𝑢, 𝑣}. We are now in a position to state the result from [153], albeit without proof. Theorem 5.32 ([153]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾t (𝑇) ≥ , 2 with equality if and only if 𝑇 ∈ F0 . Theorem 5.32 was generalized by Hajian et al. [373] in 2019, who defined the families F01 , F02 , F03 , and F0★ of trees as follows. Definition 5.33 Let F01 , F02 , and F03 be the families of trees defined as follows. • Let F01 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 by adding 𝑎 ≥ 1 new vertices and joining all of them to the same leaf of 𝑇 ′ . • Let F02 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 that contains a support vertex 𝑥 all of whose neighbors, except for exactly one neighbor, are leaves in 𝑇 ′ by removing all leaf neighbors of 𝑥. • Let F03 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 by adding a double star 𝑄 and adding an edge from a leaf of 𝑄 to a vertex of degree at least 2 in 𝑇 ′ .

Section 5.3. Total Domination in Trees

117

Definition 5.34 Let F0★ be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1 and where the tree 𝑇1 ∈ F01 ∪ F02 ∪ F03 and the tree 𝑇 = 𝑇𝑘 . Further, if 𝑘 ≥ 2, then for each 𝑖, where 2 ≤ 𝑖 ≤ 𝑘, the tree 𝑇𝑖 can be obtained from the tree 𝑇𝑖−1 by applying Operation O★ defined below. Operation O★. Add a double star 𝑄 to 𝑇𝑖−1 and add an edge between a leaf of 𝑄 and a leaf of 𝑇𝑖−1 . The following result generalizes Theorem 5.32. Theorem 5.35 ([373]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾t (𝑇) ≥ , 2 with equality if and only if 𝑇 ∈ F0 ∪ F0★. We remark that in [373], Theorem 5.35 for trees was generalized to show that if 𝐺 is aconnected graph of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves, then  𝛾t (𝐺) ≥ 12 (𝑛 − ℓ + 2) − 𝑘. The graphs 𝐺 that achieve equality for this generalized bound were also characterized in [373]. In 2004 Chellali and Haynes [152] established the following upper bound on the total domination number of a tree in terms of the number of support vertices. Theorem 5.36 ([152]) If 𝑇 is a tree of order 𝑛 ≥ 3 with 𝑠 support vertices, then 𝛾t (𝑇) ≤ 12 (𝑛 + 𝑠). In 2015 Krzywkowski [548] gave an algorithmic procedure to build trees attaining the upper bound of Theorem 5.36. Since the number of support vertices in a tree is at most the number of leaves in the tree, the following is an immediate consequence of Theorem 5.36. Corollary 5.37 ([152, 153]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, then 𝛾t (𝑇) ≤ 12 (𝑛 + ℓ). A characterization of the trees achieving equality in the bound of Corollary 5.37 can be found in the book [490]. Henning and Yeo [490] stated that this characterization was adapted from a result due to Chen and Sohn [156].

5.3.3

Vertices in All or No Minimum Total Dominating Sets in Trees

In this section, we are concerned with the vertices in all or no minimum total dominating sets. Unless otherwise stated, we use the same notation here as in the previous section.  Let these sets be defined as follows: • A𝑡 (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in every 𝛾t -set of 𝐺 , and  • N𝑡 (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in no 𝛾t -set of 𝐺 .

Chapter 5. Domination in Trees

118 For 𝑗 ∈ [3] 0 , we define

 𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿 (𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 4) . Again, the pruning of a tree is performed with respect to the root. For total domination, the pruning process is as follows: • If |𝐿 2 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If |𝐿 1 (𝑢)| ≥ 1, 𝐿 2 (𝑢) = ∅, and |𝐿 3 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If |𝐿 1 (𝑢)| ≥ 1 and 𝐿 2 (𝑢) = 𝐿 3 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 1 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = ∅ and |𝐿 3 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 3 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = 𝐿 3 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 4 to 𝑢. The steps of the pruning process are repeated as before until a tree 𝑇 𝑣 is obtained with deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇 𝑣 ) \ {𝑣}. To illustrate the pruning process for total domination, consider the tree 𝑇𝑣 in Figure 5.8(a). The vertices 𝑢 and 𝑤 are branch vertices at maximum distance 2 from 𝑣. Since |𝐿 2 (𝑢)| = 1, we delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. Since |𝐿 1 (𝑤)| = 2 and 𝐿 2 (𝑤) = 𝐿 3 (𝑤) = ∅, we delete 𝐷 (𝑤) and attach a path of length 1 to 𝑤. This pruning of 𝑇𝑣 at 𝑢 and 𝑤 produces the intermediate tree 𝑇𝑣′ shown in Figure 5.8(b). In tree 𝑇𝑣′ , the vertices 𝑡, 𝑥, 𝑦, and 𝑧 are branch vertices at maximum distance 1 from 𝑣. Since |𝐿 2 (𝑥)| = 1, we delete 𝐷 (𝑥) and attach a path of length 2 to 𝑥. Since |𝐿 1 (𝑦)| = 1, 𝐿 2 (𝑦) = ∅, and |𝐿 3 (𝑦)| = 1, we delete 𝐷 (𝑦) and attach a path of length 2 to 𝑦. Since 𝐿 1 (𝑡) = 𝐿 2 (𝑡) = 𝐿 3 (𝑡) = ∅, we delete 𝐷 (𝑡) and attach a path of length 4 to 𝑡. Since 𝐿 1 (𝑧) = 𝐿 2 (𝑧) = ∅ and |𝐿 3 (𝑧)| = 2 ≥ 1, we delete 𝐷 (𝑧) and attach a path of length 3 to 𝑧. This pruning of 𝑇𝑣′ at 𝑡, 𝑥, 𝑦, and 𝑧 produces the pruning 𝑇 𝑣 of 𝑇𝑣 . The following characterization of the sets A𝑡 (𝑇) and N𝑡 (𝑇) for an arbitrary tree 𝑇 is given by Cockayne et al. [197] in 2003. Theorem 5.38 ([197]) If 𝑣 is a vertex of a tree 𝑇 and 𝐿 𝑖 (𝑣) equals 𝐿 𝑖 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 , then the following hold: (a) 𝑣 ∈ A𝑡 (𝑇) if and only if 𝑣 is a support vertex or |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| ≥ 2. (b) 𝑣 ∈ N𝑡 (𝑇) if and only if 𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣) = ∅. To illustrate Theorem 5.38, note that in the pruning 𝑇 𝑣 of the tree 𝑇𝑣 in Figure 5.8(a), |𝐿 0 (𝑣)| = |𝐿 1 (𝑣)| = |𝐿 2 (𝑣)| = 1 and |𝐿 3 (𝑣)| = 2, that is, |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| = 2. Hence, by Theorem 5.38, we have 𝑣 ∈ A𝑡 (𝑇).

5.3.4

Unique Minimum Total Dominating Sets in Trees

We now turn our attention to trees having a unique 𝛾t -set. We need to recall the following terminology from Section 5.2.4. As before, we denote the set of leaves in

Section 5.3. Total Domination in Trees

119

𝑣

𝑦

𝑡

𝑧

𝑣

𝑥

𝑢

𝑦 𝑤

𝑡

𝑧

𝑢

𝑥 𝑤

(b) 𝑇𝑣′

(a) 𝑇𝑣

𝑣

𝑦

𝑡

𝑧

𝑥

(c) 𝑇 𝑣

Figure 5.8 The pruning 𝑇 𝑣 of the tree 𝑇𝑣

𝑇 = 𝑇𝑣 distinct from 𝑣 by 𝐿 (𝑣) that is, 𝐿 (𝑣) = 𝐷 (𝑣) ∩ 𝐿(𝑇), where 𝐿(𝑇) is the set of leaves of 𝑇. For 𝑗 ∈ [3] 0 , let  𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿(𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 4) , and let 𝐿 𝑗 (𝑣) equal 𝐿 𝑗 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 . In 2002 Haynes and Henning [421] characterized trees that have a unique 𝛾t -set as follows. Theorem 5.39 ([421]) If 𝑇 is a nontrivial tree, then the following conditions are equivalent: (a) 𝑇 has a unique 𝛾t -set. (b) 𝑇 has a 𝛾t -set 𝑆 for which every vertex 𝑣 ∈ 𝑆 is either a support vertex or satisfies |pn[𝑣, 𝑆] | ≥ 2. (c) 𝑇 has a 𝛾t -set 𝑆 for which 𝛾t (𝑇 − 𝑣) > 𝛾t (𝑇) for every 𝑣 ∈ 𝑆 that is not a support vertex. (d) For every vertex 𝑣 ∈ 𝑉 (𝑇), 𝑣 is a support vertex or |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| ≠ 1. In addition to providing the three equivalent conditions of Theorem 5.39 for a tree to have a unique 𝛾t -set, Haynes and Henning [421] gave a constructive characterization of such trees. We note that these trees were also independently characterized by Fischermann in [301] in 2004.

120

Chapter 5. Domination in Trees

5.3.5 Total Domination and Open Packing in Trees Recall that an open packing in a graph 𝐺 is a set of vertices whose open neighborhoods are pairwise disjoint. Thus, if 𝑆 is an open packing in 𝐺, then N(𝑢) ∩ N(𝑣) = ∅ for all 𝑢, 𝑣 ∈ 𝑆. The open packing number 𝜌 o (𝐺) is the maximum cardinality of an open packing in 𝐺. Recall also that in Theorem 4.46 in Chapter 4, we showed that if 𝐺 is an isolate-free graph, then the open packing number is a lower bound on the total domination number of 𝐺. We restate this result for trees. Proposition 5.40 If 𝑇 is a nontrivial tree, then 𝜌 o (𝑇) ≤ 𝛾t (𝑇). In 2005 Rall [644] showed that 𝜌 o (𝑇) = 𝛾t (𝑇) for all nontrivial trees 𝑇, using an elegant proof involving categorical products. The proof we present uses elementary properties of a tree along similar lines to the proof of Theorem 5.29 showing that the domination and packing numbers are equal for trees. We first observe the following elementary properties of a TD-set in a graph. Observation 5.41 The following properties hold in an isolate-free graph 𝐺: (a) Every TD-set in 𝐺 contains the set of support vertices of 𝐺. (b) If 𝐺 is connected and diam(𝐺) ≥ 3, then there exists a 𝛾t -set of 𝐺 that contains no leaf of 𝐺. Theorem 5.42 ([644]) If 𝑇 is a nontrivial tree, then 𝜌 o (𝑇) = 𝛾t (𝑇). Proof We proceed by induction on the order 𝑛 ≥ 2 of a tree 𝑇 to show that 𝜌 o (𝑇) = 𝛾t (𝑇). If 𝑛 ≤ 4, then 𝜌 o (𝑇) = 2 = 𝛾t (𝑇). This establishes the base cases. Let 𝑛 ≥ 5 and assume that if 𝑇 ′ is a tree of order 𝑛′ , where 2 ≤ 𝑛′ < 𝑛, then 𝜌 o (𝑇 ′ ) = 𝛾(𝑇 ′ ). Let 𝑇 be a tree of order 𝑛. Suppose that 𝑇 contains a strong support vertex 𝑣. Let 𝑢 and 𝑤 be two leaf neighbors of 𝑣, and consider the tree 𝑇 ′ = 𝑇 − 𝑢. Every TD-set of 𝑇 ′ contains the support vertex 𝑣, and is therefore a TD-set of 𝑇, implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ). Every maximum open packing in 𝑇 ′ is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇), and applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). These observations imply that 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ) ≥ 𝛾t (𝑇).

(5.8)

Consequently, we must have equality throughout Inequality (5.8). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that 𝑇 contains no strong support vertices, for otherwise the equality follows. With this assumption and since 𝑛 ≥ 5, we note that diam(𝑇) ≥ 4. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 4. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the vertex 𝑣 1 is not a strong support vertex, we have deg𝑇 (𝑣 1 ) = 2. Suppose that 𝑣 2 is a support vertex. Let 𝑢 be the (unique) leaf neighbor of 𝑣 2 , and consider the tree 𝑇 ′ = 𝑇 − 𝑢. By Observation 5.41, there is a 𝛾t -set 𝐷 ′ of 𝑇 ′ that contains no leaf of 𝑇 ′ , implying that {𝑣 1 , 𝑣 2 } ⊆ 𝐷 ′ . The set 𝐷 ′ is therefore a TD-set of 𝑇, and so 𝛾t (𝑇) ≤ |𝐷 ′ | = 𝛾t (𝑇 ′ ). Every maximum open packing in 𝑇 ′ is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ). Applying the inductive hypothesis to 𝑇 ′ ,

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121

we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). Therefore, once again equality in Inequality (5.8) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that 𝑣 2 is not a support vertex, for otherwise the desired result holds. Hence, every child of 𝑣 2 is a support vertex of degree 2 in 𝑇. Suppose that deg𝑇 (𝑣 2 ) ≥ 3. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 , and let 𝑢 0 be the child of 𝑢 1 . We now consider the tree 𝑇 ′ = 𝑇 − {𝑢 0 , 𝑢 1 }. By Observation 5.41, we can choose a 𝛾t -set 𝐷 ′ of 𝑇 ′ so that {𝑣 1 , 𝑣 2 } ⊆ 𝐷 ′ . Since the set 𝐷 ′ ∪ {𝑢 1 } is a TD-set of 𝑇, we have 𝛾t (𝑇) ≤ |𝐷 ′ | + 1 = 𝛾t (𝑇 ′ ) + 1. Let 𝑆 be a maximum open packing in 𝑇 ′ , and so 𝜌 o (𝑇 ′ ) = |𝑆|. If 𝑣 2 ∈ 𝑆, then 𝑣 0 does not belong to 𝑆. In this case, we can replace the vertex 𝑣 2 in 𝑆 with the vertex 𝑣 0 to yield a new maximum open packing in 𝑇 ′ . Hence, we can choose the maximum open packing 𝑆 in 𝑇 ′ so that 𝑣 2 ∉ 𝑆. With this choice of 𝑆, the set 𝑆 ∪ {𝑢 0 } is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ |𝑆| + 1 = 𝜌 o (𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). These observations imply that (5.9) 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 1 = 𝛾t (𝑇 ′ ) + 1 ≥ 𝛾t (𝑇). Consequently, we must have equality throughout Inequality (5.9). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that deg𝑇 (𝑣 2 ) = 2. Suppose that deg𝑇 (𝑣 3 ) = 2. In this case, if 𝑛 = 5, then 𝑇 = 𝑃5 and 𝜌 o (𝑇) = 𝛾(𝑇) = 3. Hence, we may assume that 𝑛 ≥ 6. We now consider the tree 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 , 𝑣 3 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑣 1 and 𝑣 2 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Every maximum open packing in 𝑇 ′ can be extended to an open packing in 𝑇 by adding to it the vertices 𝑣 0 and 𝑣 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). These observations imply that (5.10) 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2 = 𝛾t (𝑇 ′ ) + 2 ≥ 𝛾t (𝑇). Consequently, we must have equality throughout Inequality (5.10). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that deg𝑇 (𝑣 3 ) ≥ 3. Suppose that the vertex 𝑣 3 has a descendant 𝑢 0 at distance 3 that is different from 𝑣 0 . Let 𝑢 0 𝑢 1 𝑢 2 𝑣 3 be the path from 𝑢 0 to 𝑣 3 . Using arguments similar to those used for vertices 𝑣 1 and 𝑣 2 , we may assume that deg𝑇 (𝑢 1 ) = deg𝑇 (𝑢 2 ) = 2. We now consider the tree 𝑇 ′ = 𝑇 − {𝑢 0 , 𝑢 1 , 𝑢 2 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑢 1 and 𝑢 2 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 3 ∈ 𝑆, then we can replace 𝑣 3 in 𝑆 with the vertex 𝑣 1 . Hence, we can choose the set 𝑆 so that 𝑣 3 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertices 𝑢 0 and 𝑢 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). Therefore, once again Inequality (5.10) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Thus, we may assume that every child of 𝑣 3 different from 𝑣 2 is a leaf or a support vertex of degree 2. Suppose that the vertex 𝑣 3 has a child 𝑢 2 that is a support vertex. Let 𝑢 1 be the leaf neighbor of 𝑢 2 . We now consider the tree 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑣 1 and 𝑣 2 , implying that

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𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 3 ∈ 𝑆, then we can replace 𝑣 3 in 𝑆 with the vertex 𝑢 1 . Hence, we can choose the set 𝑆 so that 𝑣 3 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertices 𝑣 0 and 𝑣 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). Therefore, once again Inequality (5.10) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that every child of 𝑣 3 different from 𝑣 2 is a leaf, implying that deg𝑇 (𝑣 3 ) = 3. Let 𝑢 2 be the leaf child of 𝑣 3 . We now consider the tree 𝑇 ′ = 𝑇 − 𝑣 0 . Every 𝛾t -set of 𝑇 ′ contains all of its support vertices, and therefore contains both 𝑣 2 and 𝑣 3 . Such a set can therefore be extended to a TD-set of 𝑇 by adding to it the vertex 𝑣 1 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 1. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 2 ∈ 𝑆, then we can replace 𝑣 2 in 𝑆 with the vertex 𝑢 2 . Hence, we can choose the set 𝑆 so that 𝑣 2 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertex 𝑣 0 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 1. Therefore, once again Inequality (5.9) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇).

5.4

Independent Domination in Trees

In this section, we present two upper bounds on the independent domination number of a tree, and we discuss trees having unique minimum dominating sets.

5.4.1

Independent Domination Bounds in Trees

Recall that in Section 4.3.3, we presented the result due to Favaron √ [274] in 1988 that if 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛 and this bound is tight. In the special case of trees, this bound is achievable when 𝐺 is the path 𝑃4 . However, for trees of larger order, this general upper bound is far from best possible. Since every bipartite isolate-free graph is the union of two maximal independent sets, each of which forms a dominating set in the graph, we observe that the independent domination number of a bipartite isolate-free graph is at most one-half its order. In the special case when the bipartite graph is a tree, we have the following result. Proposition 5.43 If 𝑇 is a tree of order 𝑛 ≥ 2, then 𝑖(𝑇) ≤ 12 𝑛. The upper bound in Proposition 5.43 is tight. For example, if 𝑇 = 𝑇 ′ ◦ 𝐾1 is the corona of some tree 𝑇 ′ of order 𝑘, then 𝑇 has order 𝑛 = 2𝑘 and 𝑖(𝑇) = 𝑘 = 12 𝑛. As a further example, if 𝑇 is a double star 𝑆(𝑘, 𝑘), where 𝑘 ≥ 1, then 𝑇 has order 𝑛 = 2(𝑘 + 1) and 𝑖(𝑇) = 𝑘 + 1 = 12 𝑛. In 1992 Favaron [275] proved the following upper bound on the independent domination number of a tree and characterized the trees attaining it. This bound was first conjectured in 1980 by McFall and Nowakowski [587]. Theorem 5.44 ([275]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝑖(𝑇) ≤ 1 3 (𝑛 + ℓ).

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The bound in Theorem 5.44 is achieved, for example, by the path 𝑃3𝑘+1 when 𝑘 ≥ 1. Other families achieving equality include, for example, when 𝑇 = 𝑃2𝑘 ◦ 𝐾1 is the corona of a path 𝑃2𝑘 , where 𝑘 ≥ 1. In this example, 𝑇 has order 𝑛 = 4𝑘 with ℓ = 2𝑘 leaves, and satisfies 𝑖(𝑇) = 2𝑘 = 31 (𝑛 + ℓ). The full list of trees attaining this bound is provided in [275]. We remark that the proof of Theorem 5.44 provided by Favaron [275] is an inductive proof that includes the characterization of the trees attaining this bound.

5.4.2

Unique Minimum Independent Dominating Sets in Trees

We conclude this section with a characterization of trees having a unique minimum independent dominating set due to Hedetniemi [437] in 2017. For a graph 𝐺 with unique 𝑖-set 𝐼, let  X(𝐼) = 𝑣 ∈ 𝐼 : |epn(𝑣, 𝐼)| ≥ 2 ,  Y (𝐼) = 𝑣 ∈ 𝐼 : |epn(𝑣, 𝐼)| = 0 , and  Z(𝐺) = 𝑣 ∈ 𝑉 (𝐺) \ 𝐼 : |N(𝑣) ∩ 𝐼 | ≥ 2 and 𝑖(𝐺 − N[𝑣]) > 𝑖(𝐺) . Let 𝐺 1 and 𝐺 2 be graphs, each of which has a unique 𝑖-set, denoted by 𝐼1 and 𝐼2 , respectively. We now define a set of operations on graphs 𝐺 1 and 𝐺 2 . Operation 1. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 𝑗 ∈ 𝑉 (𝐺 𝑗 ) \ 𝐼 𝑗 for 𝑗 ∈ [2]. Operation 2. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding a new vertex 𝑢 and edges 𝑢𝑣 1 and 𝑢𝑣 2 , where 𝑣 𝑗 ∈ X(𝐼 𝑗 ) for 𝑗 ∈ [2]. Operation 3. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding a new vertex 𝑢 and edges 𝑢𝑣 1 and 𝑢𝑣 2 , where 𝑣 1 ∈ X(𝐼1 ) and 𝑣 2 ∈ Y (𝐼2 ). Operation 4. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 1 ∈ 𝑉 (𝐺 1 ) \ 𝐼1 is adjacent to at least two vertices in 𝐼1 , and 𝑢 2 ∈ X(𝐼2 ). Operation 5. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 1 ∈ Z(𝐺 1 ) and 𝑢 2 ∈ Y (𝐼2 ). We now state the characterization of trees having a unique 𝑖-set. To simplify the statement, we say that a star of order at least 3 is a big star. Theorem 5.45 ([437]) A tree 𝑇 has an unique 𝑖-set if and only if 𝑇 can be constructed from a disjoint union of isolated vertices and big stars by a finite sequence of Operations 1 through 5.

5.5

Equality of Domination Parameters

For any two graph parameters 𝜆 and 𝜇, a graph 𝐺 is said to be a (𝜆, 𝜇)-graph if 𝜆(𝐺) = 𝜇(𝐺). The problem of characterizing graphs for which two related domination parameters are equal has received much interest. The family of (𝛾, 𝑖)-graphs are especially challenging and to date only subsets of this family have been characterized. The family of (𝛾, 𝑖)-trees was first characterized by Harary and Livingston [385] in 1986, but this characterization is rather complex. Recall from Section 5.2.4

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that A (𝑇) denotes the set of vertices in a tree 𝑇 that are contained in every 𝛾-set of 𝑇. Similarly, let A𝑖 (𝑇) and At (𝑇) denote the sets of vertices which are contained in every 𝑖-set and in every 𝛾t -set of 𝑇, respectively. In 2000 Cockayne et al. [184] gave a different characterization in terms of the sets A (𝑇) and A𝑖 (𝑇). In 2006 Dorfling et al. [240] used a labeling method to provide a simple constructive characterization of the (𝛾, 𝑖)-trees. This labeling method also yields the (𝛾, 𝛾t )-trees. We present the labeling methods and constructions of these families here.

5.5.1

(𝜸, 𝒊)-trees

By Theorem 5.29, the (𝛾, 𝑖)-trees are precisely the (𝜌, 𝑖)-trees. Thus, the constructive characterization given by Dorfling et al. [240] of (𝜌, 𝑖)-trees yields the (𝛾, 𝑖)-trees. The key to their characterization is a labeling of the vertices that indicates the role each vertex plays in the maximum packings and minimum independent dominating sets of a tree. To aid in the construction, define a (𝜌, 𝑖)-labeling of a tree 𝑇 to be a partition 𝑆 = {𝑆 𝐴, 𝑆 𝐵 , 𝑆𝐶 , 𝑆 𝐷 } of 𝑉 such that 𝑆 𝐴 ∪ 𝑆 𝐷 is an 𝑖-set, 𝑆𝐶 ∪ 𝑆 𝐷 is a 𝜌-set, and |𝑆 𝐴 | = |𝑆𝐶 |. The label or status of a vertex 𝑣, denoted sta(𝑣), is the letter 𝑥 ∈ {𝐴, 𝐵, 𝐶, 𝐷} such that 𝑣 ∈ 𝑆 𝑥 . A labeled graph is simply one where each vertex is labeled with either 𝐴, 𝐵, 𝐶, or 𝐷. We will need two lemmas from [240]. The following lemma can be easily proven. If 𝐺 has a (𝜌, 𝑖)-labeling, then 𝜌(𝐺) = |𝑆𝐶 ∪ 𝑆 𝐷 | = |𝑆 𝐴 ∪ 𝑆 𝐷 | = 𝑖(𝐺). Conversely, if 𝜌(𝐺) = 𝑖(𝐺), then let 𝑋 be an 𝑖-set of 𝐺, 𝑌 be a 𝜌-set of 𝐺, and create a (𝜌, 𝑖)-labeling by letting 𝑆 𝐴 = 𝑋 \ 𝑌 , 𝑆 𝐵 = 𝑉 (𝐺) \ (𝑋 ∪ 𝑌 ), 𝑆𝐶 = 𝑌 \ 𝑋, and 𝑆 𝐷 = 𝑋 ∩ 𝑌 . Lemma 5.46 ([240]) A graph is a (𝜌, 𝑖)-graph if and only if it has a (𝜌, 𝑖)-labeling. Lemma 5.47 ([240]) Consider a (𝜌, 𝑖)-labeling. If 𝑣 ∈ 𝑆 𝐴 (respectively, 𝑆𝐶 ), then 𝑣 is adjacent to exactly one vertex of 𝑆𝐶 (respectively, 𝑆 𝐴), and to no vertex of 𝑆 𝐷 . Let L be the minimum family of labeled trees that: (a) contains (𝑃1 , 𝑆1 ), where the single vertex has status 𝐷, and contains (𝑃2 , 𝑆2 ), where one vertex has status 𝐴 and the other status 𝐶; and (b) is closed under the six Operations T𝑗 , where 𝑗 ∈ [6], listed below, which extend the tree 𝑇 by attaching a tree to the vertex 𝑦 ∈ 𝑉 (𝑇), called the attacher. Operation T1 . Assume sta(𝑦) ∈ {𝐴, 𝐷}. Add a vertex 𝑥 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵. Operation T2 . Assume sta(𝑦) ∈ {𝐴, 𝐵}. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵 and sta(𝑤) = 𝐷. Operation T3 . Assume sta(𝑦) = 𝐵. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐴 and sta(𝑤) = 𝐶. Operation T4 . Assume sta(𝑦) ∈ {𝐵, 𝐶}. Add a path 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = 𝐴, and sta(𝑧) = 𝐶. Operation T5 . Assume sta(𝑦) = 𝐴. Add a path 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = 𝐶, and sta(𝑧) = 𝐴. Operation T6 . Assume sta(𝑦) = 𝐵. Add a path 𝑣 𝑢 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = sta(𝑣) = 𝐶, and sta(𝑧) = sta(𝑢) = 𝐴.

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These operations are illustrated in Figure 5.9.

𝐴 or 𝐷

𝐴 or 𝐵

𝐵

(a) T1

𝐵

𝐵 or 𝐶

𝐶

𝐴

𝐵

𝐷

(b) T2

(c) T3

𝐴

𝐵

𝐵

𝐴

𝐶

𝐴

𝐶

𝐶

𝐴

(d) T4

𝐶

𝐴

𝐵

𝐵

(f) T6

(e) T5

Figure 5.9 The six T𝑗 operations

Theorem 5.48 ([240]) A labeled tree is a (𝜌, 𝑖)-tree if and only if it is in L. Proof It is straightforward to check that every element of L is a (𝜌, 𝑖)-tree. The proof that every (𝜌, 𝑖)-tree 𝑇 is in L is by induction on the order of 𝑇. By Lemma 5.46, 𝑇 has an (𝜌, 𝑖)-labeling 𝑆. For the base case, consider any star 𝑇. It follows easily that there is a construction of (𝑇, 𝑆) for any (𝜌, 𝑖)-labeling 𝑆 by starting with either the 𝑃1 or the 𝑃2 and repeatedly using T1 . So fix a (𝜌, 𝑖)-tree (𝑇, 𝑆), and assume that any smaller (𝜌, 𝑖)-tree is in L. We may assume that diam(𝑇) ≥ 3, since otherwise 𝑇 is a star, and the result holds. Let 𝐼 = 𝑆 𝐴 ∪ 𝑆 𝐷 and 𝑃 = 𝑆𝐶 ∪ 𝑆 𝐷 . We proceed further with the following claim. Claim 5.48.1 Let 𝑢 be any vertex of a rooted (𝜌, 𝑖)-tree (𝑇, 𝑆) other than the root, with 𝑣 the parent of 𝑢, and let (𝑇 ′ , 𝑆 ′ ) be the labeled tree formed by the deletion of 𝑇𝑢 . Suppose that (𝑇, 𝑆) can be obtained from (𝑇 ′ , 𝑆 ′ ) by attaching 𝑇𝑢 to 𝑣 using an Operation T𝑗 . Then (𝑇, 𝑆) ∈ L except possibly if 𝑗 = 3 and 𝑣 is not dominated by 𝐼 \ {𝑢}. Proof We want to show that (𝑇 ′ , 𝑆 ′ ) is a (𝜌, 𝑖)-tree, since then, by the inductive hypothesis, (𝑇 ′ , 𝑆 ′ ) ∈ L and can be extended to (𝑇, 𝑆) by using Operation T𝑗 . For any set 𝑍 ⊆ 𝑉 (𝑇), let 𝑍 ′ = 𝑍 ∩ 𝑉 (𝑇 ′ ). For all operations, the number of vertices of 𝑇𝑢 of status 𝐴 equals the number of vertices of 𝑇𝑢 of status 𝐶, and so ′ |. Since 𝑃 is a packing, 𝑃 ′ is a packing. Since 𝐼 is independent, 𝐼 ′ is |𝑆 ′𝐴 | = |𝑆𝐶

Chapter 5. Domination in Trees

126

independent. Since 𝐼 dominates 𝑇, 𝐼 ′ will dominate 𝑇 ′ provided 𝑣 is dominated by an element of 𝐼 other than 𝑢. If 𝑗 = 3, this is assumed. If 𝑗 ≠ 3, then 𝑢 has status 𝐵 and so this is necessarily the case. We return to the proof of Theorem 5.48. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 3. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Necessarily, 𝑣 0 and 𝑣 𝑑 are leaves of 𝑇. For ease of discussion in the current proof, we call the vertices at maximum distance from 𝑟 eccentric vertices, and so 𝑣 0 is an eccentric vertex. Suppose sta(𝑣 0 ) = 𝐵. Then since 𝑣 0 is dominated by 𝐼, the vertex 𝑣 1 has status 𝐴 or 𝐷. And so (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 0 and 𝑗 = 1. So we may assume that no eccentric vertex has status 𝐵. Suppose sta(𝑣 0 ) = 𝐷. Then by Lemma 5.47, sta(𝑣 1 ) = 𝐵. Since 𝑃 is a packing, any neighbor of 𝑣 1 has status 𝐴 or 𝐵. This means that 𝑣 1 has no other leaf neighbor (since a vertex with status 𝐴 has a neighbor with status 𝐶) and so has degree 2. Hence, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 1 and 𝑗 = 2. Thus, we may assume that every eccentric vertex has status 𝐴 or 𝐶. So, by Lemma 5.47, every vertex at distance 2 from an eccentric vertex has status 𝐵. In particular, this means that 𝑣 1 has degree 2. Suppose sta(𝑣 0 ) = 𝐶. Then, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 1 and 𝑗 = 3, unless 𝑣 2 has no neighbor in 𝐼 \ {𝑢}. So, suppose that is the case. Then sta(𝑣 3 ) ∈ {𝐵, 𝐶}. If 𝑣 2 has degree 2, then (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 4. Hence, assume deg(𝑣 2 ) ≥ 3. This means that 𝑣 2 has a neighbor 𝑢 1 ≠ 𝑣 3 that has status 𝐵 or 𝐶. Since 𝐼 dominates 𝑢 1 , the vertex 𝑢 1 has a neighbor 𝑢 0 with sta(𝑢 0 ) ∈ { 𝐴, 𝐷}. Note that 𝑢 0 is an eccentric vertex, so (as above) deg(𝑢 1 ) = 2. By Lemma 5.47 and the above assumptions, sta(𝑢 0 ) = 𝐴 and sta(𝑢 1 ) = 𝐶. But 𝑣 2 can only have one neighbor with status 𝐶, and so has degree 3. Thus, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 6. Hence, we may assume that all eccentric vertices have status 𝐴. This means that all neighbors of 𝑣 2 , apart from 𝑣 3 , have status 𝐶, and so 𝑣 2 has degree 2. It follows that sta(𝑣 3 ) = 𝐴. Thus, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 5. The next corollary follows directly from Theorem 5.42, Lemma 5.46, and Theorem 5.48. Corollary 5.49 ([240]) The (𝛾, 𝑖)-trees are precisely those trees 𝑇 such that (𝑇, 𝑆) ∈ L for some labeling 𝑆.

5.5.2

(𝜸, 𝜸t )-trees

As before by Theorem 5.29, the (𝛾, 𝛾t )-trees are precisely the (𝜌, 𝛾t )-trees. And again the key to the constructive characterization of (𝜌, 𝛾t )-trees is to find a labeling of the vertices that indicates the role each vertex plays in the sets associated with both parameters. We adopt the same terminology used for (𝜌, 𝑖)-trees. In particular, a (𝜌, 𝛾t )labeling of a graph 𝐺 = (𝑉, 𝐸) as a partition 𝑆 = (𝑆 𝐴, 𝑆 𝐵 , 𝑆𝐶 , 𝑆 𝐷 ) of 𝑉 such that 𝑆 𝐴 ∪ 𝑆 𝐷 is a 𝛾t -set, 𝑆𝐶 ∪ 𝑆 𝐷 is a 𝜌-set, and |𝑆 𝐴 | = |𝑆𝐶 |.

Section 5.5. Equality of Domination Parameters

127

Lemma 5.50 ([240]) A graph is a (𝜌, 𝛾t )-graph if and only if it has a (𝜌, 𝛾t )labeling.

The smallest (𝛾, 𝛾t )-tree is the path 𝑃4 . It has a unique labeling as a (𝜌, 𝛾t )-tree, where the leaves have status 𝐶 and the internal vertices have status 𝐴. Let Lt be the minimum family of labeled trees that: (a) contains a labeled 𝑃4 , where the leaves have status 𝐶 and the internal vertices have status 𝐴, and (b) is closed under Operations U𝑖 for 𝑖 ∈ [5], listed below, which extend the tree 𝑇 by attaching a tree to the vertex 𝑦 ∈ 𝑉 (𝑇), called the attacher. Operation U1 . Assume sta(𝑦) ∈ {𝐴, 𝐷}. Add a vertex 𝑥 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵. Operation U2 . Assume sta(𝑦) = 𝐴. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐴 and sta(𝑤) = 𝐶. Operation U3 . Take a vertex 𝑦 of status 𝐵 which has no neighbor of status 𝐶, add a labeled 𝑃4 , and join 𝑦 to a leaf of the 𝑃4 . Operation U4 . Add a labeled 𝑃4 , and join a vertex 𝑦 of status 𝐵 to an internal vertex of the 𝑃4 . Operation U5 . Add a labeled 𝑃4 and a vertex 𝑦 ′ labeled 𝐵, and attach to a vertex 𝑦 of status 𝐵 or 𝐶 the added vertex 𝑦 ′ and join 𝑦 ′ to an internal vertex of the added labeled 𝑃4 . These five operations are illustrated in Figure 5.10.

𝐴 or 𝐷

𝐴

𝐵

(a) U1

𝐵 𝑦

𝐶

𝐶

𝐴

(b) U2

𝐴

𝐴

𝐶

𝐵

𝐴

𝐴 𝐶

(c) U3 ; 𝑦 has no prior 𝑆𝐶 neighbor

𝐵 or 𝐶

(d) U4

𝐵

𝐴

𝐴

𝐶

𝐶 (e) U5

Figure 5.10 The five U𝑖 operations

𝐶

128

Chapter 5. Domination in Trees

Theorem 5.51 ([240]) A labeled tree is a (𝜌, 𝛾t )-tree if and only if it can be obtained from a labeled 𝑃4 using the Operations U𝑖 for 𝑖 ∈ [5]. The next corollary follows directly from Theorem 5.42, Lemma 5.50, and Theorem 5.48. Corollary 5.52 ([240]) The (𝛾, 𝛾t )-trees are precisely those trees 𝑇 such that (𝑇, 𝑆) ∈ Lt for some labeling 𝑆.

5.5.3 Summary We conclude this section by mentioning a couple of related concepts. As we have seen, much interest has been shown in characterizing graphs achieving equality in an inequality between two parameters. For example, the (𝛾, 𝑖)-graphs are graphs achieving equality in the relationship 𝛾(𝐺) ≤ 𝑖(𝐺). Note that if equality is reached between 𝛾(𝐺) and 𝑖(𝐺), then the graph 𝐺 has a 𝛾-set that is also an 𝑖-set. Haynes and Slater [436] considered graphs 𝐺 for which every 𝛾-set of 𝐺 is also an 𝑖-set of 𝐺, and hence introduced the concept of strong equality between parameters related by an inequality. In particular, 𝛾(𝐺) is said to be strongly equal to 𝑖(𝐺) if every 𝛾-set is also an 𝑖-set of 𝐺. Haynes et al. [428] gave a constructive characterization of the trees 𝑇 with strong equality between 𝛾(𝑇) and 𝑖(𝑇). A graph 𝐺 is said to be domination perfect if 𝛾(𝐺 ′ ) = 𝑖(𝐺 ′ ) for all subgraphs 𝐺 ′ of 𝐺. Domination perfect trees are characterized as follows: A tree is domination perfect if and only if it does not contain two adjacent vertices of degree 3 or more. (This characterization follows as a corollary of the results from any of [315, 693, 790].)

Chapter 6

Upper Bounds in Terms of Minimum Degree 6.1 Introduction As previously mentioned, since the decision problems related to the domination number, the total domination number, and the independent domination number are all NP-complete, it is of interest to determine good upper bounds on these parameters. In Chapter 4 we presented some of the more basic bounds. In this chapter we continue with bounds on these parameters in terms of the order of the graph and its minimum degree.

6.2

Bounds on the Domination Number

In this section, we present the classical bound 𝛾(𝐺) ≤ 52 𝑛 of McCuaig and Shepherd on the domination number of a connected graph 𝐺 of order 𝑛 ≥ 8 and minimum degree 𝛿(𝐺) ≥ 2 using the proof technique of what they coined 25 -minimal graphs. Using vertex-disjoint path covers, we present the important bound 𝛾(𝐺) ≤ 38 𝑛 of Reed on the domination number of a graph 𝐺 of order 𝑛 and minimum degree 𝛿(𝐺) ≥ 3. For larger minimum degrees, we present the impressive results of Bujtás using weighting arguments and discharging methods. We illustrate her proof techniques 4 to obtain the bounds 𝛾(𝐺) ≤ 11 𝑛 and 𝛾(𝐺) ≤ 13 𝑛 when the graph 𝐺 has minimum degree 𝛿(𝐺) ≥ 4 and 𝛿(𝐺) ≥ 5, respectively.

6.2.1

Minimum Degree One

As given in Theorem 4.21, the domination number of any isolate-free graph is at most half its order. A characterization of graphs obtaining this bound is presented in Theorem 4.24. For completeness, we restate these results from Chapter 4. Lemma 6.1 ([84]) Every isolate-free graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_6

129

Chapter 6. Upper Bounds in Terms of Minimum Degree

130

Theorem 6.2 ([622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 21 𝑛. Theorem 6.3 ([633]) If 𝐺 is an isolate-free graph of even order 𝑛, then 𝛾(𝐺) = 12 𝑛 if and only if every component of 𝐺 is a 4-cycle or 𝐺 = 𝐻 ◦ 𝐾1 for some graph 𝐻.

6.2.2

Minimum Degree Two

In 1973 Blank [79] proved that the 12 -bound on the domination number of an isolatefree, connected graph 𝐺 can be improved to a 25 -bound if we restrict the minimum degree to 𝛿(𝐺) ≥ 2 and the order to 𝑛 ≥ 8. Theorem 6.4 ([79]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then   𝑛+2 2 𝛾(𝐺) ≤ max , 𝑛 . 3 5 As an immediate consequence of Theorem 6.4, we have the following result. Theorem 6.5 ([79]) If 𝐺 is a connected graph of order 𝑛 ≥ 8 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛. We remark that this important 1973 paper by Blank [79] was written in Russian. Furthermore, Blank used the notation “external stability number” for the “domination number.” Hence, the paper was not easily traceable by researchers in domination theory at the time. In 1989 McCuaig and Shepherd [586] reproved Theorem 6.4. Their proof provides insight into the structure of graphs achieving the 25 -upper bound. Their result also gives a characterization of the infinite family of graphs achieving equality in the upper bound when the order 𝑛 ≥ 15. In order to present the McCuaig-Shepherd proof, the following definition is needed. Definition 6.6 ([586]) A graph 𝐺 of order 𝑛 ≥ 3 is a 25 -minimal graph if 𝐺 is edge minimal with respect to satisfying all three of the following conditions (and so, deleting any edge means that at least one of the following three conditions is no longer satisfied): (a) 𝛿(𝐺) ≥ 2. (b) 𝐺 is connected. (c) 𝛾(𝐺) ≥ 25 𝑛. Let Bdom = {𝐵1 , 𝐵2 , . . . , 𝐵7 } be the family of seven graphs (one of order four and six of order seven) shown in Figure 6.1. Each graph 𝐺 ∈ Bdom of order 𝑛 satisfies 𝛿(𝐺) = 2 and 𝛾(𝐺) > 25 𝑛. The following properties of graphs in the family Bdom will be required. Proposition 6.7 ([79]) If 𝐺 ∈ Bdom has order 𝑛, then the following hold: (a) 𝛾(𝐺) = 13 (𝑛 + 2). (b) Every vertex belongs to some 𝛾-set of 𝐺. (c) For every vertex 𝑣, 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1.

Section 6.2. Bounds on the Domination Number

(a) 𝐵1

(d) 𝐵4

131

(b) 𝐵2

(e) 𝐵5

(c) 𝐵3

(f) 𝐵6

(g) 𝐵7

Figure 6.1 The family Bdom of seven graphs 𝐺 satisfying 𝛾(𝐺) > 25 𝑛

McCuaig and Shepherd [586] constructed the following (infinite) family of graphs. We define a 5-key to be a graph of order 5 obtained from a 4-cycle by adding a new vertex and joining this vertex to exactly one vertex of the cycle. We define a unit to be a graph that is isomorphic to a 5-cycle or to a 5-key. We call a unit a cycle unit or a key unit if it is a 5-cycle or a 5-key, respectively. In each cycle unit, we choose two nonadjacent vertices in the unit and designate them as the link vertices of the unit, and in each key unit we designate the vertex of degree 1 as the link vertex of the unit. Let Fdom be the family of all graphs 𝐺 such that either 𝐺 = 𝐶5 or 𝐺 can be obtained from the disjoint union of at least two units, by adding edges between link vertices so that the resulting graph is connected and each added edge is a bridge of 𝐺. A graph 𝐺 in the family Fdom with two cycle units and two key units is shown in Figure 6.2, where the link vertices are highlighted. 2 5 -minimal

Figure 6.2 A graph 𝐺 in the family Fdom Let Fsmall = {𝐹1 , 𝐹2 , . . . , 𝐹6 } be the family of six graphs shown in Figure 6.3. We shall need the following properties of graphs in the family Fdom ∪ Fsmall . Lemma 6.8 ([79]) If 𝐺 ∈ Fdom ∪ Fsmall has order 𝑛, then the following hold: (a) 𝛾(𝐺) = 25 𝑛 and 𝐺 is a 25 -minimal graph. (b) Every vertex belongs to some 𝛾-set of 𝐺. (c) If 𝐺 ∈ Fsmall , then for every vertex 𝑣, 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1, except when 𝐺 = 𝐹1 and 𝑣 is a vertex of degree 2.

Chapter 6. Upper Bounds in Terms of Minimum Degree

132

(a) 𝐹1

(b) 𝐹2

(d) 𝐹4

(c) 𝐹3

(e) 𝐹5

(f) 𝐹6

Figure 6.3 The family Fsmall of six graphs

Proof (a) Suppose firstly that 𝐺 ∈ Fsmall has order 𝑛. If 𝐺 = 𝐹1 , then 𝑛 = 5 and 𝛾(𝐺) = 2 = 25 𝑛, while if 𝐺 ∈ Fsmall \ {𝐹1 }, then 𝑛 = 10 and 𝛾(𝐺) = 4 = 25 𝑛. Suppose next that 𝐺 ∈ Fdom has order 𝑛. Let 𝐺 have 𝑘 ≥ 2 units, and so 𝑛 = 5𝑘. Every dominating set in 𝐺 contains at least two vertices from each unit of 𝐺, implying that 𝛾(𝐺) ≥ 2𝑘. The set of all link vertices, together with a vertex from each key unit that is a distance 3 from its link vertex, forms a dominating set of 𝐺. Hence, there is a dominating set in 𝐺 that contains exactly two vertices from each unit of 𝐺, and so 𝛾(𝐺) ≤ 2𝑘. Consequently, 𝛾(𝐺) = 2𝑘 = 25 𝑛. Further, if 𝐺 ∈ Fdom ∪ Fsmall and 𝑒 is an arbitrary edge of 𝐺, then the removal of 𝑒 produces a disconnected graph or creates a vertex of degree 1. Thus, 𝐺 is a 25 -minimal graph. This completes the proof of (a). The proofs of (b) and (c) follow readily from the structure of graphs in the family Fdom ∪ Fsmall . We proceed further with a series of preliminary results that will be helpful when we present a characterization of 25 -minimal graphs. Recall that for paths 𝑃𝑛 and   cycles 𝐶𝑛 for 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) = 𝑛3 . This readily yields the following result. Proposition 6.9 A cycle 𝐶𝑛 is a 25 -minimal graph if and only if 𝑛 ∈ {4, 5, 7, 10}. A daisy with 𝑘 ≥ 2 petals is a connected graph that can be constructed from 𝑘 ≥ 2 disjoint cycles by identifying a set of 𝑘 vertices, one from each cycle, into one vertex. In particular, if the 𝑘 cycles have lengths 𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 , we denote the daisy by 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ). Further, if 𝑛 = 𝑛1 = 𝑛2 = · · · = 𝑛 𝑘 , then we write 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ) simply as 𝐷 𝑘 (𝑛). For example, the graph 𝐵2 in Figure 6.1(b) is a daisy 𝐷 2 (4) = 𝐷 (4, 4). The graphs 𝐹2 and 𝐹4 in Figure 6.3(b) and (d) are the daisies 𝐷 (4, 7) and 𝐷 3 (4) = 𝐷 (4, 4, 4), respectively. A simple proof by induction on the number of petals of a daisy yields the following upper bound on its domination number in terms of its order. Proposition 6.10 If 𝐺 is a daisy 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ) of order 𝑛 with 𝑘 ≥ 2 petals, then 𝛾(𝐺) ≤ 13 (𝑛 + 2), with equality if and only if 𝑛𝑖 ≡ 1 (mod 3) for all 𝑖 ∈ [𝑘].

Section 6.2. Bounds on the Domination Number

133

By Proposition 6.10, we can readily determine the 52 -minimal daisies.  Proposition 6.11 A daisy 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ 𝐷 2 (4), 𝐷 3 (4), 𝐷 (4, 7) . We shall need the following lemma about subdividing certain edges in a graph. Lemma 6.12 If 𝐺 is the graph obtained from a graph 𝐺 ′ by subdividing an edge three times, then 𝛾(𝐺) = 𝛾(𝐺 ′ ) + 1. Proof Let 𝐺 be obtained from a graph 𝐺 ′ by subdividing an edge 𝑒 = 𝑢𝑣 three times. Let 𝑣 1 , 𝑣 2 , and 𝑣 3 be the three new vertices, where 𝑢 𝑣 1 𝑣 2 𝑣 3 𝑣 is a path in 𝐺. Let 𝑆 ′ be a 𝛾-set of 𝐺 ′ . If {𝑢, 𝑣} ⊆ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 2 }. If 𝑢 ∈ 𝑆 ′ and 𝑣 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 3 }. If 𝑣 ∈ 𝑆 ′ and 𝑢 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 1 }. If 𝑢 ∉ 𝑆 ′ and 𝑣 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 2 }. In each case, the resulting set 𝑆 is a dominating set of 𝐺, implying that 𝛾(𝐺) ≤ |𝑆| = |𝑆 ′ | + 1 = 𝛾(𝐺 ′ ) + 1. Conversely, among all 𝛾-sets of 𝐺, let 𝑆 be chosen to contain as few vertices from the set {𝑣 1 , 𝑣 2 , 𝑣 3 } as possible. Suppose that 𝑣 1 ∈ 𝑆. If 𝑣 2 or 𝑣 3 belongs to 𝑆, then we can replace it with the vertex 𝑣, contradicting our choice of 𝑆. Hence, neither 𝑣 2 nor 𝑣 3 belongs to 𝑆, implying that 𝑣 ∈ 𝑆 in order to dominate 𝑣 3 . In this case, we let 𝑆 ′ = 𝑆 \ {𝑣 1 }. Analogously, if 𝑣 3 ∈ 𝑆, then neither 𝑣 1 nor 𝑣 2 belongs to 𝑆, implying that 𝑢 ∈ 𝑆. In this case, we let 𝑆 ′ = 𝑆 \ {𝑣 3 }. If 𝑣 2 ∈ 𝑆, then neither 𝑣 1 nor 𝑣 3 belongs to 𝑆, and in this case we let 𝑆 ′ = 𝑆 \ {𝑣 2 }. In each case, the resulting set 𝑆 ′ is a dominating set of 𝐺 ′ , implying that 𝛾(𝐺 ′ ) ≤ |𝑆 ′ | = |𝑆| − 1 = 𝛾(𝐺) − 1. Consequently, 𝛾(𝐺) = 𝛾(𝐺 ′ ) + 1. For 𝑛1 , 𝑛2 ≥ 3 and 𝑘 ≥ 1, a dumbbell 𝐷 (𝑛1 , 𝑛2 , 𝑘) is the graph obtained from two disjoint cycles 𝐶𝑛1 and 𝐶𝑛2 by adding an edge joining the two cycles and subdividing this edge 𝑘 − 1 times, resulting in vertices joined (connected) by a path of length 𝑘. The dumbbells 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), and 𝐷 (4, 5, 2) are shown in Figure 6.4.

(a) 𝐷 (5, 5, 1)

(b) 𝐷 (4, 4, 3)

(c) 𝐷 (4, 5, 2)

Figure 6.4 The dumbbells 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), and 𝐷 (4, 5, 2)   Using the fact that 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) = 𝑛3 for 𝑛 ≥ 3, it is a relatively simple exercise to determine the 25 -minimal dumbbells.  Proposition 6.13 A dumbbell 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), 𝐷 (4, 5, 2) ⊂ Fdom . Recall that the removal of an edge from a graph 𝐺 cannot decrease the domination number; that is, if 𝑒 is an edge of a graph 𝐺, then 𝛾(𝐺 − 𝑒) ≥ 𝛾(𝐺). In particular,

134

Chapter 6. Upper Bounds in Terms of Minimum Degree

if 𝐺 is a 52 -minimal graph, then the removal of an edge results in a graph that no longer satisfies condition (a) or condition (b) (or both conditions (a) and (b)) of Definition 6.6. We state this observation formally as follows. Observation 6.14 If 𝐺 is a 25 -minimal graph, then every edge of 𝐺 is a bridge of 𝐺 or is incident with a vertex of degree 2 in 𝐺, that is, if 𝑒 ∈ 𝐸 (𝐺), then 𝑒 is a bridge of 𝐺 or 𝛿(𝐺 − 𝑒) = 1. We next consider 25 -minimal graphs of order 𝑛 ≤ 7. Lemma 6.15 ([79]) Let 𝐺 be a connected graph of order 𝑛 ≤ 7 with 𝛿(𝐺) ≥ 2. Then, 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 = 𝐶5 ∈ Fdom or 𝐺 = 𝐹1 ∈ Fsmall . Proof If 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 , 𝐶5 , 𝐹1 }, then it is straightforward to check that 𝐺 is a 2 2 5 -minimal graph. To prove the necessity, suppose that 𝐺 is a 5 -minimal graph of 2 order 𝑛 ≤ 7. If 𝑛 = 3, then 𝐺 = 𝐾3 and 𝛾(𝐺) = 1 < 5 𝑛, a contradiction. If 𝑛 = 4, then since 𝛿(𝐺) ≥ 2, 𝐺 is either 𝐶4 , 𝐾4 − 𝑒 or 𝐾4 , but 𝛾(𝐾4 − 𝑒) = 𝛾(𝐾4 ) = 1. Thus, 𝐺 = 𝐶4 = 𝐵1 . If 𝑛 = 5, then 𝐺 is one of eleven possible connected graphs having minimum degree 2, but seven of these have 𝛾(𝐺) = 1. Of the remaining four, two are not edge-minimal with respect to the three properties, leaving either 𝐺 = 𝐶5 or 𝐺 = 𝐹1 . Suppose that 𝑛 = 6. If Δ(𝐺) ≥ 4, then it is immediate that 𝛾(𝐺) ≤ 2. If Δ(𝐺) = 2, then 𝐺 = 𝐶6 and 𝛾(𝐺) = 2. If Δ(𝐺) = 3, then let 𝑣 be a vertex of maximum degree 3 in 𝐺, and let 𝑢 and 𝑤 be the two vertices not adjacent to 𝑣. If 𝑢𝑤 is an edge, then 𝛾(𝐺) = 2. If 𝑢𝑤 is not an edge, then 𝑢 and 𝑤 have a common neighbor, which together with the vertex 𝑣 forms a dominating set of 𝐺, and so 𝛾(𝐺) = 2. Thus, in all cases, if 𝑛 = 6, then 𝛾(𝐺) ≤ 2 < 25 𝑛, a contradiction. If 𝑛 = 7 and Δ(𝐺) = 2, then 𝐺 = 𝐶7 = 𝐵4 . Hence, we may assume that 𝑛 = 7 and Δ(𝐺) ≥ 3, for otherwise the desired result follows. If Δ(𝐺) ≥ 5, then 𝛾(𝐺) = 2 < 25 𝑛, a contradiction. Hence, Δ(𝐺) = 3 or Δ(𝐺) = 4. Suppose that Δ(𝐺) = 4. Let 𝑣 be a vertex of maximum degree 4 and let N(𝑣) = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }. Let 𝐴 = 𝑉 \ N[𝑣] = {𝑣 5 , 𝑣 6 }. If one vertex dominates the two vertices in 𝐴, then 𝛾(𝐺) = 2, a contradiction. Hence, two vertices are needed to dominate 𝐴. Renaming vertices if necessary, this implies that N(𝑣 5 ) = {𝑣 1 , 𝑣 2 } and N(𝑣 6 ) = {𝑣 3 , 𝑣 4 }. By the edge minimality of 𝐺, the graph 𝐺 is determined. Hence, 𝐺 = 𝐵2 . Suppose that Δ(𝐺) = 3. Among all vertices of maximum degree 3 in 𝐺, let 𝑣 be chosen to have the minimum number of edges in the induced subgraph 𝐺 [N(𝑣)]. Let N(𝑣) = {𝑣 1 , 𝑣 2 , 𝑣 3 } and let 𝐵 = 𝑉 \ N[𝑣] = {𝑣 4 , 𝑣 5 , 𝑣 6 }. If one vertex dominates the set 𝐵, then 𝛾(𝐺) = 2, a contradiction. Hence, two vertices are needed to dominate 𝐵. In particular, every vertex in N(𝑣) dominates at most two vertices in 𝐵. Further, 𝐺 [𝐵] contains at most one edge. Since 𝛿(𝐺) ≥ 2, these observations imply that there are at least four edges with exactly one end in 𝐵, and therefore, by the Pigeonhole Principle, there is a vertex in N(𝑣) adjacent to exactly two vertices of 𝐵. Renaming vertices if necessary, we may assume that 𝑣 1 is adjacent to exactly two vertices of 𝐵, say 𝑣 4 and 𝑣 5 . If 𝑣 6 is adjacent to both 𝑣 2 and 𝑣 3 , then {𝑣 1 , 𝑣 6 }

Section 6.2. Bounds on the Domination Number

135

is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Thus, 𝑣 6 is adjacent to exactly one of 𝑣 2 and 𝑣 3 , say 𝑣 3 , and to exactly one of 𝑣 4 and 𝑣 5 , say 𝑣 5 . This in turn implies that there is no edge in the subgraph induced by the open neighborhood of 𝑣 1 . Hence, by our choice of the vertex 𝑣, there is no edge in 𝐺 [N(𝑣)]. The vertex 𝑣 2 is therefore adjacent to at least one of 𝑣 4 and 𝑣 5 . If 𝑣 2 is not adjacent to 𝑣 4 , then both 𝑣 2 𝑣 5 and 𝑣 3 𝑣 4 are edges. However, in this case, {𝑣 3 , 𝑣 5 } is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, 𝑣 2 𝑣 4 is an edge. However, 𝐺 now contains a 7-cycle, namely 𝑣 𝑣 3 𝑣 6 𝑣 5 𝑣 1 𝑣 4 𝑣 2 𝑣, as a proper subgraph, contradicting the fact that 𝐺 is a 52 -minimal graph. In what follows in the remaining part of this Section 6.2.2, if 𝐺 is a graph with minimum degree at least 2, then we let L be the set of all vertices of degree at least 3 in 𝐺, and let S = 𝑉 \ L, that is,   L = 𝑣 ∈ 𝑉 : deg𝐺 (𝑣) ≥ 3 and S = 𝑣 ∈ 𝑉 : deg𝐺 (𝑣) = 2 . We call a vertex in L a large vertex, and a vertex in S a small vertex. For 𝑘 ≥ 3, we define a 𝑘-handle to be a 𝑘-cycle that contains exactly one large vertex. For 𝑘 ≥ 1, a 𝑘-linkage is a path on 𝑘 + 2 vertices that starts and ends at distinct large vertices and with 𝑘 internal vertices of degree 2 in 𝐺. A handle is a 𝑘-handle for some 𝑘 ≥ 3, and a linkage is a 𝑘-linkage for some 𝑘 ≥ 1. We are now in a position to present a characterization of the 25 -minimal graphs. Theorem 6.16 ([586]) Let 𝐺 be a connected graph of order 𝑛 ≥ 3 with 𝛿(𝐺) ≥ 2. Then, 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ∈ Fdom ∪ Fsmall . Proof The sufficiency follows from Lemmas 6.8(a) and 6.15. To prove the necessity, suppose to the contrary, that the theorem is false. Among all counterexamples, let 𝐺 be chosen to be a 25 -minimal graph of minimum order 𝑛. Thus, 𝐺 ∉ {𝐵1 , 𝐵2 , 𝐵4 } and 𝐺 ∉ Fdom ∪ Fsmall . By Lemma 6.15, the result is true for 𝑛 ≤ 7. Hence, 𝑛 ≥ 8. By our choice of 𝐺, the result is true for all 25 -minimal graphs 𝐺 ′ of order 𝑛′ , where 𝑛′ < 𝑛. If 𝐺 = 𝐶𝑛 , then by Proposition 6.9, 𝐺 = 𝐶10 = 𝐹6 ∈ Fsmall , a contradiction. Hence, 𝐺 is not a cycle. Thus, Δ(𝐺) ≥ 3, and so |L| ≥ 1. If |L| = 1, then 𝐺 is a daisy and, by Proposition 6.11, we have 𝐺 = 𝐵2 or 𝐺 ∈ {𝐹2 , 𝐹4 } ⊂ Fsmall , a contradiction. If |L| = 2, then 𝐺 is a dumbbell and, by Proposition 6.13, we have 𝐺 ∈ {𝐷 (5, 5, 1), 𝐷 (4, 4, 3), 𝐷 (4, 5, 2)} ⊂ Fdom , a contradiction. Hence, |L| ≥ 3. We proceed further with the following structural properties of the graph 𝐺. Claim 6.16.1 The set L is an independent set. Proof Suppose, to the contrary, that L is not an independent set. Let 𝑒 = 𝑣 1 𝑣 2 be an edge of 𝐺, where 𝑣 1 , 𝑣 2 ∈ L. By Observation 6.14, the edge 𝑒 is a bridge of 𝐺. Let 𝐺 1 = (𝑉1 , 𝐸 1 ) and 𝐺 2 = (𝑉2 , 𝐸 2 ) be the two components of 𝐺 − 𝑒, where 𝑣 𝑖 ∈ 𝑉𝑖 for 𝑖 ∈ [2]. For 𝑖 ∈ [2], let |𝑉𝑖 | = 𝑛𝑖 , and so 𝑛 = 𝑛1 + 𝑛2 . We note that 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ). Further, for 𝑖 ∈ [2], the graph 𝐺 𝑖 is edge minimal with respect to the two conditions: (i) 𝐺 𝑖 is connected and (ii) 𝛿(𝐺 𝑖 ) ≥ 2. If 𝐺 𝑖 is not a 25 -minimal graph for some 𝑖 ∈ [2], then 𝛾(𝐺 𝑖 ) < 25 𝑛𝑖 . If neither 𝐺 1 nor 𝐺 2 is a

136 2 5 -minimal

Chapter 6. Upper Bounds in Terms of Minimum Degree

graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) < 25 𝑛1 + 25 𝑛2 = 25 𝑛, a contradiction. Hence, renaming 𝐺 1 and 𝐺 2 if necessary, we may assume that 𝐺 1 is a 25 -minimal graph. Since 𝑛1 < 𝑛, the graph 𝐺 1 cannot be a counterexample to the theorem by our choice of 𝐺. Thus, 𝐺 1 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ∈ Fdom ∪ Fsmall . Suppose that 𝐺 1 ∈ Bdom . If 𝛾(𝐺 2 ) < 25 𝑛2 , then 𝛾(𝐺) < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 2 ) ≥ 25 𝑛2 , implying that 𝐺 2 ∈ Bdom or 𝐺 2 ∈ Fdom ∪ Fsmall . By Proposition 6.7(b) and Lemma 6.8(b), every vertex 𝑣 in a graph in Bdom , Fdom , or Fsmall belongs to some 𝛾-set of 𝐺. Thus, there exists a 𝛾-set, 𝐷 2 say, of 𝐺 2 that contains the vertex 𝑣 2 . The set 𝐷 2 can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 1 − 𝑣 1 to it. Thus,
by Proposition 6.7, we have 𝛾(𝐺) ≤ |𝐷 2 | + 𝛾(𝐺 1 − 𝑣 1 ) = 𝛾(𝐺 2 ) + 𝛾(𝐺 1 ) − 1 = 𝛾(𝐺 2 ) + 13 (𝑛1 + 2) − 1 = 𝛾(𝐺 2 ) + 13 (𝑛1 − 1). If 𝐺 2 ∈ Bdom , then 𝛾(𝐺 2 ) = 13 (𝑛2 +2), implying that 𝛾(𝐺) ≤ 13 (𝑛2 +2) + 13 (𝑛1 −1) = 13 (𝑛 +1) < 25 𝑛, noting that 𝑛 ≥ 8. If 𝐺 2 ∈ Fdom ∪ Fsmall , then 𝛾(𝐺 2 ) = 25 𝑛2 , implying that 𝛾(𝐺) ≤ 2 1 2 2 2 5 𝑛2 + 3 (𝑛1 − 1) < 5 𝑛1 + 5 𝑛2 = 5 𝑛. In both cases we produce a contradiction. Hence, 𝐺 1 ∉ Bdom , and so 𝐺 1 ∈ Fdom ∪ Fsmall . Thus, 𝛾(𝐺 1 ) = 25 𝑛1 . By Lemma 6.8(b), there exists a 𝛾-set, 𝐷 1 say, of 𝐺 1 that contains the vertex 𝑣 1 . If 𝐺 2 is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) < 25 𝑛1 + 25 𝑛2 = 25 𝑛, a contradiction. Hence, 𝐺 2 is a 25 -minimal graph. Using the same argument to show that 𝐺 1 ∉ Bdom , one can show that 𝐺 2 ∉ Bdom , and so 𝐺 2 ∈ Fdom ∪ Fsmall and 𝛾(𝐺 2 ) = 25 𝑛2 . The set 𝐷 1 can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 2 − 𝑣 2 to it, and so 𝛾(𝐺) ≤ |𝐷 1 | + 𝛾(𝐺 2 − 𝑣 2 ). Suppose that 𝐺 2 ∈ Fsmall . If 𝐺 2 = 𝐹1 and 𝑣 2 is a vertex of degree 2 in 𝐺 2 , then the edge joining 𝑣 2 to a vertex of degree 3 in 𝐺 2 joins two vertices of L but is not a bridge of 𝐺, a contradiction. Hence, 𝐺 2 ≠ 𝐹1 or 𝐺 2 = 𝐹1 and 𝑣 2 is a vertex of degree 3 in 𝐺 ′ . By Lemma 6.8(c), we have 𝛾(𝐺 2 − 𝑣 2 ) = 𝛾(𝐺 2 ) − 1, implying that 𝛾(𝐺) ≤ |𝐷 1 | + 𝛾(𝐺 2 ) − 1 = 25 𝑛1 + 25 𝑛2 − 1 < 25 𝑛, a contradiction. Hence, 𝐺 2 ∈ Fdom . Using similar arguments, one can show that 𝐺 1 ∈ Fdom . If 𝐺 1 = 𝐶5 and 𝐺 2 = 𝐶5 , then 𝐺 = 𝐷 (5, 5, 1) ∈ Fdom , a contradiction. Hence, renaming 𝐺 1 and 𝐺 2 if necessary, we may assume that 𝐺 2 ≠ 𝐶5 , and so 𝐺 2 has at least two units. Suppose that 𝑣 2 is not a link vertex in 𝐺 2 . We show that 𝛾(𝐺 2 − 𝑣 2 ) ≤ 𝛾(𝐺 2 ) − 1. Let 𝑈 be the unit in 𝐺 2 that contains 𝑣 2 . If 𝑈 is a key unit, let 𝑣 2′ be the link vertex in 𝑈, while if 𝑈 is a cycle unit, let 𝑣 2′ be a link vertex in 𝑈 that is adjacent to at least one other link vertex in 𝐺 2 (that belongs to a unit different from 𝑈). Since 𝐺 2 has at least two units, the existence of such a vertex 𝑣 2′ is guaranteed by the construction of 𝐺 2 ∈ Fdom . We note that (in both cases) the graph obtained from 𝑈 by deleting 𝑣 2 and 𝑣 2′ has a dominating vertex. Let 𝐷 2 consist of such a dominating vertex in 𝑈 − {𝑣 2 , 𝑣 2′ }, together with a 𝛾-set in 𝐺 2 − 𝑈 that contains all of its link vertices. The resulting set 𝐷 2 is a dominating set of 𝐺 2 that contains two vertices from every unit of 𝐺 2 different from 𝑈, and exactly one vertex from the unit 𝑈. Hence, 𝛾(𝐺 2 − 𝑣 2 ) ≤ |𝐷 2 | = 𝛾(𝐺 2 ) − 1 = 25 𝑛2 − 1. Since 𝑣 1 ∈ 𝐷 1 , vertex 𝑣 2 is dominated by 𝐷 1 , implying that 𝐷 1 ∪ 𝐷 2 is a dominating set of 𝐺. Hence, 𝛾(𝐺) ≤ |𝐷 1 | + |𝐷 2 | = 25 𝑛1 + 25 𝑛2 − 1 = 25 𝑛 − 1 < 25 𝑛, a contradiction. Hence, 𝑣 2 is a link vertex in 𝐺 2 .

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137

If 𝐺 1 = 𝐶5 , then 𝐺 ∈ Fdom , a contradiction. Hence, 𝐺 1 has at least two units. Interchanging the roles of 𝐺 1 and 𝐺 2 , the vertex 𝑣 1 is a link vertex in 𝐺 1 . Thus, 𝐺 ∈ Fdom , a contradiction. Recall that 𝐺 is a 25 -minimal graph of minimum order 𝑛 such that 𝐺 ∉ {𝐵1 , 𝐵2 , 𝐵4 } and 𝐺 ∉ Fdom ∪ Fsmall . By our earlier observations, 𝑛 ≥ 8 and |L| ≥ 3. Claim 6.16.2 The graph 𝐺 does not contain a path on five vertices with the two ends of the path not adjacent, and with one or both ends of the path of degree at least 3 and with the internal vertices all of degree 2 in 𝐺. Proof Suppose, to the contrary, that 𝑃 : 𝑣 𝑣 1 𝑣 2 𝑣 3 𝑣 4 is a path in 𝐺, where deg𝐺 (𝑣) ≥ 3, deg𝐺 (𝑣 4 ) ≥ 2 and deg𝐺 (𝑣 𝑖 ) = 2 for all 𝑖 ∈ [3], and where 𝑣 is not adjacent to 𝑣 4 . Let 𝐺 𝑃 be the graph of order 𝑛′ = 𝑛 − 3 ≥ 5 obtained from 𝐺 by deleting the set of vertices {𝑣 1 , 𝑣 2 , 𝑣 3 }. If deg𝐺 (𝑣 4 ) = 2 or if deg𝐺 (𝑣 4 ) ≥ 3 and 𝐺 is disconnected, then let 𝐺 ′ be obtained from 𝐺 𝑃 by adding the edge 𝑣𝑣 4 , while if deg𝐺 (𝑣 4 ) ≥ 3 and 𝐺 is connected, then let 𝐺 ′ = 𝐺 𝑃 . By construction, the graph 𝐺 ′ is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. By Lemma 6.12, 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ , then 𝛾(𝐺) ≤ 25 (𝑛 − 3) + 1 < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵2 , 𝐵4 }, noting that 𝑛′ ≥ 5. If 𝐺 ′ = 𝐵4 , then |L| = 2, a contradiction. If 𝐺 ′ = 𝐵2 , then 𝐺 ′ = 𝐺 𝑃 and |L| = 3. In this case, rebuilding the graph 𝐺 back from the graph 𝐺 ′ by adding back the vertices 𝑣 1 , 𝑣 2 , 𝑣 3 and the edges of the path 𝑃, we produce a graph that is not a 25 -minimal graph, a contradiction. By Claim 6.16.1, the set L is an independent set. In particular, the large vertices at the ends of the linkage are not adjacent. Hence, as a consequence of Claim 6.16.2, we have the following structure of handles and linkages. Claim 6.16.3 The following hold in the graph 𝐺: (a) If 𝐺 contains a 𝑘-handle, then 𝑘 ∈ {3, 4, 5}. (b) If 𝐺 contains a 𝑘-linkage, then 𝑘 ∈ {1, 2}. We show next that 𝐺 contains no 4-handle or 5-handle. Claim 6.16.4 The graph 𝐺 does not contain a 4-handle. Proof Suppose, to the contrary, that 𝐺 contains a 4-handle 𝐻, given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 1 with deg𝐺 (𝑣 4 ) ≥ 3 and deg𝐺 (𝑣 𝑖 ) = 2 for 𝑖 ∈ [3]. Suppose that deg𝐺 (𝑣 4 ) ≥ 4. Let 𝐺 ′ be the graph of order 𝑛′ = 𝑛 − 3 ≥ 5 obtained from 𝐺 by deleting the vertices 𝑣 1 , 𝑣 2 , and 𝑣 3 . Every dominating set of 𝐺 ′ can be extended to a dominating set of 𝐺 by adding the vertex 𝑣 2 to it, and so 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. The graph 𝐺 ′ is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ = 25 (𝑛 − 3), then 𝛾(𝐺) ≤ 25 (𝑛 − 3) + 1 < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. Since |L| ≥ 3, this in turn implies that 𝐺 = 𝐹3 ∈ Fsmall , a contradiction. Hence, deg𝐺 (𝑣 4 ) = 3. Let 𝑣 5 be the neighbor of 𝑣 4 that does not belong to the 4-handle 𝐻. Let 𝑁5 be the set of neighbors of 𝑣 5 different from 𝑣 4 , and so 𝑁5 = N𝐺 (𝑣 5 ) \ {𝑣 4 }.

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Suppose that deg𝐺 (𝑣 5 ) ≥ 3, and so |𝑁5 | ≥ 2. Let 𝐺 ′ be the graph obtained from 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 5 } by adding edges, if necessary, between vertices in 𝑁5 so that the resulting graph is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Let 𝐺 ′ have order 𝑛′ , and so 𝑛′ = 𝑛 − 5 ≥ 3. Every 𝛾-set of 𝐺 ′ can be extended to a dominating set of 𝐺 by adding the vertices 𝑣 2 and 𝑣 5 to it, and so 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 2. If 𝛾(𝐺 ′ ) < 52 𝑛′ , then 𝛾(𝐺) < 25 (𝑛 − 5) + 2 = 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) ≥ 25 𝑛′ . By construction, the graph 𝐺 ′ is a 25 -minimal graph. Since 𝐺 ′ is not a counterexample, 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ′ ∈ Fdom ∪ Fsmall . Let 𝑣 be an arbitrary neighbor of 𝑣 5 in 𝐺 ′ , and so 𝑣 ∈ 𝑁5 . Suppose that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. The set {𝑣 2 , 𝑣 5 } can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 ′ − 𝑣 to
it. Thus, by Proposition 6.7(c), we have 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 13 (𝑛′ + 2) = 1 + 13 (𝑛 − 3) = 13 𝑛 < 25 𝑛, a contradiction. Suppose that 𝐺 ′ ∈ Fsmall . If 𝐺 ′ = 𝐹1 and 𝑣 ′ is a vertex of degree 2 in 𝐺 ′ , then the graph 𝐺 is not a 25 -minimal graph, a contradiction. If 𝐺 ′ ≠ 𝐹1 or if 𝐺 ′ = 𝐹1 and 𝑣 ′ is a not a vertex of degree 2 in 𝐺 ′ , then by Lemma 6.8(c),
we have 𝛾(𝐺 ′ − 𝑣) = 𝛾(𝐺 ′ ) − 1, implying that 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 25 𝑛′ = 1 + 25 (𝑛 − 5) < 25 𝑛, a contradiction. Suppose that 𝐺 ′ ∈ Fdom . If it is possible to construct 𝐺 ′ in such a way that all the vertices of 𝑁5 are link vertices of 𝐺 ′ , then 𝐺 ∈ Fdom (with the vertex 𝑣 5 a link vertex in 𝐺), a contradiction. Hence, there is no construction of 𝐺 ′ where all the neighbors of 𝑣 5 in 𝐺 ′ are link vertices of 𝐺 ′ . Thus, we can choose the vertex 𝑣 so that 𝑣 is not a link vertex of 𝐺 ′ . If 𝐺 ′ = 𝐶5 , then |L| = 2, a contradiction. Hence, 𝐺 ′ contains at least two units. As shown in the proof of Claim 6.16.1, 𝛾(𝐺 ′ − 𝑣) ≤ 𝛾(𝐺 ′ ) − 1. The set {𝑣 2 , 𝑣 5 } can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 ′ − 𝑣 to it. Thus, by Proposition 6.7(c), we have
𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 25 𝑛′ = 1 + 25 (𝑛 − 5) < 25 𝑛, a contradiction. Hence, deg𝐺 (𝑣 5 ) = 2, and so |𝑁5 | = 1. Let 𝑁5 = {𝑣 6 }. If deg𝐺 (𝑣 6 ) ≥ 3, then letting 𝐺 ′ = 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 5 }, we reach a contradiction, as above. Hence, deg𝐺 (𝑣 6 ) = 2. Let 𝑣 7 be the neighbor of 𝑣 6 different from 𝑣 5 . By Claim 6.16.3, the graph 𝐺 contains no 𝑘-linkage for 𝑘 ≥ 3, implying that deg𝐺 (𝑣 7 ) ≥ 3. We now consider the graph 𝐺 ′ = 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 6 } of order 𝑛′ = 𝑛 − 6. Every 𝛾-set of 𝐺 ′ can be extended to dominating set of 𝐺 by adding the set {𝑣 2 , 𝑣 5 } to it, and so 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ ). If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ , then 𝛾(𝐺) ≤ 2 + 25 𝑛′ = 2 + 25 (𝑛 − 6) < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. Since |L| ≥ 3, this in turn implies that 𝐺 ′ = 𝐵2 . Thus, 𝑛 = 13 and 𝛾(𝐺) = 5 < 25 𝑛, a contradiction. Claim 6.16.5 The graph 𝐺 does not contain a 5-handle. Proof Suppose, to the contrary, that 𝐺 contains a 5-handle 𝐻, given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 with 𝑣 5 as its (unique) vertex of degree at least 3. Thus, each vertex 𝑣 𝑖 has degree 2 in 𝐺 for 𝑖 ∈ [4]. By supposition, deg𝐺 (𝑣 5 ) ≥ 3. Let 𝑁5 be the set of neighbors of 𝑣 5 that do not belong to the 5-handle 𝐻. If deg𝐺 (𝑣 5 ) ≥ 4, then |𝑁5 | ≥ 2 and proceeding as in the proof of Claim 6.16.4, we obtain a contradiction. Hence, deg𝐺 (𝑣 5 ) = 3, and so |𝑁5 | = 1. Let 𝑁5 = {𝑣 6 }. Proceeding once again exactly as in the proof of Claim 6.16.4 we obtain a contradiction.

Section 6.2. Bounds on the Domination Number

139

By Claims 6.16.3, 6.16.4, and 6.16.5, we have the following structural result in 𝐺. Claim 6.16.6 Every handle in 𝐺, if any, is a 3-handle, and every linkage in 𝐺 is a 1-linkage or a 2-linkage. Let |L| = ℓ and |S| = 𝑠, and so 𝑛 = ℓ+𝑠. By Claim 6.16.6, the set L is a dominating set of 𝐺. Thus, since 𝐺 is a 52 -minimal graph, ℓ = |L| ≥ 𝛾(𝐺) ≥ 25 𝑛 = 25 (ℓ + 𝑠), and so 3ℓ ≥ 2𝑠. However, counting edges between L and S, we have 3ℓ ≤ 2𝑠, noting that each vertex in L is adjacent to at least three vertices in S and each vertex in S is adjacent to at most two neighbors in L. Consequently, 3ℓ = 2𝑠. Thus, ℓ = 23 𝑠 = 23 (𝑛 − ℓ), or equivalently, ℓ = 25 𝑛. Further, each vertex in L has degree exactly 3 and the set S is an independent set, and so every edge is incident with exactly one vertex of degree 2. This in turn implies that 𝐺 has no 3-handles and that every linkage in 𝐺 is a 1-linkage. Thus, 𝐺 is a bipartite graph with partite sets L and S, where each vertex in L has degree 3 and each vertex in S has degree 2. Since 𝑛 ≥ 8, we note that 𝐺 ≠ 𝐾2,3 = 𝐹1 and |L| ≥ 3. Hence, there exist two
vertices 𝑢 and 𝑣 in L with exactly one common neighbor 𝑤. The set L \ {𝑢, 𝑣} ∪ {𝑤} is a dominating set of 𝐺 of cardinality |L| − 1 = 25 𝑛 − 1 < 25 𝑛, a contradiction. This completes the proof of Theorem 6.16. As an immediate consequence of Theorem 6.16, we have the following result. Corollary 6.17 ([586]) If 𝐺 is a 25 -minimal graph of order 𝑛 > 10, then 𝐺 ∈ Fdom . We are now in a position to state the McCuaig-Shepherd result. Theorem 6.18 ([586]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 is one of the seven exceptional graphs in the family Bdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. Let 𝐺 ′ be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Since removing edges from a graph cannot decrease its domination number, 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ). If 𝐺 ′ is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) < 25 𝑛. If 𝐺 ′ is a 25 -minimal graph, then by Theorem 6.16, 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 } or 𝐺 ′ ∈ Fdom ∪ Fsmall . If 𝐺 ′ ∈ Fdom ∪ Fsmall , then 𝑛 ≥ 5 and 𝛾(𝐺 ′ ) = 25 𝑛, implying that 𝛾(𝐺) ≤ 25 𝑛. If 𝐺 ′ = 𝐵1 , then 𝑛 = 4 and either 𝛾(𝐺) = 1 < 25 𝑛 or 𝐺 = 𝐺 ′ and 𝛾(𝐺) = 2 = 15 (2𝑛 + 2). If 𝐺 ′ ∈ {𝐵2 , 𝐵4 }, then 𝑛 = 7 and 𝛾(𝐺 ′ ) = 3 = 15 (2𝑛 + 1). Further, in this case either 𝛾(𝐺) ≤ 2 < 25 𝑛 or 𝐺 ∈ Bdom \ {𝐵1 }, in which case 𝛾(𝐺) = 15 (2𝑛 + 1). Thus, 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 ∈ Bdom . Let Gdom be the family of all graphs that can be obtained from a graph in the family Fdom by adding edges, including the possibility of none, joining link vertices. For example, for the graph 𝐹 ∈ Fdom shown in Figure 6.5(a) with the link vertices of 𝐹 given by the highlighted vertices, a graph 𝐺 ∈ Gdom obtained from 𝐹 is shown in Figure 6.5(b). We note that Fdom ⊂ Gdom . Theorem 6.19 ([79]) If 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, with equality if and only if 𝐺 ∈ Gdom .

Chapter 6. Upper Bounds in Terms of Minimum Degree

140

(a) 𝐹 ∈ Fdom

(b) 𝐺 ∈ Gdom

Figure 6.5 A graph 𝐺 ∈ Gdom obtained from a graph 𝐹 ∈ Fdom Proof Let 𝐺 ∈ Gdom have order 𝑛 > 10. Every dominating set of 𝐺 contains at least two vertices from each unit of 𝐺. However, there exists a dominating set of 𝐺 that contains exactly two vertices from each unit of 𝐺. Thus, 𝛾(𝐺) = 25 𝑛. Conversely, suppose that 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2 satisfying 𝛾(𝐺) = 25 𝑛. Let 𝐹 be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐹 is connected and (ii) 𝛿(𝐹) ≥ 2. If 𝐹 is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐹) < 25 𝑛, a contradiction. Hence, 𝐹 is a 25 -minimal graph of order 𝑛 > 10. Thus, by Corollary 6.17, 𝐹 ∈ Fdom and 𝛾(𝐹) = 25 𝑛. If there is an edge 𝑒 ∈ 𝐸 (𝐺) \ 𝐸 (𝐹) such that there is no construction of 𝐹 in which both ends of 𝑒 are link vertices, then analogous methods used in the proof of Claim 6.16.1 show that 𝛾(𝐺) < 𝛾(𝐹) = 25 𝑛, a contradiction. Hence, 𝐺 can be obtained from the graph 𝐹 ∈ Fdom by adding edges, including the possibility of none, joining link vertices of 𝐹, implying that 𝐺 ∈ Gdom . As remarked in [477], there are infinitely many 2-connected graphs that achieve equality in the bound of Theorem 6.18. One such family can be constructed as follows: Let 𝑘 ≥ 2 be an integer and let F2conn be the family of all graphs that can be obtained from a 2-connected graph 𝐹 of order 2𝑘 that contains a perfect matching 𝑀 as follows. Replace each edge 𝑢𝑣 ∈ 𝑀 by a 5-cycle containing 𝑢 and 𝑣 as nonadjacent vertices on the cycle. Let 𝐺 denote the resulting graph of order 𝑛 = 5𝑘. Then, 𝛾(𝐺) = 2𝑘 = 25 𝑛. A graph in the family F2conn with 𝑘 = 4 that is obtained from an 8-cycle 𝐹 is shown in Figure 6.6.

𝑢

𝑣

Figure 6.6 A graph in the family F2conn

6.2.3

Minimum Degree Three

In 1996 Reed [655] proved that the 25 -bound in Theorem 6.18 can be improved to a 3 8 -bound if we restrict the minimum degree to be at least 3. In this section, we present

Section 6.2. Bounds on the Domination Number

141

Reed’s proof of this classical result. For this purpose, recall that a vertex-disjoint path cover or just path cover, abbreviated vdp-cover, of a graph 𝐺 is set of vertex-disjoint paths 𝑄 1 , 𝑄 2 , . . . , 𝑄 𝑘 (not necessarily induced) that cover the vertices of 𝐺, that is, 𝑉=

𝑘 Ø

𝑉 (𝑄 𝑖 ).

𝑖=1

Let |𝑃| denote the number of vertices in a path 𝑃. A path 𝑃 is called a 0-, 1- or 2-path if |𝑃| is congruent to 0, 1, or 2 modulo 3, respectively. Thus, if 𝑃 is an 𝑖-path where 𝑖 ∈ [2] 0 , then |𝑃| ≡ 𝑖 (mod 3). If 𝑃 is a (𝑢, 𝑣)-path, then we call 𝑢 and 𝑣 the ends of the path 𝑃. Every vertex of a path different from its ends we call an internal vertex of the path. We now present the main ideas of the proof of Reed’s result. However, we omit some of the more technical details of the proof. Theorem 6.20 ([655]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 83 𝑛. Proof Sketch Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3. Let 𝑆 be a vdp-cover of 𝐺 and let 𝑆𝑖 be the set of 𝑖-paths in 𝑆 for 𝑖 ∈ [2] 0 . If 𝑣 is an internal vertex on a path 𝑃 and 𝑃 − 𝑣 consists of an 𝑖-path and a 𝑗-path, then we call 𝑣 an (𝑖, 𝑗)-vertex of 𝑃. An end 𝑣 of a path 𝑃 ∈ 𝑆 is an out-endvertex of the path 𝑃 if 𝑣 has a neighbor which is not on 𝑃. If 𝑃 ∈ 𝑆2 , and so 𝑃 is a 2-path, then an end of 𝑃 is a (2, 2)-endvertex of 𝑃 if it is not an out-endvertex and is adjacent to a (2, 2)-vertex of 𝑃. We define an optimal vdp-cover of 𝐺 to be a vdp-cover 𝑆 for which the following hold: (a) |𝑆| is minimized. (b) Subject to (a), 2|𝑆1 | + |𝑆2 | is minimized. (c) Subject to (b), |𝑆2 | is minimized. Í (d) Subject to (c), Í𝑃 ∈𝑆0 |𝑃| is minimized. (e) Subject to (d), 𝑃 ∈𝑆1 |𝑃| is minimized. (f) Subject to (e), the total number of out-endvertices is maximized. We proceed further with the following properties of an optimal vdp-cover 𝑆. Claim 6.20.1 If 𝑆 is an optimal vdp-cover of 𝐺 and 𝑥 is an out-endvertex of a path 𝑃 ∈ 𝑆1 ∪ 𝑆2 that is adjacent to a vertex 𝑦 in some path 𝑄 ∈ 𝑆 distinct from 𝑃, then the following hold: (a) 𝑄 is not a 1-path, that is, 𝑄 ∈ 𝑆0 ∪ 𝑆2 . (b) If 𝑄 ∈ 𝑆0 , then 𝑦 is a (1, 1)-vertex of 𝑄. (c) If 𝑄 ∈ 𝑆2 , then 𝑦 is a (2, 2)-vertex of 𝑄. Proof Sketch Let 𝑄 = 𝑄 𝑖 𝑦 𝑄 𝑗 , where 𝑄 𝑖 is an 𝑖-path and 𝑄 𝑗 is a 𝑗-path, and so 𝑦 is an (𝑖, 𝑗)-vertex of the path 𝑄. Interchanging 𝑄 𝑖 and 𝑄 𝑗 if necessary, we may assume 𝑖 ≤ 𝑗. By Condition (a) of the optimal vdp-cover 𝑆, vertex 𝑦 is not an end of 𝑄, and so 𝑦 is an internal vertex of 𝑄. Let 𝑆 ′ and 𝑆 ′′ be the vdp-covers obtained from 𝑆 as follows:
𝑆 ′ = 𝑆 \ {𝑃, 𝑄} ∪ {𝑄 𝑖 , 𝑃 𝑦 𝑄 𝑗 },
𝑆 ′′ = 𝑆 \ {𝑃, 𝑄} ∪ {𝑃 𝑦 𝑄 𝑖 , 𝑄 𝑗 }.

142

Chapter 6. Upper Bounds in Terms of Minimum Degree

(a) Suppose, to the contrary, that 𝑄 ∈ 𝑆1 . Then, 𝑦 is either a (0, 0)-vertex or a (1, 2)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 and 𝑦 is a (0, 0)-vertex, then 𝑆 ′ replaces two 1-paths with a 0-path and a 2-path, while if 𝑃 ∈ 𝑆1 and 𝑦 is a (1, 2)-vertex, then 𝑆 ′′ replaces two 1-paths with a 0-path and a 2-path. In both cases, we contradict the choice of 𝑆 as an optimal vdp-cover noting that we violate Condition (b). If 𝑃 ∈ 𝑆2 and 𝑦 is a (0, 0)-vertex, then 𝑆 ′ replaces a 1-path and a 2-path with two 0-paths, once again violating Condition (b) of the optimal vdp-cover 𝑆. If 𝑃 ∈ 𝑆2 and 𝑦 is a (1, 2)-vertex, then 𝑆 ′ has lengthened a 2-path and shortened a 1-path, violating Condition (e) of the optimal vdp-cover 𝑆. Both cases therefore contradict the choice of 𝑆 as an optimal vdp-cover. This proves part (a). (b) Suppose that 𝑄 ∈ 𝑆0 . Then, 𝑦 is either a (0, 2)-vertex or a (1, 1)-vertex of 𝑄. Suppose, to the contrary, that 𝑦 is a (0, 2)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 , then 𝑆 ′ has lengthened a 1-path and shortened a 0-path, while if 𝑃 ∈ 𝑆2 , then 𝑆 ′ has lengthened a 2-path and shortened a 0-path. In both cases, we violate Condition (d) of the optimal vdp-cover 𝑆. Hence, 𝑦 is a (1, 1)-vertex of 𝑄. This proves part (b). (c) Suppose that 𝑄 ∈ 𝑆2 . Then, 𝑦 is either a (0, 1)-vertex or a (2, 2)-vertex of 𝑄. Suppose, to the contrary, that 𝑦 is a (0, 1)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 , then 𝑆 ′ replaces a 1-path and a 2-path with two 0-paths, violating Condition (b) of the optimal vdp-cover 𝑆. If 𝑃 ∈ 𝑆2 , then 𝑆 ′ replaces two 2-paths with a 0-path and a 1-path, violating Condition (c) of the optimal vdp-cover 𝑆. Hence, 𝑦 is a (2, 2)-vertex of 𝑄. This proves part (c) and completes the proof of Claim 6.20.1. For each 1-path 𝑃 ∈ 𝑆 that has an out-endvertex 𝑥, we choose a vertex 𝑦 not in 𝑃 that is adjacent to 𝑥. For each 2-path 𝑃 ∈ 𝑆 that has two out-endvertices, we choose for each out-endvertex 𝑥 of 𝑃 a vertex 𝑦 not in 𝑃 that is adjacent to 𝑥. For each 2-path 𝑃 ∈ 𝑆 on five vertices that induces the graph 𝐹 shown in Figure 6.7 and which has precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹, we choose a vertex 𝑦 not in 𝑃 that is adjacent to the out-endvertex 𝑥 of 𝑃. In all the above cases, we designate the vertex 𝑦 as an acceptor for 𝑥, and we call 𝑦 an acceptor for the path 𝑃 associated with the vertex 𝑥.

𝑥 Figure 6.7 The graph 𝐹 We call a path in the optimal vdp-cover 𝑆 of the graph 𝐺 an accepting path if at least one of its vertices was designated as an acceptor for some path in 𝑆. Let 𝐴 ⊆ 𝑆 be the set of accepting 2-paths in 𝑆. For each out-endvertex 𝑥 of a path 𝑃 in 𝐴 for which we have not yet chosen an acceptor (this occurs because the path has only one out-endvertex), we choose a vertex adjacent to this out-endvertex in 𝐺 − 𝑉 (𝑃) and designate this vertex to be an acceptor for 𝑥. If this new acceptor is on a previously non-accepting 2-path 𝑃′ , then we add 𝑃′ to 𝐴. Continue this process until there is an acceptor for every out-endvertex of the paths in 𝐴. Furthermore, for

Section 6.2. Bounds on the Domination Number

143

every (2, 2)-endvertex 𝑥 of a path 𝑃 ∈ 𝐴, we choose a (2, 2)-vertex 𝑦 of 𝑃 which is adjacent to 𝑥 and designate it as an in-acceptor for 𝑥. By Claim 6.20.1, every acceptor vertex and every in-acceptor vertex that belongs to an accepting 2-path is a (2, 2)-vertex. Each accepting 2-path 𝑃 ∈ 𝑆 can therefore be written in the form 𝑄 1 𝑄 2 𝑄 3 , where 𝑄 1 and 𝑄 3 are both 1-paths containing no acceptors or in-acceptors, and are maximal with respect to this property. By Claim 6.20.1 and our earlier observations, the second and penultimate (that is second to last) vertices of 𝑄 2 are acceptors or in-acceptors. The paths 𝑄 1 and 𝑄 3 are called tips of 𝑃, while 𝑄 2 is called the central path of 𝑃. We next construct a dominating set 𝐷 of 𝐺 as follows. Initially, we let 𝐷 = ∅. We now build our dominating set 𝐷 in the following fashion: Step 1. Let 𝐷 0 be the set of all (1, 1)-vertices of 0-paths, and add all vertices in 𝐷 0 to the set 𝐷. Step 2. Let 𝐷 2 be the set of all (2, 2)-vertices of accepting 2-path 𝑃 that belong to the central path of 𝑃, and add all vertices in 𝐷 2 to the set 𝐷. To illustrate Steps 1 and 2, the (2, 2)-vertices of a 2-path 𝑃 and the three (1, 1)vertices of a 0-path 𝑃 are illustrated in Figure 6.8(a) and (b), respectively, by the highlighted vertices. By Claim 6.20.1, if 𝑥 is an out-endvertex of a path 𝑃 ∈ 𝑆1 ∪ 𝑆2 , then the vertex 𝑥 is dominated by the set 𝐷 0 ∪ 𝐷 2 . Thus, the set 𝐷 0 ∪ 𝐷 2 dominates all the out-endvertices of paths in 𝑆1 ∪ 𝑆2 . (2, 2)

(2, 2) (a) A 2-path

(1, 1)

(1, 1)

(1, 1)

(b) A 0-path

Figure 6.8 A 2-path and a 0-path   Step 3. For each 1-path with at least one out-endvertex, choose | 𝑃3 | vertices of 𝑃 which dominate all the vertices of 𝑃, except possibly for the out-endvertex 𝑥 of 𝑃 that is adjacent to the acceptor of 𝑃. As observed earlier, vertex 𝑥 is dominated   by the set 𝐷 0 ∪ 𝐷 2 . We now add these | 𝑃3 | vertices of 𝑃 to the set 𝐷. Step 4. For each non-accepting 2-path 𝑃 with both ends of 𝑃 either an outendvertex or a (2,2)-endvertex, we add all (2, 2)-vertices of 𝑃 to the set 𝐷. We  note that there are | 𝑃3 | such (2, 2)-vertices and these vertices dominate all vertices of 𝑃, except possibly for the ends of 𝑃 which are dominated by their acceptors or in-acceptors in the set 𝐷 0 ∪ 𝐷 2 . Step 5. For each 2-path 𝑃 on five vertices whose vertices induce the graph 𝐹 in Figure 6.6 and that has precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹, we add to the set 𝐷 a vertex in 𝐹 not adjacent to the out-endvertex 𝑥 of 𝐹. We note that such an added vertex dominates all vertices of 𝐹, except for the vertex 𝑥

144

Chapter 6. Upper Bounds in Terms of Minimum Degree

which is dominated by its acceptor that belongs to the set 𝐷 0 ∪ 𝐷 2 . Suppose that the 2-path 𝐹 is an accepting 2-path with acceptor 𝑦 for some path 𝑃′ associated with an out-endvertex 𝑥 ′ . In this case, 𝑦 is a (2, 2)-vertex of 𝐹 and so, by the structure of 𝐹, we could replace the two paths 𝑃′ and 𝐹 with one new path (that contains the edge 𝑥 ′ 𝑦, and has the vertex 𝑥 as an out-endvertex). This violates Condition (a) of the optimal vdp-cover 𝑆. Hence, the 2-path 𝐹 is a non-accepting 2-path. Step 6. For each 1-path 𝑃 with no out-endvertex, choose a 𝛾-set of 𝐺 [𝑉 (𝑃)] and add the vertices of this set to 𝐷. Let 𝐹 be the graph in Figure 6.7. For each non-accepting 2-path 𝑃 with at most one out-endvertex, which either does not induce the graph 𝐹 or induces the graph 𝐹 but does not have the vertex of degree 2 in 𝐹 as its out-endvertex, choose a 𝛾-set of 𝑃 and add the vertices of this set to 𝐷. In the   above cases, we note that the 𝛾-set of 𝐺 [𝑉 (𝑃)] has at most | 𝑃3 | vertices. Step 7. For each tip 𝑃1 of an accepting 2-path 𝑃, if the common end 𝑥 of 𝑃1 and 𝑃 is an out-endvertex or a (2, 2)-endvertex, then add to the  set 𝐷 all (2, 2)-vertices of 𝑃 that belong to the tip 𝑃1 . We note that there are | 𝑃3 | such (2, 2)-vertices and these vertices dominate all vertices of 𝑃1 , except possibly for 𝑥, which is dominated by the set 𝐷 0 ∪ 𝐷 2 . Step 8. For each tip 𝑃1 of an accepting 2-path 𝑃, if the common end 𝑥 of 𝑃1 and 𝑃 is neither an out-endvertex nor a (2, 2)-endvertex, then choose a 𝛾-set of 𝐺 [𝑉 (𝑃1 )] and add the vertices of this set to 𝐷. We note that the 𝛾-set of 𝐺 [𝑉 (𝑃)] has at most  |𝑃|  3 vertices. We show next that the set 𝐷 constructed above is a dominating set of 𝐺. Claim 6.20.2 The set 𝐷 constructed by Steps 1–8 is a dominating set of 𝐺. Proof Sketch As observed earlier, every acceptor belongs to the set 𝐷 0 ∪ 𝐷 2 , and therefore belongs to the set 𝐷. By construction, the set 𝐷 0 chosen in Step 1 dominates the vertices on all 0-paths. Hence, the vertices of the 0-paths are dominated by 𝐷. The vertices chosen in Step 3 ensure that all vertices of a 1-path 𝑃 with an out-endvertex are dominated by 𝐷 since all acceptors are in 𝐷, while the vertices chosen in Step 6 ensure that all vertices of a 1-path 𝑃 with no out-endvertex are dominated by 𝐷. Hence, the vertices of the 1-paths are dominated by 𝐷. The vertices chosen in Steps 4, 5, and 6 ensure that all vertices of non-accepting 2-paths 𝑃 are dominated by 𝐷, since all acceptors are in 𝐷. Hence, the vertices of all non-accepting 2-paths are dominated by 𝐷. The vertices in 𝐷 2 chosen in Step 2 dominate all vertices that belong to a central path of accepting 2-paths, while the vertices chosen in Steps 7 and 8 ensure that all vertices that belong to the tips of accepting 2-paths are dominated by 𝐷. Hence, the vertices of all accepting 2-paths are dominated by 𝐷. Thus, the vertices of the 2-paths are dominated by 𝐷. By Claim 6.20.2, the set 𝐷 is a dominating set of 𝐺. It remains to show that |𝐷| ≤ 38 𝑛. For this purpose, we define the following sets: • 𝑂 1 : the set of 1-paths 𝑃 which either have an out-endvertex or contain a dominating   set of cardinality | 𝑃3 | . • 𝑂 2 : the set of non-accepting 2-paths which either have two out-endvertices or a   dominating set of cardinality | 𝑃3 | and all 2-paths which induce the graph 𝐹 of

Section 6.2. Bounds on the Domination Number

145

Figure 6.7 and which have precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹. • 𝐼1 : the set of 1-paths not in 𝑂 1 . • 𝐼2 : the set of non-accepting 2-paths not in 𝑂 2 . • 𝐸: a tip 𝑇 of an accepting 2-path 𝑃 is in 𝐸 if and only if the corresponding end of 𝑃 is neither  an out-endvertex nor a (2, 2)-endvertex, and 𝑇 cannot be dominated by the | 𝑃3 | vertices of the central path of 𝑃. • 𝑊: the set of (2, 2)-endvertices of accepting 2-paths for which we have chosen an in-acceptor. Recall that 𝐴 ⊆ 𝑆 is the set of accepting 2-paths in 𝑆. The cardinality of the dominating set 𝐷 is given by ∑︁ |𝑃| − 1 ∑︁ |𝑃| − 2 ∑︁ |𝑃| + 2 |𝐷| = + + 3 3 3 𝑃∈ 𝐼1 𝑃∈𝑂1 𝑃∈𝑂2 ∑︁ |𝑃| + 1 ∑︁ |𝑃| ∑︁ |𝑃| − 2 + + + + |𝐸 |. 3 3 3 𝑃∈ 𝐼 𝑃∈𝑆 𝑃∈ 𝐴 2

0

Equivalently, |𝐷| =

𝑛 3

− 13 |𝑂 1 | − 23 |𝑂 2 | + 23 |𝐼1 | + 13 |𝐼2 | − 23 | 𝐴| + |𝐸 |.

(6.1)

Note that each accepting 2-path corresponds to an out-endvertex of some path in 𝑂 1 ∪ 𝑂 2 or to an out-endvertex of an accepting 2-path of 𝐴 which is not in 𝐸 ∪ 𝑊. Thus, | 𝐴| ≤ |𝑂 1 | + 2|𝑂 2 | + 2| 𝐴| − |𝐸 | − |𝑊 |, or equivalently, |𝐸 | ≤ |𝑂 1 | + 2|𝑂 2 | + | 𝐴| − |𝑊 |. (6.2) We note that |𝐸 | ≤ 2| 𝐴| − |𝑊 |. Hence, by Inequality (6.2), |𝐸 | ≤ 23 |𝑂 1 | + 43 |𝑂 2 | + 43 | 𝐴| − |𝑊 |, or equivalently, − 23 | 𝐴| ≤ 13 |𝑂 1 | + 23 |𝑂 2 | − 12 |𝐸 | − 12 |𝑊 |.

(6.3)

Substituting Inequality (6.3) into Equation (6.1), |𝐷 | ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 | − 12 |𝑊 |.

To each element of 𝐸 there corresponds an accepting 2-path 𝑃𝑇 such that 𝑇 is the tip of 𝑃𝑇 . With this notation, we define 𝐸 ′ ⊆ 𝐸 as follows: 𝐸 ′ = {𝑇 ∈ 𝐸 : the end of 𝑃𝑇 not in 𝑇 is not an element of 𝑊 }. Clearly, |𝐸 ′ | ≥ |𝐸 | − |𝑊 |, and so |𝐷| ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 ′ |.

(6.4)

We proceed further with the following two claims. We omit the proofs of these claims, which are technical in parts, and can also be checked using a computer proof.

Chapter 6. Upper Bounds in Terms of Minimum Degree

146

Claim 6.20.3 The following hold: (a) Each 1-path 𝑃 ∈ 𝐼1 satisfies |𝑃| ≥ 16. (b) Each 2-path 𝑃 ∈ 𝐼2 satisfies |𝑃| ≥ 8. Claim 6.20.4 If a path 𝑇 is a tip in 𝐸 ′ of an accepting 2-path 𝑃, and 𝑐 0 is the end of the central path in 𝑃 that is adjacent to an end of 𝑇, then the following hold: (a) If 𝐺 is a cubic graph, then either |𝑇 | ≥ 10 or 𝐺 is isomorphic to the graph 𝐻1 shown in Figure 6.9. (b) If 𝐺 is not a cubic graph, then |𝑇 | ≥ 7, with equality if and only if the subgraph of 𝐺 induced by 𝑉 (𝑇) ∪ {𝑐 0 } is isomorphic to the graph 𝐻1 , where in addition the vertex 𝑐 0 is the only vertex in 𝐻1 whose degree in 𝐺 is greater than 3. (c) If 𝐺 is not a cubic graph, then |𝑇 | ≥ 10, with the exception of at most two tips 𝑇 in 𝐸 ′ (that satisfy |𝑇 | = 7).

𝑐0

Figure 6.9 The graph 𝐻1 in the statement of Claim 6.20.4

It follows from Claim 6.20.3(a) and (b) that ∑︁ ∑︁ |𝑃| ≥ 16|𝐼1 | and |𝑃| ≥ 8|𝐼2 |. 𝑃∈ 𝐼1

(6.5)

𝑃∈𝐼2

We note that the central path of an accepting 2-path has at least three vertices. Assume first that 𝐺 is a cubic graph. If 𝐺 = 𝐻1 , then 𝛾(𝐺) = 3 = 38 𝑛, as desired. Hence, we may assume that 𝐺 ≠ 𝐻1 . Each element 𝑇 of 𝐸 ′ is the tip of some accepting 2-path 𝑃, and by Claim 6.20.4(a) satisfies |𝑇 | ≥ 10. If the other tip 𝑇 ′ of 𝑃 is also in 𝐸 ′ , then by Claim 6.20.4, we have |𝑃| ≥ 23. Otherwise, |𝑃| ≥ 14, or equivalently, |𝑃| − 1 ≥ 13. In addition, we know that there is some out-endvertex whose acceptor is on 𝑃. For an accepting 2-path 𝑃, let 𝑡 (𝑃) = (the number of tips of 𝑃 in 𝐸 ′ ) 𝜓1 (𝑃) = (the number of out-endvertices whose acceptor is on 𝑃) 𝜓2 (𝑃) = (the number of out-endvertices of 𝑃). Hence, 𝜓1 (𝑃) ≥ 1 and 𝑡 (𝑃) ∈ {0, 1, 2}. If 𝑡 (𝑃) = 0, then |𝑃| ≥ 5 and 𝜓2 (𝑃) ≤ 2. If 𝑡 (𝑃) = 1, then |𝑃| ≥ 14 and 𝜓2 (𝑃) ≤ 1. If 𝑡 (𝑃) = 2, then |𝑃| ≥ 23 and 𝜓2 (𝑃) = 0. Thus, for every accepting 2-path 𝑃, |𝑃| + 𝜓1 (𝑃) − 𝜓2 (𝑃) ≥ 12𝑡 (𝑃).

(6.6)

Section 6.2. Bounds on the Domination Number

147

Let 𝜑( 𝐴) = (the number of out-endvertices of paths in 𝐴) 𝜑(𝑂 1 ) = (the number of out-endvertices of paths in 𝑂 1 that have a designated acceptor) 𝜑(𝑂 2 ) = (the number of out-endvertices of paths in 𝑂 2 that have a designated acceptor) 𝜑(𝑆) = (the number of out-endvertices of paths in 𝑆 that have a designated acceptor). Then, ∑︁

∑︁

𝜓1 (𝑃) = 𝜑(𝑆),

𝑃∈ 𝐴

𝜓2 (𝑃) = 𝜑( 𝐴),

∑︁

and

𝑡 (𝑃) = |𝐸 ′ |.

𝑃∈ 𝐴

𝑃∈ 𝐴

Thus, by Inequality (6.6), summing over all paths 𝑃 ∈ 𝐴, we have   ∑︁ |𝑃| + 𝜑(𝑆) − 𝜑( 𝐴) ≥ 12|𝐸 ′ |.

(6.7)

𝑃∈ 𝐴

For each path 𝑃 ∈ 𝑆, let 𝜑(𝑃) = (the number of out-endvertices of 𝑃 that have a designated acceptor). For each path 𝑃 ∈ 𝑂 1 ∪ 𝑂 2 , |𝑃| ≥ 𝜑(𝑃). Thus, ∑︁ |𝑃| ≥ 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ). (6.8) 𝑃∈𝑂1 ∪𝑂2

Since 𝜑( 𝐴) + 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ) = 𝜑(𝑆), we have by Inequalities (6.7) and (6.8) that ∑︁ (6.9) |𝑃| ≥ 𝜑( 𝐴) + 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ) − 𝜑(𝑆) + 12|𝐸 ′ | = 12|𝐸 ′ |. 𝑃∈ 𝐴∪𝑂1 ∪𝑂2

Since 𝑆 = 𝑂 1 ∪ 𝑂 2 ∪ 𝐼1 ∪ 𝐼2 ∪ 𝐴, we have by Inequalities (6.5) and (6.9) that ∑︁ 𝑛= |𝑃| ≥ 16|𝐼1 | + 8|𝐼2 | + 12|𝐸 ′ |, 𝑃 ∈𝑆

or equivalently, 2 3 |𝐼1 |

+ 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 24 .

(6.10)

Substituting Inequality (6.10) into Inequality (6.4) yields 𝛾(𝐺) = |𝐷| ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 3

+

𝑛 24

= 38 𝑛.

This completes the proof of Theorem 6.20 in the case when 𝐺 is a cubic graph. If 𝐺 is not a cubic graph, then analogous arguments as above show, using Claim 6.20.4, that ∑︁ |𝑃| ≥ 12|𝐸 ′ | − 6 𝑃∈ 𝐴∪𝑂1 ∪𝑂2

148

Chapter 6. Upper Bounds in Terms of Minimum Degree

and 𝑛+6=

∑︁

|𝑃| ≥ 16|𝐼1 | + 8|𝐼2 | + 12|𝐸 ′ |,

𝑃∈𝑆

or equivalently, 2 3 |𝐼1 |

+ 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 24

+ 14 .

(6.11)

Substituting Inequality (6.11) into Inequality (6.4) yields 𝛾(𝐺) ≤ |𝐷| ≤ 38 𝑛 + 14 . With additional work using similar techniques, Reed [655] showed that the 14 -term can be omitted in Inequality (6.11) to yield the desired bound 𝛾(𝐺) ≤ 38 𝑛. That the bound in Theorem 6.20 is tight may be seen as follows. Let F≥3 be the family of connected graphs with minimum degree at least 3 that can be obtained from a connected graph 𝐹 by adding a vertex-disjoint copy of the cubic graph 𝐻1 shown in Figure 6.9 for each vertex 𝑣 of 𝐹 and identifying the vertex 𝑣 of 𝐹 with the vertex 𝑐 0 of 𝐻1 . Let 𝐺 denote the resulting graph of order 𝑛 = 8|𝑉 (𝐹)| and minimum degree 3. In the special case when the graph 𝐹 is the cycle 𝐶4 , then the associated graph 𝐺 constructed from 𝐹 is illustrated in Figure 6.10. No two vertices in each copy of the graph 𝐻1 dominate all but the top, highlighted vertex in that copy. Every dominating set in 𝐺 therefore contains at least three vertices from each copy of the graph 𝐻1 , and so 𝛾(𝐺) ≥ 3|𝑉 (𝐹)| = 38 𝑛. By Theorem 6.20, 𝛾(𝐺) ≤ 38 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛.

Figure 6.10 A graph in the family F≥3 In 2009 Shan et al. [673] modified the proof techniques employed by Reed [655] to obtain the following generalization of the result in Theorem 6.20 by allowing vertices of degree 2 back into the mix. Theorem 6.21 ([673]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and 𝑛2 vertices of degree 2, then 𝛾(𝐺) ≤ 38 𝑛 + 18 𝑛2 . If 𝑛2 = 0, then Theorem 6.21 reduces to Reed’s result given by Theorem 6.20. As an immediate consequence of Theorem 6.21, we have the following. Corollary 6.22 ([673]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and fewer than vertices of degree 2, then 𝛾(𝐺) < 25 𝑛.

𝑛 5

By Corollary 6.22, if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and fewer than 𝑛5 vertices of degree 2, then the result of Theorem 6.21 is better than that of Theorem 6.18.

Section 6.2. Bounds on the Domination Number

149

As a special case of Theorem 6.20, we have the following result. Theorem 6.23 ([655]) If 𝐺 is a cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 83 𝑛. The two non-planar cubic graphs 𝐺 of order 𝑛 = 8 shown in Figure 6.11 satisfy 𝛾(𝐺) = 3 = 38 𝑛, showing that the upper bound in Theorem 6.23 is achievable.

(a)

(b)

Figure 6.11 The two non-planar cubic graphs of order 𝑛 = 8 We remark that Reed’s proof of Theorem 6.20 (and Theorem 6.23), which uses ingenious counting arguments, is technical in parts. Indeed some cases in the proof (namely, Claims 6.20.3 and 6.20.4) are best checked using a computer. In 2015 Dorbec et al. [236] presented a completely different proof to establish the 38 -bound on the domination number of a cubic graph that does not need a computer to check. A subcubic graph is a graph with maximum degree at most 3. Theorem 6.24 ([236]) If 𝐺 is a subcubic graph with 𝑛𝑖 vertices of degree 𝑖 for 𝑖 ∈ [3] 0 , then 8𝛾(𝐺) ≤ 8𝑛0 + 5𝑛1 + 4𝑛2 + 3𝑛3 . As a special case in Theorem 6.24 when 𝐺 is a cubic graph, we have the important result given in Theorem 6.23, noting that in this case 𝑛0 = 𝑛1 = 𝑛2 = 0. If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then we note that 8𝑛0 + 5𝑛1 + 4𝑛2 + 3𝑛3 = 6𝑛 − 2𝑚 + 2𝑖. Hence, as an immediate consequence of Theorem 6.24, we have the following result from 1999 due to Fisher et al. [303] and Rautenbach [649]. Corollary 6.25 ([303, 649]) If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 4𝛾(𝐺) ≤ 3𝑛 − 𝑚 + 𝑖. As observed earlier, the two non-planar cubic graphs 𝐺 of order 𝑛 = 8 shown in Figure 6.11 satisfy 𝛾(𝐺) = 3 = 38 𝑛. In 2009 Kostochka and Stocker [537] proved that these two non-planar cubic graphs are the only connected cubic graphs that achieve the three-eights bound in Theorem 6.23. Excluding these graphs gave the following improved bound. 5 Theorem 6.26 ([537]) If 𝐺 is a connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 14 𝑛, unless 𝐺 is one of the two non-planar cubic graphs of order 𝑛 = 8 (shown in Figu1 re 6.11) in which case 𝛾(𝐺) = 3 = 14 (5𝑛 + 2).

For 𝑝 ≥ 3, 𝑝 > 𝑘 ≥ 1, and gcd( 𝑝, 𝑘) = 1, a generalized Petersen graph 𝑃( 𝑝, 𝑘) is the graph with vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑝−1 } ∪ {𝑤 0 , 𝑤 1 , . . . , 𝑤 𝑝−1 } and edges 𝑣 𝑖 𝑤 𝑖 ,

150

Chapter 6. Upper Bounds in Terms of Minimum Degree

𝑣 𝑖 𝑣 𝑖+1 and 𝑤 𝑖 𝑤 𝑖+𝑘 for 𝑖 ∈ [ 𝑝 − 1] 0 and the subscript sum is taken modulo 𝑝. In the special case when 𝑝 = 5 and 𝑘 = 2, the graph 𝑃( 𝑝, 𝑘) is the famous Petersen graph, illustrated in Figure 6.12.

Figure 6.12 The Petersen graph 𝑃(5, 2) If 𝐺 is the generalized Petersen graph 𝑃(7, 2) shown in Figure 6.13, then 𝐺 5 is a connected cubic graph of order 𝑛 = 14 satisfying 𝛾(𝐺) = 5 = 14 𝑛. Hence, 5 the 14 -upper bound of Theorem 6.26 is achievable. More generally, Kostochka and 5  Stocker [537] showed that the upper bound 14 𝑛 on the domination number of a connected cubic graph of order 𝑛 is achievable for 𝑛 ∈ {10, 12, 14, 16, 18}.

Figure 6.13 The generalized Petersen graph 𝑃(7, 2) In 1996 Reed [655] conjectured that his upper bound given in Theorem   6.23 on the domination number of a cubic graph can be improved to 𝛾(𝐺) ≤ 13 𝑛 . In 2005 Kostochka and Stodolsky [538] disproved his conjecture by constructing a connected cubic graph 𝐺 on 60 vertices with 𝛾(𝐺) = 21 and presented a sequence {𝐺 𝑘 }∞ 𝑘=1 of connected cubic graphs with lim

𝑘→∞

𝛾(𝐺 𝑘 ) 8 1 1 ≥ = + . |𝑉 (𝐺 𝑘 )| 23 3 69

In 2006 Kelmans [521] constructed a connected cubic graph 𝐺 on 54 vertices with 𝛾(𝐺) = 19 and gave an infinite series of 2-connected cubic graphs 𝐻 𝑘 with lim

𝑘→∞

𝛾(𝐻 𝑘 ) 1 1 ≥ + . |𝑉 (𝐻 𝑘 )| 3 60

Section 6.2. Bounds on the Domination Number

151

𝑛 denotes the family of all connected cubic graphs of order 𝑛, then as a If Gcubic consequence of Theorem 6.26 and the above result due to Kelmans,   1 1 𝛾(𝐺) 5 1 1 0.35 = + ≤ lim sup ≤ = + ≈ 0.3571. 𝑛 3 60 𝑛→∞ 𝐺 ∈ Gcubic 𝑛 14 3 42

It remains, however, an open problem to determine the limit of the supremum of the ratio of the domination number over the order of a connected cubic graph. It is not 5 known if there are graphs of large order that achieve the 14 -bound in Theorem 6.26. The problem of determining a tight upper bound on the domination number of a connected cubic graph of sufficiently large order, in terms of its order, remains one of the major outstanding problems in domination theory.

6.2.4

Minimum Degree Four

We remark that the results of Ore given in Theorem 6.2, Blank [79] and McCuaig and Shepherd [586] given in Theorem 6.18, and Reed [655] given in Theorem 6.20, all have the same form which we present in Theorem 6.27. Theorem 6.27 If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 where 𝛿 ∈ [3], then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1 unless 𝛿 = 2 and 𝐺 is one of the seven exceptional graphs in the family Bdom shown in Figure 6.1. Motivated by the form of the bound on the domination number given in Theorem 6.27, Haynes et al. [417] conjectured in 1998 that this upper bound on the domination number holds for any minimum degree 𝛿 ≥ 1. Conjecture 6.28 ([417]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1 unless 𝛿 = 2 and 𝐺 is one of the seven graphs in the family Bdom shown in Figure 6.1. Conjecture 6.28 is true as the following results, namely Theorems 6.29 and 6.35, show. In fact, we will give an improved bound in Chapter 7. In 2009 Sohn and Xudong [682] proved that Conjecture 6.28 holds for 𝛿 = 4, that 4 is, they proved that the 38 -bound in Theorem 6.20 can be improved to a 11 -bound if the minimum degree is at least 4. Theorem 6.29 ([682]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤

4 11 𝑛.

The proof of Sohn and Xudong [682] uses the vertex-disjoint path cover method first employed by Reed [655], as demonstrated in the proof of Theorem 6.20.

152

Chapter 6. Upper Bounds in Terms of Minimum Degree

4 In 2021 Bujtás [123] gave a simpler proof of the 11 -bound on the domination number of graph with minimum degree 4 that combines an algorithmic approach using weighting arguments together with discharging methods. In order to present her proof, we first define the concept of a residual graph.

Definition 6.30 Given a graph 𝐺 with vertex set 𝑉 = 𝑉 (𝐺) and a subset 𝐷 ⊆ 𝑉, the residual graph denoted by 𝐺 𝐷 is the graph obtained from 𝐺 by assigning colors to the vertices and deleting certain edges according to the following rules: (a) A vertex 𝑣 is white if 𝑣 ∉ N[𝐷]. (b) A vertex 𝑣 is blue if 𝑣 ∈ N[𝐷] and N[𝑣] ⊈ N[𝐷]. (c) A vertex 𝑣 is red if N[𝑣] ⊆ N[𝐷]. (d) The edges of 𝐺 𝐷 consist of all edges of 𝐺 that are incident with at least one white vertex. Thus, in a residual graph 𝐺 𝐷 associated with a set 𝐷 of vertices of 𝐺, a vertex not dominated by 𝐷 is colored white, a vertex dominated by 𝐷 but with at least one neighbor not yet dominated by 𝐷 (and therefore colored white) is colored blue, and a vertex whose closed neighborhood is dominated by 𝐷 is colored red. We denote the set of white, blue, and red vertices by 𝑊, 𝐵, and 𝑅, respectively. We note that 𝐷 ⊆ 𝑅 and 𝑉 = 𝑊 ∪ 𝐵 ∪ 𝑅. As an illustration, consider the graph 𝐺 and the set 𝐷 of vertices shown in Figure 6.14(a), where the vertices in 𝐷 are highlighted. Applying Bujtás’s coloring rules, we color the vertices of 𝐺 with the colors white, blue, and red as shown in Figure 6.14(b). Thereafter, we delete all edges that are not incident with at least one white vertex from the graph, that is, we remove from the graph all edges incident with two red vertices or with two blue vertices or with one red vertex and one blue vertex. The resulting graph is the residual graph 𝐺 𝐷 shown in Figure 6.14(c).

(a) A set 𝐷 of vertices in a graph 𝐺

(b) The associated coloring of the vertices in 𝐺

(c) The residual graph 𝐺 𝐷

Figure 6.14 The residual graph 𝐺 𝐷 of a graph 𝐺 with a given set 𝐷 ⊂ 𝑉 (𝐺) We define the white-degree of a vertex 𝑣 in 𝐺 𝐷 , denoted by deg𝑊 (𝑣), as the number of white neighbors of the vertex 𝑣, and we define the blue-degree of a (white) vertex 𝑣 in 𝐺 𝐷 , denoted by deg 𝐵 (𝑣), as the number of blue neighbors of the vertex 𝑣.

Section 6.2. Bounds on the Domination Number

153

The maximum white-degree over all white and blue vertices, respectively, we denote by Δ𝑊 (𝑊) and Δ𝑊 (𝐵), respectively. By Definition 6.30, we have the following observation. Observation 6.31 ([123]) The following hold in the residual graph 𝐺 𝐷 : (a) Every white vertex 𝑣 has no red neighbor, and so deg𝐺 (𝑣) = deg𝑊 (𝑣) + deg 𝐵 (𝑣) ≥ 𝛿(𝐺). (b) A blue vertex 𝑣 has at least one white neighbor, and no blue or red neighbors. Further, if 𝑣 is a blue vertex, then at least one neighbor of 𝑣 in 𝐺 is colored red or blue in 𝐺 𝐷 , implying that deg𝑊 (𝑣) < deg𝐺 (𝑣). (c) Every red vertex is isolated in 𝐺 𝐷 . (d) If 𝐷 ⊂ 𝐷 ′ ⊆ 𝑉, then every red vertex in 𝐺 𝐷 remains red in 𝐺 𝐷 ′ , while every blue vertex in 𝐺 𝐷 is colored blue or red in 𝐺 𝐷 ′ . The number of white neighbors of a vertex in 𝐺 𝐷 ′ is therefore at most the number of its white neighbors in 𝐺 𝐷 . (e) The set 𝐷 is a dominating set in 𝐺 if and only if every vertex of 𝐺 𝐷 is colored red, that is, 𝑅 = 𝑉. 4 We now present the simpler proof due to Bujtás [123] of the 11 -bound on the domination number given in Theorem 6.29, which we restate here.

Theorem 6.29 ([682]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤

4 11 𝑛.

Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 and let 𝐷 ⊆ 𝑉. We define 𝐵4 as the set of blue vertices of degree at least 4 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [3]. Thus, every (blue) vertex in 𝐵4 has at least four white neighbors, while every (blue) vertex in 𝐵𝑖 has exactly 𝑖 white neighbors for 𝑖 ∈ [3]. A vertex is a blue leaf if it belongs to 𝐵1 . For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.1.

set containing 𝑣

𝑊

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

16

10

9

8

7

0

Table 6.1 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 16|𝑊 | + 10|𝐵4 | + 9|𝐵3 | + 8|𝐵2 | + 7|𝐵1 |. 𝑣 ∈𝑉

By Observation 6.31(e), w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. Given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 )

154

Chapter 6. Upper Bounds in Terms of Minimum Degree

as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. We define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 44| 𝐴|. For 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝐷 ′ = 𝐴 ∪ 𝐷, and we define 𝑊 ′ , 𝐵′ and 𝑅 ′ as the set of white, blue and red vertices, respectively, in 𝐺 𝐷 ′ . We define 𝐵4′ as the set of blue vertices of degree at least 4 in 𝐺 𝐷 ′ , and 𝐵𝑖′ as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 ′ for 𝑖 ∈ [3]. We proceed further by proving that the following property holds. Claim 6.29.1 If w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. Proof Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 but the graph 𝐺 does not contain a 𝐷-desirable set. In what follows, we present a series of subclaims describing some structural properties of 𝐺, which culminate in the existence of a 𝐷-desirable set. Subclaim 6.29.1.1 Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. Proof Suppose that Δ𝑊 (𝑊) ≥ 5. In this case, let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Thus, the weight of 𝑣 decreases by 16, while the weight of each white neighbor of 𝑣 decreases by at least 16 − 10 = 6. Thus, 𝜓( 𝐴) ≥ 16 + 5 · 6 = 46 > 44 = 44| 𝐴|, and so the set 𝐴 is a 𝐷-desirable set, contradicting our supposition that no such set exists. Hence, Δ𝑊 (𝑊) ≤ 4. Suppose that Δ𝑊 (𝑊) = 4. As before, let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Let 𝑢 be an arbitrary white neighbor of 𝑣 in 𝐺 𝐷 . The vertex 𝑢 has at most four white neighbors in 𝐺 𝐷 , and therefore has at most three white neighbors in 𝐺 𝐷 ′ . Thus, 𝑢 ∈ 𝐵3′ ∪ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 9 = 7. Therefore, 𝜓( 𝐴) ≥ 16 + 4 · 7 = 44 = 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝑊) ≤ 3. Suppose that Δ𝑊 (𝐵) ≥ 5. Let 𝑣 be a blue vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝐵), and let 𝐴 = {𝑣}. We note that 𝑣 ∈ 𝐵4 and 𝑣 ∈ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 10. Let 𝑢 be an arbitrary (white) neighbor of 𝑣 in 𝐺 𝐷 . Since Δ𝑊 (𝑊) ≤ 3, vertex 𝑢 has at most three white neighbors in 𝐺 𝐷 . Thus, 𝑢 ∈ 𝐵3′ ∪ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 9 = 7. Therefore, 𝜓( 𝐴) ≥ 10 + 5 · 7 = 45 > 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝐵) ≤ 4. Suppose that Δ𝑊 (𝑊) = 3. Since 𝛿(𝐺) ≥ 4, it follows from Observation 6.31(a) and (d) that every white vertex has at least one blue neighbor. Let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Let 𝑢 be an arbitrary white neighbor of 𝑣 in 𝐺 𝐷 . The vertex 𝑢 has at most two white neighbors in 𝐺 𝐷 ′ . Thus, 𝑢 ∈ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 8 = 8. Let 𝑁 𝑣 be the set consisting of 𝑣 and its three white neighbors in 𝐺 𝐷 . By our earlier observations, every vertex in 𝑁 𝑣 has a blue neighbor. Thus, there are ℓ ≥ 4 edges joining vertices in 𝑁 𝑣 to blue vertices. Since Δ𝑊 (𝐵) ≤ 4, these ℓ edges

Section 6.2. Bounds on the Domination Number

155

result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑁 𝑣 by at least ℓ ≥ 4. Therefore, 𝜓( 𝐴) ≥ 16 + 3 · 8 + 4 = 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝑊) ≤ 2. Suppose that Δ𝑊 (𝐵) = 4. Let 𝑣 be a blue vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝐵), and let 𝐴 = {𝑣}. We note that 𝑣 ∈ 𝐵4 and 𝑣 ∈ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 10. Let 𝑢 be an arbitrary (white) neighbor of 𝑣 in 𝐺 𝐷 . Since Δ𝑊 (𝑊) ≤ 2, the vertex 𝑢 has at most two white neighbors in 𝐺 𝐷 and at least one blue neighbor different from 𝑣. Thus, 𝑢 ∈ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 8 = 8. Further, there are ℓ ≥ 4 edges joining white neighbors of 𝑣 in 𝐺 𝐷 to blue vertices. These ℓ edges result in a decrease in the weight contribution of the blue neighbors of vertices in 𝑁 𝑣 by at least ℓ ≥ 4. Therefore, 𝜓( 𝐴) ≥ 10 + 4 · 8 + 4 = 46 > 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝐵) ≤ 3. This completes the proof of Subclaim 6.29.1.1. By Subclaim 6.29.1.1, Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. Subclaim 6.29.1.2 Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 or 𝐶7 . Proof Since Δ𝑊 (𝑊) ≤ 2, every component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path or a cycle. Suppose that 𝐺 𝐷 [𝑊] contains a path component 𝑣 1 𝑣 2 . . . 𝑣 𝑗 where 𝑗 ≥ 3. We let 𝐴 = {𝑣 2 }. In the graph 𝐺 𝐷 ′ , both 𝑣 1 and 𝑣 2 belong to 𝑅 ′ , and so each has a weight decrease of 16. The vertex 𝑣 3 belongs to the set 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ . If 𝑣 3 ∈ 𝐵1′ , then its weight decrease is 16 − 7 = 9, while if 𝑣 3 ∈ 𝑅 ′ , then its weight decrease is 16. Let 𝑋 = {𝑣 1 , 𝑣 2 , 𝑣 3 } and note that there are at least 3 + 2 + 2 = 7 edges that join vertices in 𝑋 in 𝐺 𝐷 to blue vertices. These edges result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑋 by at least 7. Therefore, 𝜓( 𝐴) ≥ 2 · 16 + 9 + 7 = 48 > 44| 𝐴|, a contradiction. Hence, every path component in 𝐺 𝐷 [𝑊] is either a path 𝑃1 or 𝑃2 . Suppose next that 𝐺 𝐷 [𝑊] contains a cycle component 𝐶 : 𝑣 1 𝑣 2 . . . 𝑣 𝑞 for some 𝑞 ≥ 3. In this case, we let 𝐴 = {𝑣 𝑞 } ∪

⌊𝑞/3⌋ Ø

! {𝑣 3𝑖 } .

𝑖=1

Suppose firstly that 𝑞 ≡ 0 (mod 3). Thus, 𝑞 = 3𝑘 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 } and | 𝐴| = 𝑘. In the graph 𝐺 𝐷 ′ , all vertices of 𝐶 belong to 𝑅 ′ , and therefore result in a decrease in the weight by 16 · 3𝑘 = 48𝑘. We note that there are at least 2 · 3𝑘 = 6𝑘 edges that join vertices in 𝑉 (𝐶) in 𝐺 𝐷 to blue vertices. These edges result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑉 (𝐶) by at least 6𝑘. Therefore, 𝜓( 𝐴) ≥ 48𝑘 + 6𝑘 = 54𝑘 > 44𝑘 = 44| 𝐴|, a contradiction. Suppose secondly that 𝑞 ≡ 2 (mod 3). Thus, 𝑞 = 3𝑘 + 2 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 , 𝑣 3𝑘+2 } and | 𝐴| = 𝑘 + 1. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 16 · (3𝑘 + 2) + 2 · (3𝑘 + 2) = 54𝑘 + 36 ≥ 44𝑘 + 44 = 44| 𝐴|, a contradiction.

156

Chapter 6. Upper Bounds in Terms of Minimum Degree

Suppose thirdly that 𝑞 ≡ 1 (mod 3). Thus, 𝑞 = 3𝑘 + 1 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 , 𝑣 3𝑘+1 } and | 𝐴| = 𝑘 + 1. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 16 · (3𝑘 + 1) + 2 · (3𝑘 + 1) = 54𝑘 + 18. If 𝑘 ≥ 3, then 𝜓( 𝐴) ≥ 54𝑘 + 18 > 44𝑘 + 44 = 44| 𝐴|, a contradiction. Hence, 𝑘 ∈ [2], that is, the component 𝐶 is either 𝐶4 or 𝐶7 . This completes the proof of Subclaim 6.29.1.2. Let 𝑊𝑖 be the set of white vertices having exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We note that 𝑊0 consists of all vertices of 𝑊 that belong to 𝑃1 -components of 𝐺 𝐷 [𝑊], while 𝑊1 consists of all vertices of 𝑊 that belong to 𝑃2 -components of 𝐺 𝐷 [𝑊]. By Subclaim 6.29.1.2, the set 𝑊2 consists of all vertices of 𝑊 that belong to a 𝐶4 -component or a 𝐶7 -component of 𝐺 𝐷 [𝑊]. Subclaim 6.29.1.3 No vertex of 𝐵3 is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Proof Suppose, to the contrary, that there is a vertex 𝑣 ∈ 𝐵3 that has a neighbor 𝑢 ∈ 𝑊0 . Let 𝑢 1 and 𝑢 2 denote the other two white neighbors of 𝑣 in 𝐺 𝐷 , and let 𝐴 = {𝑣}. In the graph 𝐺 𝐷 ′ we note that 𝑣 ∈ 𝑅 ′ and 𝑢 ∈ 𝑅 ′ , and so the weight decrease of the vertices 𝑣 and 𝑢 are 9 and 16, respectively. The vertices 𝑢 1 and 𝑢 2 belong to the set 𝐵1′ ∪ 𝐵2′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , and therefore have a combined weight decrease of at least 2 · (16 − 8) = 16. Moreover, there are at least three edges that join 𝑢 to blue vertices different from 𝑣, and at least one edge that joins each of 𝑢 1 and 𝑢 2 to a blue vertex different from 𝑣. Thus, there are at least five edges that join vertices in {𝑢, 𝑢 1 , 𝑢 2 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 5, implying that 𝜓( 𝐴) ≥ 9 + 16 + 16 + 5 = 46 > 44𝑘 = 44| 𝐴|, a contradiction. By Subclaim 6.29.1.3, every neighbor of a vertex in 𝑊0 belongs to the set 𝐵1 ∪ 𝐵2 in the graph 𝐺 𝐷 . Adopting the notation of Bujtás [123], we call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Subclaim 6.29.1.4 The following hold: (a) No special vertex is adjacent to two vertices of 𝑊0 in 𝐺 𝐷 . (b) No special vertex is adjacent to a vertex of 𝑊2 in 𝐺 𝐷 . (c) If 𝑣 1 and 𝑣 2 are two adjacent vertices in 𝑊1 , then at most one of them has a special (blue) neighbor. Proof (a) Suppose, to the contrary, that a special vertex 𝑣 ∈ 𝐵2 has two neighbors 𝑢 1 and 𝑢 2 that belong to the set 𝑊0 . We let 𝐴 = {𝑣}. In the graph 𝐺 𝐷 ′ we note that {𝑢 1 , 𝑢 2 , 𝑣} ⊆ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 8, while the weight decrease of each of 𝑢 1 and 𝑢 2 is 16. There are at least six edges that join vertices in {𝑢 1 , 𝑢 2 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 6, implying that 𝜓( 𝐴) ≥ 8 + 2 · 16 + 6 = 46 > 44𝑘 = 44| 𝐴|, a contradiction. This proves part (a). (b) Suppose, to the contrary, that a special vertex 𝑣 ∈ 𝐵2 has a neighbor 𝑢 0 that belongs to 𝑊0 and a neighbor 𝑢 1 that belongs to 𝑊2 . By our earlier observations, 𝑢 1 belongs to a 𝐶4 -component or a 𝐶7 -component 𝐶 of 𝐺 𝐷 [𝑊]. Suppose firstly that 𝐶 is a 4-cycle given by 𝐶 : 𝑢 1 𝑢 2 𝑢 3 𝑢 4 𝑢 1 . In this case, we let 𝐴 = {𝑣, 𝑢 3 }. In the graph 𝐺 𝐷 ′ we note that {𝑣, 𝑢 0 , 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 } ⊆ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 8, while the weight decrease of each vertex in

Section 6.2. Bounds on the Domination Number

157

𝑉 (𝐶) ∪ {𝑢 0 } is 16. There are at least three edges that join 𝑢 0 to blue vertices different from 𝑣, at least one edge that joins 𝑢 1 to a blue vertex different from 𝑣, and at least two edges that join 𝑢 𝑖 to a blue vertex different from 𝑣 for 𝑖 ∈ [4] \ {1}. Thus, there are at least ten edges that join vertices in {𝑢 0 , 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 10, implying that 𝜓( 𝐴) ≥ 8 + 5 · 16 + 10 = 98 > 44| 𝐴|, a contradiction. Suppose secondly that 𝐶 is a 7-cycle given by 𝐶 : 𝑢 1 𝑢 2 . . . 𝑢 7 𝑢 1 . In this case, we let 𝐴 = {𝑣, 𝑢 3 , 𝑢 6 }. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 8+8·16+16 = 152 > 44| 𝐴|, a contradiction. This proves part (b). (c) Suppose, to the contrary, that there is a 𝑃2 -component of 𝐺 𝐷 [𝑊] in which both vertices 𝑣 1 and 𝑣 2 have special (blue) neighbors. Let 𝑢 𝑖 be a special (blue) neighbor of 𝑣 𝑖 and let 𝑥𝑖 be the second white neighbor of 𝑢 𝑖 for 𝑖 ∈ [2]. We note that {𝑥 1 , 𝑥2 } ⊆ 𝑊0 , {𝑣 1 , 𝑣 2 } ⊆ 𝑊1 , {𝑢 1 , 𝑢 2 } ⊆ 𝐵2 , and 𝑢 1 ≠ 𝑢 2 . We now let 𝐴 = {𝑢 1 , 𝑢 2 }. In the graph 𝐺 𝐷 ′ we note that {𝑢 1 , 𝑢 2 , 𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } ⊆ 𝑅 ′ , and so the weight decrease of each vertex in {𝑢 1 , 𝑢 2 } is 8, while the weight decrease of each vertex in {𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } is 16. There are at least six edges that join vertices in {𝑥 1 , 𝑥2 } to blue vertices different from 𝑢 1 and 𝑢 2 , and there are at least four edges that join vertices in {𝑣 1 , 𝑣 2 } to blue vertices different from 𝑢 1 and 𝑢 2 . Thus, there are at least ten edges that join vertices in {𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } to blue vertices different from 𝑢 1 and 𝑢 2 . These edges result in a decrease in the total weight of at least 10, implying that 𝜓( 𝐴) ≥ 2 · 8 + 4 · 16 + 10 = 90 > 44| 𝐴|, a contradiction. This proves part (c), and completes the proof of Subclaim 6.29.1.4. We now return to the proof of Claim 6.29.1. With the structure of the residual graph 𝐺 𝐷 given by Subclaims 6.29.1.1, 6.29.1.2, 6.29.1.3, and 6.29.1.4, we apply the following discharging arguments that distribute the weights of the vertices. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 16, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 9, 8, and 7, respectively. Since Δ𝑊 (𝐵) ≤ 3, we note that the sum of the charges equals w(𝐺 𝐷 ). Thereafter, we distribute the charge of every blue vertex that is not a special vertex equally amongst its white neighbors, while we distribute the charge of a special vertex unequally amongst its two neighbors by giving a charge of 7 to its (unique) neighbor in 𝑊0 and a charge of 1 to its other neighbor. Thus, • Every vertex of 𝐵3 gives 3 to each of its three white neighbors. • Every non-special vertex of 𝐵2 gives 4 to both of its white neighbors. • Every vertex of 𝐵1 gives 7 to its (unique) white neighbor. • Every special vertex of 𝐵2 gives 7 to its white neighbor in 𝑊0 and gives 1 to its other white neighbor. Subclaim 6.29.1.5 The following hold in the graph 𝐺 𝐷 [𝑊]: (a) Every 𝑃1 -component has a charge of at least 44. (b) Every 𝑃2 -component has a charge of at least 44. (c) Every 𝐶4 -component has a charge of at least 88. (d) Every 𝐶7 -component has a charge of at least 154.

158

Chapter 6. Upper Bounds in Terms of Minimum Degree

Proof Subclaim 6.29.1.3 implies that the open neighborhood of a vertex from a 𝑃1 -component of 𝐺 𝐷 [𝑊] is contained in 𝐵1 ∪ 𝐵2 . Thus, upon completion of the discharging process, every vertex from a 𝑃1 -component of 𝐺 𝐷 [𝑊] has a charge of at least 16 + 4 · 7 = 44. By Subclaim 6.29.1.4(c), if 𝑣 1 and 𝑣 2 are the two vertices of a 𝑃2 -component of 𝐺 𝐷 [𝑊], then at least one of 𝑣 1 and 𝑣 2 , say 𝑣 1 , has three non-special blue neighbors, and therefore receives a charge of at least 3 · 3 = 9, while 𝑣 2 receives a charge of at least 3 · 1. Hence, every 𝑃2 -component of 𝐺 𝐷 [𝑊] has a charge of at least 2 · 16 + 3 · 3 + 1 · 3 = 44. By Subclaim 6.29.1.4(b), no special vertex is adjacent to a vertex in a 𝐶4 - or 𝐶7 -component of 𝐺 𝐷 [𝑊]. Hence, if 𝐶 is a 𝐶4 -component of 𝐺 𝐷 [𝑊], then there are at least eight edges that join vertices in 𝑉 (𝐶) to blue vertices, implying that the vertices of 𝐶 receive a charge of at least 8 · 3 from blue neighbors of 𝐶. If 𝐶 is a 𝐶7 -component of 𝐺 𝐷 [𝑊], then there are at least fourteen edges that join vertices in 𝑉 (𝐶) to blue vertices, implying that the vertices of 𝐶 receive a charge of at least 14 · 3 from blue neighbors of 𝐶. Thus, every 𝐶4 -component of 𝐺 𝐷 [𝑊] has a charge of at least 4 · 16 + 8 · 3 = 88, while every 𝐶7 -component of 𝐺 𝐷 [𝑊] has a charge of at least 7 · 16 + 14 · 3 = 154. Let 𝑝 1 , 𝑝 2 , 𝑐 4 , and 𝑐 7 be the number of 𝑃1 -, 𝑃2 -, 𝐶4 -, and 𝐶7 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴 be a 𝛾-set of 𝐺 [𝑊]. We note that | 𝐴| = 𝑝 1 + 𝑝 2 + 2𝑐 4 + 3𝑐 7 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺 𝐷 ′ , and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier, the weight of the blue vertices in 𝐺 𝐷 is charged to the white vertices. Thus, by Subclaim 6.29.1.5, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 44𝑝 1 + 44𝑝 2 + 88𝑐 4 + 154𝑐 7 ≥ 44( 𝑝 1 + 𝑝 2 + 2𝑐 4 + 3𝑐 7 ) = 44| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists. By Claim 6.29.1, if w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. We now return to the proof of Theorem 6.29. Recall that 𝐺 is an arbitrary graph of order 𝑛 with 𝛿(𝐺) ≥ 4. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . We note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 16𝑛. By Claim 6.29.1, there exists a 𝐷 0 -desirable set 𝐴1 . In this case, if we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 (noting that 𝐷 0 = ∅) and 𝐺 1 = 𝐺 𝐷1 , then w(𝐺 0 ) − w(𝐺 1 ) ≥ 44| 𝐴1 |, that is,

Section 6.2. Bounds on the Domination Number

159

w(𝐺 1 ) ≤ w(𝐺 0 ) − 44| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.29.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then w(𝐺 1 ) − w(𝐺 2 ) ≥ 44| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 44| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 44| 𝐴 𝑘 | ≤ w(𝐺 0 ) + 44

𝑘 ∑︁

| 𝐴𝑖 |

𝑖=1

≤ 16𝑛 − 44

𝑘 ∑︁

| 𝐴𝑖 |

𝑖=1

= 16𝑛 − 44|𝐷|. Consequently, 𝛾(𝐺) ≤ |𝐷 | ≤

16 44 𝑛

=

4 11 𝑛.

At the 9th Cracow Conference on Graph Theory held in Ryto, Poland, in June 2022, 4 71 Bujtás improved the 11 -bound in Theorem 6.29 to essentially a 200 -bound. More precisely, she proved that if 𝐺 is a connected graph with minimum degree at least 4 71𝑛 of order 𝑛, then 𝛾(𝐺) ≤ 71𝑛+5 200 . Moreover, if 𝑛 is large enough, then 𝛾(𝐺) ≤ 200 . Henning [460] posed the following conjecture. Conjecture 6.32 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤ 13 𝑛. If 𝐺 is the graph obtained from a complete graph 𝐾6 by removing the edges of a perfect matching, then 𝐺 is a 4-regular graph of order 𝑛 = 6 satisfying 𝛾(𝐺) = 2 = 13 𝑛. Hence, if Conjecture 6.32 is correct, then the bound is best possible.

6.2.5

Minimum Degree Five

In 2006 Xing et al. [754] proved that Conjecture 6.28 holds for 𝛿 = 5, that is, they 4 5 proved that the 11 -bound in Theorem 6.29 can be improved to a 14 -bound if we restrict the minimum degree to be at least 5. Theorem 6.33 ([754]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤ 5 14 𝑛 < 0.3572 𝑛. The proof of Xing et al. [754] once again uses the vertex-disjoint path cover method first employed by Reed [655]. In 2016 Bujtás and Klavžar [125] improved this bound as follows.

Chapter 6. Upper Bounds in Terms of Minimum Degree

160

Theorem 6.34 ([125]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤

2671 𝑛 < 0.3439 𝑛. 7766

In 2021 Bujtás [123] proved a breakthrough result that the domination number in graphs with minimum degree at least 5 is at most the magical threshold of 13 𝑛. The proof Bujtás presented of this result uses the same approach as her proof of Theorem 6.29 presented in the previous section, except that as expected in this case when the minimum degree is at least 5, the vertex weighting arguments, as well as the discharging methods employed, are more intricate and involved, and a more detailed case analysis is needed. We present here only an outline of her proof, where we adopt the notation defined in Section 6.2.4. Theorem 6.35 ([123]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤ 13 𝑛. Proof Sketch Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, and let 𝐷 ⊆ 𝑉. We define 𝐵5 as the set of blue vertices of degree at least 5 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [4]. Thus, every (blue) vertex in 𝐵5 has at least five white neighbors, while every (blue) vertex in 𝐵𝑖 has exactly 𝑖 white neighbors for 𝑖 ∈ [4]. For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.2.

set containing 𝑣

𝑊

𝐵5

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

35

23

21

19

17

14

0

Table 6.2 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 35|𝑊 | + 23|𝐵5 | + 21|𝐵4 | + 19|𝐵3 | + 17|𝐵2 | + 14|𝐵1 |. 𝑣 ∈𝑉

By Observation 6.31(e), w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. As in the proof of Theorem 6.29, given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 ) as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. However, we now define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 105| 𝐴|. We proceed further by proving that the following property holds. Claim 6.35.1 If w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. Proof Sketch Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 but the graph 𝐺 does not contain a 𝐷-desirable set. As before, we let 𝑊𝑖 be the set of white vertices having

Section 6.2. Bounds on the Domination Number

161

exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We note that 𝑊0 consists of all vertices of 𝑊 that belong to 𝑃1 -components of 𝐺 𝐷 [𝑊]. We call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . In what follows, we present a series of subclaims (without proof) describing some structural properties of 𝐺, which show that 𝐺 has a 𝐷-desirable set. Subclaim 6.35.1.1 The following hold: (a) Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. (b) Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 , 𝐶5 , 𝐶7 , or 𝐶10 . (c) No vertex in 𝐵3 is adjacent to a vertex in 𝑊0 in 𝐺 𝐷 . (d) No special vertex is adjacent to two vertices in 𝑊0 in 𝐺 𝐷 . (e) No special vertex is adjacent to a vertex in a 𝐶4 - or 𝐶7 -component in 𝐺 𝐷 [𝑊]. (f) If 𝑣 1 and 𝑣 2 are two adjacent vertices in 𝑊1 , then at most one of them has a special (blue) neighbor. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 35, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 19, 17, and 14, respectively. Since Δ𝑊 (𝐵) ≤ 3, we note that the sum of the charges equals w(𝐺). Thereafter, we distribute the charge of every blue vertex that is not a special vertex equally amongst its white neighbors, while we distribute the charge of a special vertex unequally amongst its two neighbors by giving a charge of 14 to its (unique) neighbor in 𝑊0 and a charge of 3 to its other neighbor. Thus, • Every vertex in 𝐵3 gives 19 3 to each of its three white neighbors. • Every non-special vertex in 𝐵2 gives 17 2 to both of its white neighbors. • Every vertex in 𝐵1 gives 14 to its (unique) white neighbor. • Every special vertex in 𝐵2 gives 14 to its white neighbor in 𝑊0 and gives 3 to its other white neighbor. Upon completion of the discharging process, the charge of each white component is given as follows. Subclaim 6.35.1.2 The following hold in the graph 𝐺 𝐷 [𝑊]: (a) Every 𝑃1 -component has a charge of at least 105. (b) Every 𝑃2 -component has a charge of at least 107. (c) Every 𝐶4 -component has a charge of at least 216. (d) Every 𝐶5 -component has a charge of at least 220. (e) Every 𝐶7 -component has a charge of at least 378. (f) Every 𝐶10 -component has a charge of at least 440. Let 𝑝 1 , 𝑝 2 , 𝑐 4 , 𝑐 5 , 𝑐 7 , and 𝑐 10 be the number of 𝑃1 -, 𝑃2 -, 𝐶4 -, 𝐶5 -, 𝐶7 -, and 𝐶10 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴 be a 𝛾-set of 𝐺 [𝑊]. We note that | 𝐴| = 𝑝 1 + 𝑝 2 + 2𝑐 4 + 2𝑐 5 + 3𝑐 7 + 4𝑐 10 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺 𝐷 ′ , and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier,

Chapter 6. Upper Bounds in Terms of Minimum Degree

162

the weight of the blue vertices in 𝐺 𝐷 is charged to the white vertices. Thus, by Subclaim 6.35.1.2, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 105𝑝 1 + 107𝑝 2 + 216𝑐 4 + 220𝑐 5 + 378𝑐 7 + 440𝑐 10 ≥ 105( 𝑝 1 + 𝑝 2 + 2𝑐 4 + 2𝑐 5 + 3𝑐 7 + 4𝑐 10 ) = 105| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists. By Claim 6.35.1, if w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. We now return to the proof of Theorem 6.35. Recall that 𝐺 is an arbitrary graph of order 𝑛 with 𝛿(𝐺) ≥ 5. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . We note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 35𝑛. By Claim 6.35.1, there exists a 𝐷 0 -desirable set 𝐴1 . If we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 and 𝐺 1 = 𝐺 𝐷1 , then w(𝐺 0 ) − w(𝐺 1 ) ≥ 105| 𝐴1 |, that is, w(𝐺 1 ) ≤ w(𝐺 0 ) − 105| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.35.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then w(𝐺 1 ) − w(𝐺 2 ) ≥ 105| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 105| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 105| 𝐴 𝑘 | ≤ 35𝑛 − 105

𝑘 ∑︁

| 𝐴𝑖 | = 35𝑛 − 105|𝐷|.

𝑖=1

Consequently, 𝛾(𝐺) ≤ |𝐷| ≤

35 105 𝑛

= 13 𝑛.

The 13 -upper bound on the domination number of a graph with minimum degree at least 5 given in Theorem 6.35 is currently the best known bound. However, it is not known if this bound is achievable. We also note that Theorem 6.35 proves Conjecture 6.28 for minimum degree at least 5 and, combined with Theorems 6.27 and 6.29, proves the conjecture for all 𝛿 ≥ 1.

6.2.6

Minimum Degree Six

In 2016 Bujtás and Klavžar [125] established the best known upper bound up to that time on the domination number of a graph with minimum degree at least 6.

Section 6.2. Bounds on the Domination Number

163

Theorem 6.36 ([125]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤

1702 𝑛 ≈ 0.3158 𝑛. 5389

The upper bound given in Theorem 6.36 was subsequently improved in 2021 by Bujtás and Henning [124]. Their proof once again uses Bujtás approach of vertex weighting arguments and discharging methods, combined with a detailed case analysis. We present here a brief outline of the proof, which has the same flavor as the earlier proofs presented using this approach, but uses some new ideas and is more intricate and involved than the proofs of Theorems 6.29 and 6.35. Theorem 6.37 ([124]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤

127 𝑛 ≈ 0.3038 𝑛. 418

Proof Sketch Since removing edges from a graph cannot decrease its domination number, it suffices for us to prove the result for graphs 𝐺 = (𝑉, 𝐸) that are edge minimal with respect to the two conditions: (i) 𝐺 is connected and (ii) 𝛿(𝐺) ≥ 6. With this assumption, every edge is incident with at least one vertex of degree 6, and so the set of vertices of degree at least 7 in 𝐺 is an independent set. Let 𝐷 ⊆ 𝑉 (𝐺). We define 𝐵6 as the set of blue vertices of degree at least 6 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [5]. For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.3.

set containing 𝑣

𝑊

𝐵6

𝐵5

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

508

508

317.4

297

268

236

194

0

Table 6.3 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 508|𝑊 | + 508|𝐵6 | + 317.4|𝐵5 | 𝑣 ∈𝑉 (𝐺) + 297|𝐵4 | + 268|𝐵3 | + 236|𝐵2 | + 194|𝐵1 |. We observe that w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. As in the proof of Theorem 6.29, given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 ) as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. However, we now define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 1672| 𝐴|. Further, we define 𝐷 ′ = 𝐴 ∪ 𝐷 and denote by 𝑊 ′ , 𝐵′ , and 𝑅 ′ the set of white, blue, and red vertices, respectively, in 𝐺 𝐷 ′ . We define 𝐵6′ as the set of blue vertices of

164

Chapter 6. Upper Bounds in Terms of Minimum Degree

degree at least 6 in 𝐺 𝐷 ′ , and 𝐵𝑖′ as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 ′ for 𝑖 ∈ [5]. A key property of the graph 𝐺 is given by the following claim. Claim 6.37.1 If w(𝐺 𝐷 ) > 0 and 𝐵6 = ∅ in 𝐺 𝐷 , then the graph 𝐺 contains a 𝐷-desirable set 𝐴 such that 𝐵6′ remains empty in 𝐺 𝐷 ′ . Proof Sketch Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 and 𝐵6 = ∅, but the graph 𝐺 does not contain a 𝐷-desirable set with the given property. By supposition, 𝐵6 = ∅ in 𝐺 𝐷 , and so Δ𝑊 (𝐵) ≤ 5. In what follows, we present a series of subclaims (without proof) describing some structural properties of 𝐺, which prove that 𝐺 has a 𝐷-desirable set. Subclaim 6.37.1.1 The following hold: (a) Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. (b) Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 or 𝐶5 . Let 𝑊𝑖 be the set of white vertices having exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Subclaim 6.37.1.2 The following hold: (a) No vertex of 𝐵3 is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . (b) No special vertex is adjacent to two vertices of 𝑊0 in 𝐺 𝐷 . (c) No special vertex is adjacent to a vertex in a 𝐶4 -component in 𝐺 𝐷 [𝑊]. (d) If 𝑢 ∈ 𝑊0 and 𝑣 ∈ 𝑊1 , then there are at most two (special) vertices that have 𝑢 and 𝑣 as common neighbors in 𝐺 𝐷 . We next specify a set P ★ of 6-paths. A path 𝑥1 𝑥 2 . . . 𝑥6 of 𝐺 𝐷 is called a 𝑃★-path if 𝑥1 , 𝑥6 ∈ 𝑊0 , 𝑥2 , 𝑥5 ∈ 𝐵2 , and 𝑥 3 , 𝑥4 ∈ 𝑊1 . Let P ★ be a maximal set of pairwise vertex-disjoint 𝑃★-paths in 𝐺 𝐷 , where maximality means that every 𝑃★-path that does ★ not belong to P ★ shares Ð a vertex with at least one path in P . We say that a vertex 𝑣 is covered if 𝑣 ∈ 𝑃 ∈ P★ 𝑉 (𝑃). A special vertex 𝑣 ∈ 𝐵2 is called C-special and UC-special if its unique neighbor 𝑣 ′ in 𝑊0 is covered or uncovered, respectively. The maximum possible number of UC-special neighbors of an uncovered 𝑃2 -component of 𝐺 𝐷 [𝑊] is given by the following subclaim. Subclaim 6.37.1.3 If 𝑣 1 and 𝑣 2 form an uncovered 𝑃2 -component in 𝐺 𝐷 [𝑊], then there are at least five edges between {𝑣 1 , 𝑣 2 } and the set of blue vertices which are not UC-special. With the structure of the residual graph 𝐺 𝐷 given by Subclaims 6.37.1.1, 6.37.1.2, and 6.37.1.3, we apply the following discharging arguments that distribute the weights of the vertices. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 508, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 268, 236, and 194, respectively. Since Δ𝑊 (𝐵) ≤ 3, the

Section 6.2. Bounds on the Domination Number

165

sum of the charges equals w(𝐺 𝐷 ). Thereafter, we distribute the charge of every blue vertex that is not a UC-special vertex equally amongst its white neighbors, while we distribute the charge of a UC-special vertex unequally amongst its two neighbors by giving a charge of 194 to its (unique) neighbor in 𝑊0 and a charge of 42 to its other neighbor. Thus, • Every vertex in 𝐵3 gives 89 31 to each of its three (white) neighbors. • Every non-special or C-special vertex in 𝐵2 gives 118 to both its (white) neighbors. • Every UC-special vertex in 𝐵2 gives 194 to its (white) neighbor in 𝑊0 and gives 42 to its other (white) neighbor. • Every vertex in 𝐵1 gives 194 to its (unique white) neighbor. Subclaim 6.37.1.4 The charge of each component in 𝐺 𝐷 [𝑊] and that of a path 𝑃 ∈ P ★ are given by the values in Table 6.4.

component

𝐶5

𝐶4

uncovered 𝑃2

uncovered 𝑃1

charge

≥ 3380

≥ 3461 13

≥ 1672 23

≥ 1672

component

covered 𝑃2

covered 𝑃1

path 𝑃 ∈ P ★

charge

≥ 1588

≥ 1216

≥ 4020

Table 6.4 The charge of each component in 𝐺 𝐷 [𝑊] Let 𝑝 = |P ★ | and let 𝑐 4 , 𝑐 5 , 𝑝 1 , and 𝑝 2 be the number of 𝐶4 -, 𝐶5 -, uncovered 𝑃1 -, and uncovered 𝑃2 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴′ be a 𝛾-set of the uncovered components of 𝐺 [𝑊] and let 𝐴′′ be the set of the covered blue vertices in 𝐺 𝐷 . We note that 𝐴 = 𝐴′ ∪ 𝐴′′ dominates all white vertices of 𝐺 𝐷 and | 𝐴| = 2𝑝 + 2𝑐 5 + 2𝑐 4 + 𝑝 1 + 𝑝 2 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺, and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier, the weight of the blue vertices in 𝐺 𝐷 is distributed among the white vertices. Thus, by Subclaim 6.37.1.4, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 4020𝑝 + 3380𝑐 5 + 3461 13 𝑐 4 + 1672 23 𝑝 2 + 1672𝑝 1 ≥ 1672(2𝑝 + 2𝑐 4 + 2𝑐 5 + 𝑝 1 + 𝑝 2 ) = 1672| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists.

166

Chapter 6. Upper Bounds in Terms of Minimum Degree

We now return to the proof of Theorem 6.37. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . Note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 508𝑛. Further, 𝐵 = 𝑅 = ∅. In particular, we note that 𝐵6 = ∅. By Claim 6.37.1, there exists a 𝐷 0 -desirable set 𝐴1 . If we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 and 𝐺 1 = 𝐺 𝐷1 , then the set 𝐵6 remains empty in 𝐺 1 and w(𝐺 0 ) − w(𝐺 1 ) ≥ 1672| 𝐴1 |, that is, w(𝐺 1 ) ≤ w(𝐺 0 ) − 1672| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.37.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then the set 𝐵6 remains empty in 𝐺 2 and w(𝐺 1 ) − w(𝐺 2 ) ≥ 1672| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 1672| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 𝑘 ∑︁ | 𝐴𝑖 | 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 1672| 𝐴 𝑘 | ≤ w(𝐺 0 ) − 1672 𝑖=1

= 508𝑛 − 1672|𝐷|. Consequently, 𝛾(𝐺) ≤ |𝐷| ≤

508 127 𝑛= 𝑛. 1672 418

The upper bound on the domination number of a graph with minimum degree at least 6 given in Theorem 6.37 is currently the best known bound. However, it is not known if this bound is achievable. Henning [460] posed the following conjecture. Conjecture 6.38 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤ 14 𝑛. If 𝐺 is the graph obtained from a complete graph 𝐾8 by removing the edges of a perfect matching, then 𝐺 is a 6-regular graph of order 𝑛 = 8 satisfying 𝛾(𝐺) = 2 = 14 𝑛. Hence, if Conjecture 6.38 is correct, then the bound is best possible. As a further example given in [460], if 𝐺 is the 6-regular graph of order 𝑛 = 16 shown in Figure 6.15, then 𝛾(𝐺) = 4 = 14 𝑛, where the four highlighted vertices form a 𝛾-set of 𝐺. Unfortunately, there is a large gap between the best known lower and upper bounds to date on the domination number of a graph with minimum degree at least 6. A
natural 3 1 question is whether the best known upper bound of 𝛾(𝐺) ≤ 127 418 𝑛 < 10 + 261 𝑛 given in Theorem 6.37 can be improved ever-so-slightly to the more aesthetically pleasing 3 bound 𝛾(𝐺) ≤ 10 𝑛. If so, can the vertex weighting arguments and discharging methods presented in the proof of Theorem 6.37 be sufficiently refined to achieve such an improved bound? Or is a radically new idea needed to achieve a further breakthrough yielding an improved bound?

Section 6.2. Bounds on the Domination Number

167

Figure 6.15 A 6-regular graph of order 𝑛 = 16 with 𝛾(𝐺) = 4 = 14 𝑛

We summarize the known upper bounds to date on the domination number of a graph 𝐺 with small minimum degree 𝛿 ∈ [6] in terms of its order 𝑛 in Table 6.5. As remarked earlier, the bounds for minimum degree 𝛿 ∈ {1, 2, 3} are tight, while the bounds for minimum degree 𝛿 ∈ {4, 5, 6} are currently the best known bounds to date. However, it is not known if the given bounds for minimum degree 𝛿 ∈ {4, 5, 6} are achievable. Year

𝛿(𝐺) ≥

1962

𝛿(𝐺) ≥ 1

𝛾(𝐺) ≤

1973

𝛿(𝐺) ≥ 2

𝛾(𝐺) ≤

1996

𝛿(𝐺) ≥ 3

𝛾(𝐺) ≤

2009

𝛿(𝐺) ≥ 4

𝛾(𝐺) ≤

2021

𝛿(𝐺) ≥ 5

𝛾(𝐺) ≤

2021 a b

𝛿(𝐺) ≥ 6

𝛾(𝐺) ≤

𝛾(𝐺) ≤

1 a 2𝑛 2 b 5𝑛 3 8𝑛 4 11 𝑛 1 3𝑛 127 418 𝑛

Citation [622] [79, 586] [655] [682] [123]


10778 = 𝑠, and so condition (b) holds. Further, 𝑏 6 ≤ 𝑎, 𝑏 6 − 𝑏 5 = 99, 𝑏 5 − 𝑏 4 = 127, 𝑏 4 − 𝑏 3 = 170, 𝑏 3 − 𝑏 2 = 252, and 𝑏 2 − 𝑏 1 = 397. Hence, condition (a) holds. Thus, by Theorem 6.39, 𝛾(𝐺) ≤

𝑎 3404 1702 𝑛= 𝑛= 𝑛. 𝑠 10778 5389

This is precisely the result stated formally in Theorem 6.36. More generally, Bujtás and Klavžar [125] computed the upper bounds on 𝛾(𝐺) for graphs 𝐺 with minimum degree 5 ≤ 𝛿 ≤ 50. In Table 6.7, we present their computed bounds for values of 𝛿 ∈ {7, 8, . . . , 15}, where we list in this table the values 𝑎𝑠 obtained by applying Theorem 6.39. For example, if 𝛿(𝐺) ≥ 7, then by Table 6.7, we have 𝛾(𝐺) ≤ 0.2926 𝑛, while if 𝛿(𝐺) ≥ 8, then 𝛾(𝐺) ≤ 0.2732 𝑛. The upper bounds on the domination number of

Section 6.3. Bounds on the Total Domination Number 𝛿(𝐺)

7

8

9

10

11

12

169 13

14

15

𝛾(𝐺) ≤ 0.2926 0.2732 0.2565 0.2421 0.2294 0.2182 0.2082 0.1992 0.1910 Table 6.7 Upper bounds on 𝛾(𝐺) in terms of its order with given minimum degree 𝛿(𝐺)

a graph given in Table 6.7 are currently the best known bounds on the domination number of a graph with minimum degree 𝛿 where 𝛿 ∈ {7, 8, . . . , 15}. However, it is not known if these bounds are achievable.

6.3

Bounds on the Total Domination Number

In this section, we present the bound 𝛾t (𝐺) ≤ 47 𝑛 of Henning on the total domination number of a connected graph 𝐺 of order 𝑛 ≥ 11 and minimum degree 𝛿(𝐺) ≥ 2 using the proof technique of 47 -minimal graphs. We present the pleasing result due to Archdeacon et al. that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛 using counting arguments and Brooks’ Coloring Theorem. We discuss the interplay between total domination in graphs and transversals in hypergraphs, an idea first explored by Thomassé and Yeo. Using this powerful proof technique, we present their result that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛. We close this section with a heuristic algorithm due to Henning and Yeo that yields an upper bound on the total domination of a graph in terms of its order and minimum degree, namely if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, 𝛿)
then 𝛾t (𝐺) ≤ 1+ln( 𝑛. 𝛿

6.3.1

Minimum Degree One

The situation for minimum degree one is handled in Chapter 4. We repeat the results here for completeness. Theorem 6.40 ([182]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛. Theorem 6.41 ([117]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) = 23 𝑛 if and only if 𝐺 is 𝐶3 , 𝐶6 , or 𝐹 ◦ 𝑃2 for some connected graph 𝐹.

6.3.2

Minimum Degree Two

If 𝐺 is a graph of order 𝑛, each component of which is a 3-cycle or a 6-cycle, then 𝛾t (𝐺) = 23 𝑛. Hence, the upper bound in Theorem 6.40 is best possible for graphs of minimum degree 2. However, the upper bound can be improved if we impose the additional restriction that the graph 𝐺 is connected, as shown in 1995 by Sun [697]. Theorem 6.42 ([697]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then

170

Chapter 6. Upper Bounds in Terms of Minimum Degree 𝛾t (𝐺) ≤

4

7 (𝑛

 + 1) .

In 2000 Henning [453] showed that the upper bound in Theorem 6.42 can be improved slightly if we forbid six graphs of small orders. In order to state this result, we shall need the following definition. Definition 6.43 ([453]) A graph of order 𝑛 ≥ 3 is a 47 -minimal graph if 𝐺 is edge minimal with respect to satisfying the following three conditions: (a) 𝛿(𝐺) ≥ 2. (b) 𝐺 is connected. (c) 𝛾t (𝐺) ≥ 47 𝑛. Let Gtdom be the family of graphs 𝐺 that can be constructed from a connected graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 6-cycle 𝐶𝑣 and join 𝑣 either to one vertex of 𝐶𝑣 or to two vertices at distance 2 on the cycle 𝐶𝑣 . We call the graph 𝐹 the underlying graph of 𝐺. An example of a graph 𝐺 in the family Gtdom , in the case when the underlying graph 𝐹 of 𝐺 is a 4-cycle, is illustrated in Figure 6.16. 𝐹 = 𝐶4

Figure 6.16 A graph in the family Gtdom Let 𝐺 ∈ Gtdom and let 𝐹 be the underlying graph used to construct 𝐺. For each vertex 𝑣 of 𝐹, the subgraph of 𝐺 induced by 𝑣 and the vertices of its associated 6-cycle is called a unit of 𝐺. There are two types of units. The first type of unit is obtained from a 6-cycle by adding a new vertex and joining it to two vertices at distance 2 on the cycle. In this case, we call the resulting graph 𝐺 7 , which is shown in Figure 6.17(a), and the resulting unit a 𝐺 7 -unit. The second type of unit we call a key unit, which is obtained from a 6-cycle by adding a new vertex and joining it to exactly one vertex on the cycle, as shown in Figure 6.17(b). For example, the graph in the family Gtdom shown in Figure 6.16 contains four units, two of which are key units and two of which are 𝐺 7 -units. We note that if 𝐺 ∈ Gtdom has order 𝑛 and 𝑘 units, then 𝑛 = 7𝑘. Every TD-set of the graph 𝐺 ∈ Gtdom must contain at least four vertices from each unit of 𝐺, implying that 𝛾t (𝐺) ≥ 4𝑘. However, every four consecutive vertices from the cycle 𝐶𝑣 , starting at a neighbor of 𝑣 on 𝐶𝑣 , totally dominates the vertices of the unit containing 𝑣, implying that 𝛾t (𝐺) ≤ 4𝑘. Consequently, 𝛾t (𝐺) = 4𝑘. This yields the following observation. Proposition 6.44 If 𝐺 ∈ Gtdom has order 𝑛, then 𝛾t (𝐺) = 47 𝑛.

Section 6.3. Bounds on the Total Domination Number

(a) A 𝐺 7 -unit

171

(b) A key unit

Figure 6.17 The two types of units in a graph in Gtdom

If the underlying graph 𝐹 of a graph 𝐺 ∈ Gtdom is a tree and if every unit of 𝐺 is a key unit, then we denote the resulting subfamily by Gtdom,tree . A graph 𝐺 ∈ Gtdom,tree whose underlying tree 𝐹 is a path 𝑃3 is shown in Figure 6.18.

Figure 6.18 A graph in the family Gtdom,tree Let Etdom = {𝐶3 , 𝐶5 , 𝐶6 , 𝐶7 , 𝐶10 , 𝐶14 } ∪ {𝐺 7 } be the family of seven exceptional graphs (consisting of six small cycles and an additional graph of order seven). We are now in a position to present the characterization of 47 -minimal graphs given in [453]. Since the proof of Theorem 6.45 given in [453] uses the same approach employed by McCuaig and Shepherd [586] to prove Theorem 6.16, we omit the proof. Theorem 6.45 ([453]) A graph 𝐺 is a 47 -minimal graph if and only if 𝐺 ∈ Etdom ∪ Gtdom,tree . ′ and 𝐶 ′′ be the two graphs that are obtained from a 10-cycle by adding Let 𝐶10 10 one or two chords as shown in Figure 6.19(a) and (b), respectively.

′ (a) 𝐶10

′′ (b) 𝐶10

′ and 𝐶 ′′ Figure 6.19 The graphs 𝐶10 10

We next present an improvement of the upper bound in Theorem 6.42. Let  ′ ′′ Btdom = 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 , 𝐶10

172

Chapter 6. Upper Bounds in Terms of Minimum Degree

be the family of four small cycles and the two exceptional graphs obtained from a 10-cycle by adding one or two chords. Theorem 6.46 ([453]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 74 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. Let 𝐺 ′ be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Since removing edges from a graph cannot decrease its total domination number, 𝛾t (𝐺) ≤ 𝛾t (𝐺 ′ ). If 𝐺 ′ is not a 47 -minimal graph, then 𝛾t (𝐺 ′ ) < 47 𝑛, implying that 𝛾(𝐺) ≤ 47 𝑛. If 𝐺 ′ is a 47 -minimal graph, then by Theorem 6.45, we have 𝐺 ′ ∈ Etdom ∪ Gtdom . If 𝐺 ′ ∈ Gtdom , then by Proposition 6.44, we have 𝛾(𝐺 ′ ) = 47 𝑛, while if 𝐺 ′ ∈ {𝐶7 , 𝐶14 , 𝐺 7 }, then 𝛾(𝐺 ′ ) = 47 𝑛, and in these cases we once again have 𝛾(𝐺) ≤ 47 𝑛. If 𝐺 ′ = 𝐶3 , then 𝐺 = 𝐶3 and 𝛾t (𝐺) = 2 = 17 (4𝑛 + 2). If 𝐺 ′ = 𝐶5 , then either 𝐺 = 𝐶5 and 𝛾t (𝐺) = 3 = 17 (4𝑛 + 1) or 𝛾t (𝐺) = 2 < 47 𝑛. If 𝐺 ′ = 𝐶6 , then either 𝐺 = 𝐶6  and 𝛾t (𝐺) = 4 = 47 (𝑛 + 1) or 𝛾t (𝐺) ≤ 3 < 47 𝑛. If 𝐺 ′ = 𝐶10 , then either 𝐺 ∈ 𝐶10 , ′ , 𝐶 ′′ and 𝛾 (𝐺) = 6 = 1 (4𝑛 + 2) or 𝛾 (𝐺) ≤ 5 < 4 𝑛. Thus, 𝛾 (𝐺) ≤ 4 𝑛, unless 𝐶10 t t t 7 7 7  10 ′ , 𝐶 ′′ = B 𝐺 ∈ 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 . tdom 10 Using the characterization of 47 -minimal graphs given in Theorem 6.45, the graphs of large order achieving equality in the upper bound of Theorem 6.46 were characterized by Henning in [453]. Theorem 6.47 ([453]) If 𝐺 is a connected graph of order 𝑛 ≥ 11 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 47 𝑛, with equality if and only if 𝐺 = 𝐶14 or 𝐺 ∈ Gtdom .

6.3.3

Minimum Degree Three

In 2000 Favaron et al. [282] showed that the 47 -bound on the total domination number 7 given in Theorem 6.46 can be improved to a 13 -bound if the minimum degree requirement is increased to at least 3. Theorem 6.48 ([282]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 7 𝛾t (𝐺) ≤ 13 𝑛. However, it was conjectured in [282] that the total domination number of a graph with minimum degree at least 3 is at most one-half its order. The authors in [282] wrote that if the conjecture is true, then the bound is tight as may be seen as follows. Consider the generalized Petersen graph 𝑃(8, 3) shown in Figure 6.20(a). Proposition 6.49 If 𝐺 = 𝑃(8, 3), then 𝐺 has order 𝑛 = 16 and 𝛾t (𝐺) = 12 𝑛. Proof Let 𝐺 = 𝑃(8, 3), and let 𝑋 and 𝑌 be the partite sets of the bipartite graph 𝐺, as illustrated in Figure 6.20(b), where the vertices in 𝑋 and 𝑌 are colored red and blue, respectively. Thus, |𝑋 | = |𝑌 | = 8. Let 𝐷 be an arbitrary TD-set of 𝐺, and let 𝐷 𝑋 = 𝐷 ∩ 𝑋 and let 𝐷𝑌 = 𝐷 ∩ 𝑌 . The vertices in 𝑋 and 𝑌 are totally dominated by the sets 𝐷𝑌 and 𝐷 𝑋 , respectively.

Section 6.3. Bounds on the Total Domination Number

173

𝑦2 𝑥1

𝑥2 𝑦1

(a)

(b)

Figure 6.20 The generalized Petersen graph 𝑃(8, 3)

Suppose that |𝐷 𝑋 | ≤ 3. Since |𝑌 | = 8 and each vertex totally dominates three vertices, we have |𝐷 𝑋 | = 3. If every two vertices in 𝐷 𝑋 have a common neighbor, then 𝐷 𝑋 would totally dominate at most seven vertices in 𝑌 , a contradiction. Hence, the set 𝐷 𝑋 contains two vertices 𝑥1 and 𝑥2 that have no common neighbor in 𝑌 . Every vertex in 𝑋 has a common neighbor with every other vertex in 𝑋, except for the unique vertex in 𝑋 at distance 4 from it. Thus, 𝑑𝐺 (𝑥 1 , 𝑥2 ) = 4. By symmetry, we may assume that 𝑥1 and 𝑥2 are the vertices labeled in Figure 6.20(b). The two vertices in 𝑌 not totally dominated by {𝑥1 , 𝑥2 } are at distance 4 in 𝐺. These vertices are labeled 𝑦 1 and 𝑦 2 in Figure 6.20(b). The third vertex of 𝐷 𝑋 , which is different from 𝑥1 and 𝑥2 , therefore cannot totally dominate both 𝑦 1 and 𝑦 2 , contradicting the fact that 𝐷 𝑋 totally dominates the set 𝑌 . Hence, |𝐷 𝑋 | ≥ 4. Using similar arguments, |𝐷𝑌 | ≥ 4. Hence, |𝐷| = |𝐷 𝑋 | + |𝐷𝑌 | ≥ 4 + 4 = 8. Since 𝐷 is an arbitrary TD-set of 𝐺, this implies that 𝛾t (𝐺) ≥ 8. The set of vertices of the outer 8-cycle is a TD-set of 𝐺, implying that 𝛾t (𝐺) ≤ 8. Consequently, 𝛾t (𝐺) = 8 = 12 𝑛. The authors in [282] constructed the following two infinite families Gcubic and Hcubic of cubic graphs with total domination numbers one-half their orders. For 𝑘 ≥ 1, let 𝐺 𝑘 be the graph constructed as follows. Consider two copies of the path 𝑃2𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑎 2 𝑏 2 . . . 𝑎 𝑘 𝑏 𝑘 and 𝑐 1 𝑑1 𝑐 2 𝑑2 . . . 𝑐 𝑘 𝑑 𝑘 . For each 𝑖 ∈ [𝑘], join 𝑎 𝑖 to 𝑑𝑖 and 𝑏 𝑖 to 𝑐 𝑖 . To complete the construction of the graph 𝐺 𝑘 join 𝑎 1 to 𝑐 1 and 𝑏 𝑘 to 𝑑 𝑘 . We note that 𝐺 1 = 𝐾4 . Let Gcubic = {𝐺 𝑘 : 𝑘 ≥ 1}. For 𝑘 ≥ 2, let 𝐻 𝑘 be obtained from 𝐺 𝑘 by deleting the two edges 𝑎 1 𝑐 1 and 𝑏 𝑘 𝑑 𝑘 and adding the two edges 𝑎 1 𝑏 𝑘 and 𝑐 1 𝑑 𝑘 . Let Hcubic = {𝐻 𝑘 : 𝑘 ≥ 2}. The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic are illustrated in Figure 6.21(a) and (b), respectively. Proposition 6.50 If 𝐺 ∈ Gcubic ∪ Hcubic has order 𝑛, then 𝛾t (𝐺) = 12 𝑛. Proof Let 𝐺 ∈ Gcubic have order 𝑛, and so 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1. Thus, 𝑛 = 4𝑘. Let 𝑉 (𝐺) = 𝑉1 ∪𝑉2 ∪ · · · ∪𝑉𝑘 , where 𝑉𝑖 = {𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 , 𝑑𝑖 } for 𝑖 ∈ [𝑘]. The set {𝑎 1 , 𝑎 2 , . . . , 𝑎 𝑘 } ∪ {𝑏 1 , 𝑏 2 , . . . , 𝑏 𝑘 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 2𝑘. Hence, it suffices for us to show that 𝛾t (𝐺) ≥ 2𝑘. Suppose, to the contrary, that 𝛾t (𝐺) < 2𝑘. Thus, for

Chapter 6. Upper Bounds in Terms of Minimum Degree

174 𝑎1

𝑐1

𝑏1

𝑑1

𝑎2

𝑐2

𝑏2

𝑑2

𝑎3

𝑐3

𝑏3

𝑑3

𝑎4

𝑐4 𝑑4

𝑏4 (a) 𝐺 4

(b) 𝐻4

Figure 6.21 The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic

every 𝛾t -set 𝑋 of 𝐺, we have |𝑋 ∩ 𝑉𝑖 | ≤ 1 for at least one 𝑖 ∈ [𝑘]. For each such set 𝑋, let  𝐼 𝑋 = 𝑖 ∈ [𝑘] : |𝑋 ∩ 𝑉𝑖 | ≤ 1 . Among all 𝛾t -sets of 𝐺, let 𝑋 be chosen so that |𝐼 𝑋 | is a minimum. By supposition, |𝐼 𝑋 | ≥ 1, that is, 𝑖 ∈ 𝐼 𝑋 for at least one 𝑖 ∈ [𝑘]. Suppose firstly that |𝑋 ∩ 𝑉1 | ≤ 1. If 𝑋 ∩ 𝑉1 = ∅, then 𝑎 1 and 𝑐 1 are not totally dominated by the set 𝑋, a contradiction. Hence, |𝑋 ∩ 𝑉1 | ≥ 1. Suppose that |𝑋 ∩ 𝑉1 | = 1. In order to totally dominate the vertices 𝑎 1 and 𝑐 1 , either 𝑏 1 or 𝑑1 belongs to the set 𝑋. By symmetry, we may assume that 𝑏 1 ∈ 𝑋. In order to totally dominate the vertices 𝑏 1 and 𝑑1 , we have 𝑎 2 ∈ 𝑋 and 𝑐 2 ∈ 𝑋, respectively. In order to totally dominate vertex 𝑐 2 , at least one of 𝑏 2 or 𝑑2
belongs to the set 𝑋. But then 𝑌 = 𝑋 \ {𝑎 2 } ∪ {𝑎 1 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. Hence, |𝑋 ∩ 𝑉1 | ≥ 2, and so 𝑖 ∉ 𝐼 𝑋 . By symmetry, 𝑘 ∉ 𝐼 𝑋 . Thus, 2 ≤ 𝑖 ≤ 𝑘 − 1. Suppose that 𝑋 ∩ 𝑉𝑖 = ∅. In order to totally dominate the vertices in the set 𝑉𝑖 , we have {𝑏 𝑖−1 , 𝑐 𝑖−1 , 𝑎 𝑖+1 , 𝑑𝑖+1 } ⊆ 𝑋. In order to totally dominate 𝑏 𝑖−1 , at least one of 𝑎 𝑖−1 or 𝑐 𝑖−1 belongs to the set 𝑋. In order to totally dominate vertex 𝑎
𝑖+1 , at least one of 𝑏 𝑖+1 or 𝑑𝑖+1 belongs to the set 𝑋. But then 𝑌 = 𝑋 \ {𝑎 𝑖+1 , 𝑏 𝑖−1 } ∪ {𝑎 𝑖 , 𝑏 𝑖 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. Hence, |𝑋 ∩ 𝑉𝑖 | ≥ 1, implying by supposition that |𝑋 ∩ 𝑉𝑖 | = 1. By symmetry we may assume that 𝑋 ∩ 𝑉𝑖 = {𝑎 𝑖 }. In order to totally dominate vertex 𝑐 𝑖 , we have 𝑑𝑖−1 ∈ 𝑋. In order to totally dominate vertex 𝑑𝑖−1 , at least one of 𝑎 𝑖−1 or 𝑐 𝑖−1 belongs to the set 𝑋. In
order to totally dominate vertex 𝑎 𝑖 , we have 𝑏 𝑖−1 ∈ 𝑋. But then 𝑌 = 𝑋 \ {𝑏 𝑖−1 } ∪ {𝑏 𝑖 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. We deduce, therefore, that 𝛾t (𝐺) ≥ 2𝑘. As observed earlier, 𝛾t (𝐺) ≤ 2𝑘. Consequently, 𝛾t (𝐺) = 2𝑘 = 12 𝑛. Let 𝐺 ∈ Hcubic have order 𝑛, and so 𝐺 = 𝐻 𝑘 for some 𝑘 ≥ 2. The proof in this case is similar to the previous case, except simpler since we need only consider the case when |𝑋 ∩ 𝑉𝑖 | ≤ 1 for some 𝑖 ∈ [𝑘] (and not consider separately the case when |𝑋 ∩ 𝑉1 | ≤ 1 or |𝑋 ∩ 𝑉𝑘 | ≤ 1, due to the symmetry of 𝐺).

Section 6.3. Bounds on the Total Domination Number

175

This conjecture in [282] attracted considerable interest. In 2004 a group of mathematicians from the USA, the Czech Republic, China, and Israel, combined forces and presented the following remarkable proof of this conjecture that the total domination of a graph with minimum degree at least 3 is at most one-half its order. In order to prove this result, Archdeacon et al. [35] surprisingly used Brooks’ Coloring Theorem. The key result they use to prove the conjecture is the following lemma. Lemma 6.51 ([35]) If 𝐻 is a bipartite graph with partite sets 𝑋 and 𝑌 whose vertices in 𝑌 are of degree at least 3, then there exists a set 𝐴 ⊆ 𝑋 such that 𝐴 dominates 𝑌 and | 𝐴| ≤ 41 |𝑋 ∪ 𝑌 |. Proof Let |𝑋 | = 𝑥 and |𝑌 | = 𝑦. We proceed by induction on |𝑉 (𝐻)| + |𝐸 (𝐻)|. The smallest graph described by the lemma is 𝐻 = 𝐾1,3 , where 𝑌 consists of the central vertex of the star and 𝑋 the set of three leaves. In this case, choosing the set 𝐴 to consist of any arbitrary vertex in 𝑋 yields the desired result. This establishes the base case. For the inductive hypothesis, assume the result is true for all bipartite graphs 𝐻 ′ with partite sets (𝑋 ′ , 𝑌 ′ ) whose vertices in 𝑌 ′ are of degree at least 3 in 𝐻 ′ . In what follows, we let |𝑋 ′ | = 𝑥 ′ and |𝑌 ′ | = 𝑦 ′ . Suppose there exists a vertex 𝑣 ∈ 𝑌 of degree at least 4. Let 𝑒 be any edge incident with 𝑣 and consider the bipartite graph 𝐻 ′ = 𝐻 − 𝑒 with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 and 𝑌 ′ = 𝑌 . We note that every vertex in 𝑌 ′ has degree at least 3 in 𝐻 ′ . Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) = 14 (𝑥 + 𝑦). The set 𝐴 = 𝐴′ is the desired set. Hence, we may assume that every vertex in 𝑌 has degree exactly 3 in 𝐻. Suppose there exists an isolated vertex 𝑣 ∈ 𝑋. Consider the bipartite graph 𝐻 ′ = 𝐻 − 𝑣 with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣} and 𝑌 ′ = 𝑌 . Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) < 14 (𝑥 + 𝑦). The set 𝐴 = 𝐴′ is the desired set. Hence, we may assume that every vertex in 𝑋 has degree at least 1 in 𝐻. Suppose there exists a vertex 𝑣 in 𝑋 of degree at least 3. If 𝑣 dominates 𝑌 , then the desired result is immediate by choosing 𝐴 = {𝑣} and noting that 𝑥 ≥ 3 and 𝑦 ≥ 3, and so | 𝐴| < 14 (𝑥 + 𝑦). Hence, we may assume that 𝑣 does not dominate 𝑌 . We consider the bipartite graph 𝐻 ′ = 𝐻 − N[𝑣] with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣} and 𝑌 ′ = 𝑌 \ N(𝑣). We note that 𝑥 ′ = 𝑥 − 1 and 𝑦 ′ ≤ 𝑦 − 3. Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) ≤ 14 (𝑥 + 𝑦) − 1. The set 𝐴 = 𝐴′ ∪ {𝑣} is the desired set. Hence, we may assume that every vertex in 𝑋 has degree at most 2 in 𝐻. With our current assumptions, every vertex in 𝑋 has degree 1 or 2 in 𝐻 and every vertex in 𝑌 has degree 3 in 𝐻. Let 𝑋𝑖 be the set of vertices in 𝑋 of degree 𝑖 and let |𝑋𝑖 | = 𝑥𝑖 for 𝑖 ∈ [2], and so 𝑥 = 𝑥1 + 𝑥2 and 𝑥1 + 2𝑥 2 = |𝐸 (𝐻)| = 3𝑦. Suppose that two vertices 𝑣 1 and 𝑣 2 in 𝑋2 have two common neighbors 𝑤 1 and 𝑤 2 in 𝑌 . We consider the bipartite graph 𝐻 ′ = 𝐻 − {𝑣 1 , 𝑣 2 , 𝑤 1 , 𝑤 2 } with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣 1 , 𝑣 2 } and 𝑌 ′ = 𝑌 \ {𝑤 1 , 𝑤 2 }. We note that 𝑥 ′ = 𝑥 − 2 and 𝑦 ′ = 𝑦 − 2. Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) = 14 (𝑥 + 𝑦) − 1. The set 𝐴 = 𝐴′ ∪ {𝑣 1 }

Chapter 6. Upper Bounds in Terms of Minimum Degree

176

is the desired set. Hence, we may assume that every two vertices in 𝑋2 have at most one common neighbor in 𝑌 . Let 𝐹 be the graph with 𝑉 (𝐹) = 𝑋2 , where two vertices are adjacent in 𝐹 if and only if they have a common neighbor in 𝐻. Let 𝑣 ∈ 𝑉 (𝐹). Let 𝑤 1 and 𝑤 2 be the two neighbors of 𝑣 in 𝐻 that belong to 𝑌 , and let N 𝐻 (𝑤 𝑖 ) = {𝑢 𝑖 , 𝑣, 𝑣 𝑖 } for 𝑖 ∈ [2]. Thus, N𝐹 (𝑣) = {𝑢 1 , 𝑣 1 } ∪ {𝑢 2 , 𝑣 2 }. If {𝑢 1 , 𝑣 1 } ∩ {𝑢 2 , 𝑣 2 } = ∅, then 𝑣 is the only common neighbor of 𝑤 1 and 𝑤 2 in 𝐻, and deg𝐹 (𝑣) = 4. If {𝑢 1 , 𝑣 1 } ∩ {𝑢 2 , 𝑣 2 } ≠ ∅, then deg𝐹 (𝑣) ≤ 3. Thus, Δ(𝐹) ≤ 4. Further, if Δ(𝐹) = 4, then renaming vertices if necessary, we may assume that deg𝐹 (𝑣) = 4. However, in this case by our earlier observations, neither 𝑢 𝑖 nor 𝑣 𝑖 is adjacent to 𝑤 3−𝑖 in the graph 𝐻 for 𝑖 ∈ [2], implying that there is no edge between the vertices of {𝑢 1 , 𝑣 1 } and the vertices of {𝑢 2 , 𝑣 2 } in the graph 𝐹. Therefore, no component of 𝐹 is a complete graph 𝐾5 . Applying Brooks’ Coloring Theorem, the graph 𝐹 is 4-colorable, and so 𝐹 has an independent set 𝐼 such that |𝐼 | ≥ 41 |𝑉 (𝐹)| = 14 𝑥 2 . In the graph 𝐻, the vertices in the set 𝐼 have vertex-disjoint neighborhoods, and so |N 𝐻 (𝐼)| = 2|𝐼 |. We now consider the set 𝑌 = 𝑌 \ N 𝐻 (𝐼). For each vertex 𝑤 ∈ 𝑌 , we choose an adjacent vertex 𝑤 ′ ∈ 𝑋 and we let Ø 𝐼′ = {𝑤 ′ }, 𝑤 ∈𝑌

𝐼′

|𝐼 ′ |

and so ⊆ 𝑋 \ 𝐼 and ≤ |𝑌 | = 𝑦 − 2|𝐼 |. We now let 𝐴 = 𝐼 ∪ 𝐼 ′ . Since 𝑥 = 𝑥1 + 𝑥2 5 and 𝑥1 + 2𝑥2 = |𝐸 (𝐻)| = 3𝑦, we have 𝑦 = 13 (𝑥1 + 2𝑥 2 ) and 14 (𝑥 + 𝑦) = 13 𝑥1 + 12 𝑥2 . Thus, | 𝐴| = |𝐼 | + |𝐼 ′ | ≤ 𝑦 − |𝐼 | ≤ 𝑦 − 14 𝑥2 = 13 (𝑥 1 + 2𝑥2 ) − 14 𝑥2 = 13 𝑥1 +

5 12 𝑥 2

= 14 (𝑥 + 𝑦).

By construction, the set 𝐴 ⊆ 𝑋 dominates the set 𝑌 in the graph 𝐻. As a consequence of Lemma 6.51, we have the main result of Archdeacon et al. [35]. Theorem 6.52 ([35]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, and let 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }. Let 𝐻 be the bipartite graph with partite sets 𝑋 = {𝑥1 , 𝑥2 , . . . , 𝑥 𝑛 } and 𝑌 = {𝑦 1 , 𝑦 2 , . . . , 𝑦 𝑛 }, and where 𝑥 𝑖 𝑦 𝑗 is an edge in 𝐻 if and only if 𝑣 𝑖 𝑣 𝑗 is an edge in 𝐺, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. By construction, 𝛿(𝐻) = 𝛿(𝐺) ≥ 3. In particular, every vertex in 𝑌 has degree at least 3 in 𝐻. By Lemma 6.51, there exists a set 𝐴 ⊆ 𝑋 such that 𝐴 dominates 𝑌 and | 𝐴| ≤ 14 |𝑋 ∪ 𝑌 | = 12 𝑛. Every vertex 𝑦 𝑗 ∈ 𝑌 is adjacent to a vertex 𝑥𝑖 ∈ 𝐴 in 𝐻, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. Thus, every vertex 𝑣 𝑗 ∈ 𝑉 is adjacent to a vertex 𝑣 𝑖 ∈ 𝐴 in 𝐺, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. This implies that 𝐴 is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ | 𝐴| ≤ 12 𝑛. A natural problem is to characterize the connected graphs that achieve equality in the upper bound of Theorem 6.52. By Proposition 6.49, the generalized Petersen graph 𝑃(8, 3) achieves equality in the bound of Theorem 6.52, and by Proposition 6.50, we have two infinite classes of cubic graphs that achieve equality in the

Section 6.3. Bounds on the Total Domination Number

177

bound. Hence, this bound is tight. However, for several years it remained an open problem to give a complete characterization of the graphs that achieve equality in the bound of Theorem 6.52. Using a graph theoretic approach, this appears to be a difficult problem. In the next section, however, we discuss an interplay with total dominating sets in graphs and transversals in hypergraphs that will enable us to provide such a characterization, showing that the generalized Petersen graph 𝑃(8, 3) and the two infinite families, Gcubic and Hcubic , constructed in [282], are precisely the extremal graphs. The result of Theorem 6.52 was strengthened in 2007 by Lam and Wei [553]. Theorem 6.53 ([553]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 such that deg(𝑢) + deg(𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺, then 𝛾t (𝐺) ≤ 21 𝑛. As a special case of Theorem 6.53, we have the following result. Theorem 6.54 ([553]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 such that the set of vertices of degree 2 in 𝐺 form an independent set, then 𝛾t (𝐺) ≤ 12 𝑛. The result of Theorem 6.53 was further strengthened in 2007 by Henning and Yeo [481] who relaxed the condition that every two adjacent vertices have degree sum at least 5. In order to state their result, we introduce some additional terminology. Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and Δ(𝐺) ≥ 3. Let L be the set of vertices of 𝐺 of degree at least 3 in 𝐺, and let 𝑃 be any path component of 𝐺 − L. If |𝑉 (𝑃)| ≡ 0 (mod 4) and either the two ends of 𝑃 are adjacent in 𝐺 to a common vertex of L or the two ends of 𝑃 are adjacent in 𝐺 to different, but adjacent, vertices of L, then we call 𝑃 a 0★-path. If |𝑉 (𝑃)| ≥ 5 and |𝑉 (𝑃)| ≡ 1 (mod 4) with the two ends of 𝑃 adjacent in 𝐺 to a common vertex of L, we call 𝑃 a 1★-path. If |𝑉 (𝑃)| ≡ 3 (mod 4), we call 𝑃 a 3★-path. For 𝑖 ∈ {0, 1, 3}, we denote the number of 𝑖★-paths in 𝐺 by 𝑝 𝑖 . Theorem 6.55 ([481]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and Δ(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 (𝑛 + 𝑝 0 + 𝑝 1 + 𝑝 3 ).

6.3.4

An Interplay with Transversals in Hypergraphs

If the minimum degree of a graph is at least 4, then establishing good upper bounds on the total domination number of a graph using a purely graph theory approach is often challenging. Thankfully, we do have a powerful tool at our disposal, namely a translation of the problem of determining the total domination number of a graph to the problem of determining the transversal number of a hypergraph that can be associated with the graph. Indeed, much of the interest in total domination in graphs dating back to 2003 arises from its interplay with transversals in hypergraphs. Before we describe this interplay between graphs and hypergraphs, which is explained more comprehensively in the book [490], we present the basic hypergraph terminology and notation that we shall use. Hypergraphs are systems of sets which are conceived as natural extensions of graphs. A hypergraph 𝐻 is a finite set 𝑉 (𝐻) of elements, called vertices, together with a finite multiset 𝐸 (𝐻) of subsets of 𝑉 (𝐻),

178

Chapter 6. Upper Bounds in Terms of Minimum Degree

called hyperedges. Abusing terminology, we often refer to a “hyperedge” of a hypergraph simply as an “edge” of the hypergraph. The notation we use for hypergraphs in what follows is similar to that we use for graphs. Specifically, the order of 𝐻 is 𝑛(𝐻) = |𝑉 (𝐻)| and the size of 𝐻 is 𝑚(𝐻) = |𝐸 (𝐻)|. A 𝑘-edge in 𝐻 is an edge of size 𝑘. The hypergraph 𝐻 is 𝑘-uniform if every edge of 𝐻 is a 𝑘-edge. Every (simple) graph is a 2-uniform hypergraph. Thus, graphs are special hypergraphs. Two vertices in 𝐻 are adjacent if they belong to a common edge of 𝐻. The degree of a vertex 𝑣 in 𝐻, denoted by deg 𝐻 (𝑣), is the number of edges of 𝐻 which contain 𝑣. The minimum and maximum degrees among the vertices of 𝐻 are denoted by 𝛿(𝐻) and Δ(𝐻), respectively. We refer the reader to Appendix A for additional hypergraph terminology. The Fano plane, shown in Figure 6.22, is an example of a 3-uniform, 3-regular hypergraph of order 𝑛(𝐻) = 7 and size 𝑚(𝐻) = 7. As with graphs, we write 𝑛 for 𝑛(𝐻) and 𝑚 for 𝑚(𝐻) when the graph 𝐻 is clear from the context.

Figure 6.22 The Fano plane A transversal (also called a hitting set in the literature) in a hypergraph 𝐻 is a set 𝑇 of vertices that have a nonempty intersection with every edge of 𝐻. Such a set 𝑇 is said to cover or hit every edge of 𝐻. The transversal number 𝜏(𝐻) of 𝐻 is the minimum cardinality of a transversal in 𝐻. We note the traversal number is called the vertex cover number when 𝐻 is a graph and is denoted by 𝛽(𝐻). A transversal in 𝐻 of cardinality 𝜏(𝐻) is called a 𝜏-transversal of 𝐻. For example, if 𝐻 is the Fano plane shown in Figure 6.22, then 𝜏(𝐻) = 3. Moreover, if 𝑒 is an arbitrary edge of 𝐻, then the set consisting of the three vertices that belong to the edge 𝑒 is a 𝜏-transversal of 𝐻. Let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. Let ONH(𝐺) be the open neighborhood hypergraph of 𝐺, where the hypergraph ONH(𝐺) has the same vertex set as 𝐺, namely 𝑉, and whose edge set consists of the open neighborhoods of the vertices in the graph 𝐺. Thus, ONH(𝐺) has exactly 𝑛 edges corresponding to the open neighborhoods N𝐺 (𝑣) of the vertices 𝑣 in the graph 𝐺. By our supposition that 𝐺 has minimum degree 𝛿 ≥ 1, every open neighborhood of a vertex in 𝐺 has cardinality at least 𝛿, and therefore each edge of ONH(𝐺) has size at least 𝛿. To illustrate the open neighborhood hypergraph of a graph, if 𝐺 = 𝐺 14 is the well-known Heawood graph shown in Figure 6.23(a), then the open neighborhood hypergraph of 𝐺 is shown in Figure 6.23(b). We note that in this case, ONH(𝐺) consists of two vertex-disjoint copies of the Fano plane.

Section 6.3. Bounds on the Total Domination Number 7

𝑎

1

𝑔

179

1

𝑎

𝑏

6

2

3

6

2

𝑔

𝑏 𝑑

𝑐

𝑓 3

5 𝑒

4 (a) 𝐺 14

𝑑

7

5

4

𝑒

𝑐

𝑓

(b) ONH(𝐺 14 )

Figure 6.23 The Heawood graph and its open neighborhood hypergraph

We are now in a position to describe the interplay between total domination in graphs and transversals in hypergraphs. As before, let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. Every TD-set in 𝐺 contains a vertex from the open neighborhood of every vertex in 𝐺, and is therefore a transversal in ONH(𝐺). In particular, if 𝐷 is a 𝛾t -set of 𝐺, then 𝐷 is a transversal in ONH(𝐺), and so 𝜏(ONH(𝐺)) ≤ |𝐷| = 𝛾t (𝐺). On the other hand, every transversal in ONH(𝐺) contains a vertex from the open neighborhood of every vertex of 𝐺, and is therefore a TD-set in 𝐺. In particular, if 𝑇 is a 𝜏-transversal of ONH(𝐺), then 𝑇 is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ |𝑇 | = 𝜏(ONH(𝐺)). Consequently, 𝛾t (𝐺) = 𝜏(ONH(𝐺)). Thus, the transversal number of the open neighborhood hypergraph of a graph is precisely the total domination number of the graph. We state this formally as follows. Observation 6.56 If 𝐺 is an isolate-free graph and ONH(𝐺) is the open neighborhood hypergraph of 𝐺, then 𝛾t (𝐺) = 𝜏(ONH(𝐺)). To illustrate Observation 6.56, consider our earlier example when 𝐺 = 𝐺 14 is the Heawood graph shown in Figure 6.23(a). Using purely graph theory arguments, one can show (with some work) that 𝛾t (𝐺) = 6. One can also observe, as done earlier, that the open neighborhood hypergraph ONH(𝐺) of 𝐺 consists of two components, both isomorphic to the Fano plane (see Figure 6.23(b)). If 𝐻 is the Fano plane, then 𝜏(𝐻) = 3, implying that 𝜏(ONH(𝐺)) = 6. Thus, by Observation 6.56, we can immediately deduce that 𝛾t (𝐺) = 6. The idea of using transversals in hypergraphs to obtain bounds on the total domination number of a graph first appeared in a paper by Thomassé and Yeo [709] submitted in 2003 (but only published in 2007). Their paper served to open the floodgates for further improved bounds on the total domination number of a graph with given minimum degree. Without wishing to detract from the elegant graph theoretic proof of Theorem 6.52 given in 2004 by Archdeacon et al. [35], we note that the result itself can readily be deduced from the following 1990 hypergraph result due to Tuza [720] and Chvátal and McDiarmid [176]. However, as remarked earlier, the interplay between total domination in graphs and transversals in hypergraphs seemed to pass

180

Chapter 6. Upper Bounds in Terms of Minimum Degree

by unnoticed until announced in 2003 by Thomassé and Yeo [709]. We omit a proof of Theorem 6.57, which can be found in [490]. Theorem 6.57 ([176, 720]) If 𝐻 is a hypergraph of order 𝑛 and size 𝑚 where all edges have size at least 3, then 𝜏(𝐻) ≤ 41 (𝑛 + 𝑚). As an immediate consequence of Theorem 6.57, we have the result of Theorem 6.52, which we restate as a corollary (of Theorem 6.57). Corollary 6.58 ([176, 720]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3. The open neighborhood hypergraph ONH(𝐺) of 𝐺 has 𝑛 vertices and 𝑛 edges where all edges have size at least 𝛿(𝐺) ≥ 3. Hence, by Observation 6.56 and Theorem 6.57, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 14 (𝑛 + 𝑚) = 12 𝑛. In order to characterize the connected graphs achieving equality in the upper bound of Theorem 6.52, in 2008 Henning and Yeo [483] first characterized the hypergraphs that achieve equality in the upper bound of Theorem 6.57. We omit the details of their characterization, which are discussed in [490]. As a consequence of this hypergraph characterization, the authors in [483] obtained the desired graph theory characterization of the extremal graphs achieving equality in the bound of Theorem 6.52. We omit the proof, which can be found in [483] and relies heavily on the interplay between total domination in graphs and transversals in hypergraphs. Theorem 6.59 ([483]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) = 12 𝑛 if and only if 𝐺 ∈ Gcubic ∪ Hcubic or 𝐺 is the generalized Petersen graph 𝑃(8, 3).

6.3.5

Minimum Degree Four

In 1992 Chvátal and McDiarmid [176] established the following upper bound on the transversal number of uniform hypergraphs in terms of their order and size. Theorem 6.60 ([176]) For every integer 𝛿 ≥ 2, if 𝐻 is a 𝛿-uniform hypergraph of order 𝑛 and size 𝑚, then   𝑛 + 2𝛿 𝑚 𝜏(𝐻) ≤  3 𝛿  . 2

As a consequence of Theorem 6.60, we have the following upper bound on the total domination number of a graph in terms of its order 𝑛 and minimum degree 𝛿. Theorem 6.61 ([176]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 2, then  ! 1 + 2𝛿 𝛾t (𝐺) ≤  3 𝛿  𝑛. 2

Section 6.3. Bounds on the Total Domination Number

181

Proof Let 𝐺 be a graph with minimum degree 𝛿(𝐺) = 𝛿 ≥ 2 and order 𝑛. Consider the hypergraph 𝐻 obtained from the open neighborhood hypergraph ONH(𝐺) by “shrinking” all edges of ONH(𝐺), if necessary, to edges of size 𝛿, that is, if 𝑒 is an edge of ONH(𝐺) of size 𝑘 for 𝑘 > 𝛿, then we select a subset 𝑒 ′ ⊂ 𝑒 of 𝑘 − 𝛿 vertices arbitrarily, and remove the vertices in 𝑒 ′ from the edge 𝑒. The resulting edge 𝑒 \ 𝑒 ′ of size 𝛿 is the edge of 𝐻 that replaces the edge 𝑒 of ONH(𝐺). By construction, the hypergraph 𝐻 is a 𝛿-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, by Observation 6.56, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻). The desired result now follows from Theorem 6.60. In Table 6.8, we list the bounds in Theorem 6.61 for 𝛿 ∈ [10].

𝛿 𝛿 !

1+ 2  3𝛿 

𝑛

1

2

3

4

5

6

7

8

9

10

𝑛

2 3𝑛

1 2𝑛

1 2𝑛

3 7𝑛

4 9𝑛

2 5𝑛

5 12 𝑛

5 13 𝑛

2 5𝑛

2

Table 6.8 Upper bounds on 𝛾t (𝐺) in Theorem 6.61 for 𝛿 ∈ [10] The bounds in Theorem 6.61 for small 𝛿 ∈ [3] are best possible. For 𝛿 ∈ [2], this may be seen by taking 𝐺 to be the disjoint union of copies of 𝐾 𝛿+1 , while for 𝛿 = 3 the bound given in Theorem 6.61 is precisely the tight bound of Theorem 6.52. However, for 𝛿 ≥ 4, the upper bounds in Theorem 6.61 are not best possible. In their seminal 2007 paper, Thomassé and Yeo [709] improved the upper bound given in Theorem 6.61 when 𝛿 = 4. Theorem 6.62 ([709]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛. In order to prove Theorem 6.62, Thomassé and Yeo [709] proved the following result on transversals in 4-uniform hypergraphs. Theorem 6.63 ([709]) If 𝐻 is a 4-uniform hypergraph of order 𝑛 and size 𝑚, then 5 4 𝜏(𝐻) ≤ 21 𝑛 + 21 𝑚. Theorem 6.63 is a special case of a stronger, more general result proven in [709]. However, the statement of Theorem 6.63 suffices for our purposes. To see this, let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4. The open neighborhood hypergraph ONH(𝐺) of 𝐺 has 𝑛 vertices and 𝑛 edges where all edges have size at least 𝛿(𝐺) ≥ 4. As before, we consider the hypergraph 𝐻 obtained from ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 4 to produce a 4-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, by Observation 6.56 1 and Theorem 6.63, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤ 21 (5𝑛 + 4𝑚) = 37 𝑛. Hence, Theorem 6.62 is an immediate consequence of Theorem 6.63. In order to characterize the connected graphs that achieve equality in the upper bound of Theorem 6.62, we first need a characterization of the 4-uniform hypergraphs that achieve equality in the bound of Theorem 6.63. The complement 𝐹 of a

182

Chapter 6. Upper Bounds in Terms of Minimum Degree

hypergraph 𝐹 is the hypergraph with the same vertex set 𝑉 (𝐹) and where 𝑒 is an edge in the complement 𝐹 if and only if 𝑉 (𝐹) \ 𝑒 is an edge in 𝐹, that is, 𝑒 ∈ 𝐸 (𝐹) if and only if 𝑉 (𝐹) \ 𝑒 ∈ 𝐸 (𝐹). If 𝐻 is the complement of the Fano plane, then 𝐻 is a 4-uniform hypergraph with 𝑛 = 7 vertices and 𝑚 = 7 edges satisfying 1 𝜏(𝐻) = 3 = 63 21 = 21 (5𝑛 + 4𝑚). With a more detailed analysis, Yeo [765] (see also [490]) showed that the complement of the Fano plane is the only hypergraph that achieves equality in the bound of Theorem 6.63. The well-known Heawood graph is shown in Figure 6.24(a). Theorem 6.64 ([765]) If 𝐻 is a 4-uniform hypergraph of order 𝑛 and size 𝑚, then 5 4 𝜏(𝐻) ≤ 21 𝑛 + 21 𝑚, with equality if and only if 𝐻 is the complement of the Fano plane. The bipartite complement of the Heawood graph, shown in Figure 6.24(b), is the bipartite graph formed by taking the two partite sets of the Heawood graph and joining a vertex from one partite set to a vertex from the other partite set by an edge whenever they are not joined in the Heawood graph. The bipartite complement of the Heawood graph can also be seen as the incidence bipartite graph of the complement of the Fano plane. The incidence bipartite graph 𝐺 of a hypergraph 𝐻 is the bipartite graph with partite sets (𝑋, 𝑌 ) where 𝑋 = 𝑉 (𝐻) and 𝑌 = 𝐸 (𝐻), where a vertex 𝑣 ∈ 𝑋 is joined to a vertex 𝑒 ∈ 𝑌 in the bipartite graph 𝐺 if the vertex 𝑣 ∈ 𝑉 (𝐻) is contained in the edge 𝑒 ∈ 𝐸 (𝐻) in the hypergraph 𝐻.

(a) The Heawood graph

(b) Bipartite complement of the Heawood graph

Figure 6.24 The Heawood graph and its bipartite complement Using the interplay between total domination in graphs and transversals in hypergraphs, as a consequence of Theorem 6.64, we have the following result, a proof of which is given in [490]. Theorem 6.65 ([490]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛, with equality if and only if 𝐺 is the bipartite complement of the Heawood graph.

6.3.6

Minimum Degree Five

Thomassé and Yeo [709] posed the following conjecture about the total domination number of a graph with minimum degree at least 5.

Section 6.3. Bounds on the Total Domination Number

183

Conjecture 6.66 (Thomassé-Yeo Conjecture [709]) If 𝐺 is a graph of order 𝑛 with 4 𝛿(𝐺) ≥ 5, then 𝛾t (𝐺) ≤ 11 𝑛. If Conjecture 6.66 is true, then the bound is best possible, as shown by Thomassé and Yeo [709], who constructed a 5-uniform hypergraph 𝐻 as follows. Let 𝐻 be the hypergraph with vertex set 𝑉 (𝐻) = [10] 0 and edge set 𝐸 (𝐻) = {𝑒 0 , 𝑒 1 , . . . , 𝑒 10 }, where the edge 𝑒 𝑖 = 𝑄 + 𝑖 for 𝑖 ∈ [10] 0 and where 𝑄 = {1, 3, 4, 5, 9} is the set of non-zero quadratic residues modulo 11. Thus, 𝑒 0 = {1, 3, 4, 5, 9}, 𝑒 1 = {2, 4, 5, 6, 10}, 𝑒 2 = {0, 3, 5, 6, 7}, . . . , 𝑒 10 = {0, 2, 3, 4, 8}. The hypergraph 𝐻 is a 5uniform, 5-regular hypergraph with 𝑛 = 11 vertices and 𝑚 = 11 edges, and with 4 transversal number 𝜏(𝐻) = 4 = 11 𝑛. Let 𝐺 22 be the incidence bipartite graph of the hypergraph 𝐻. The graph 𝐺 22 , illustrated in Figure 6.25, is a 5-regular (bipartite) 4 graph of order 𝑛 = 22 that satisfies 𝛾t (𝐺 22 ) = 8 = 11 𝑛.

Figure 6.25 A graph 𝐺 22 of order 𝑛 = 22 and 𝛾t (𝐺 22 ) = 8 =

4 11 𝑛

Several attempts were made to settle Conjecture 6.66, posed by Thomassé and Yeo. In 2015 Dorfling and Henning [243] established the following upper bound on the transversal number of a 5-uniform hypergraph in terms of its order and size. Theorem 6.67 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛 and size 𝑚, then 1 𝜏(𝐻) ≤ 44 (10𝑛 + 7𝑚). From Observation 6.56 and Theorem 6.67, one can prove the following upper bound on the total domination number of a graph with minimum degree at least 5. Theorem 6.68 ([243]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then
4 1 𝛾t (𝐺) ≤ 17 44 𝑛 = 11 + 44 𝑛. The upper bound in Theorem 6.67 was subsequently improved in 2016 by Eustis et al. [263]. To state their result, for 𝑑 ≥ 0 an integer, let 𝑓 (𝑑) be the function defined as follows. For small 𝑑 ∈ [4] 0 , the values of 𝑓 (𝑑) are given in Table 6.9. For 𝑑 ≥ 5, let 𝑓 (𝑑) be defined recursively by 𝑓 (𝑑) = 𝑓 (𝑑 − 1) −

2 𝑓 (𝑑 − 1) − 𝑓 (𝑑 − 2) . 4𝑑

(6.12)

Chapter 6. Upper Bounds in Terms of Minimum Degree

184

𝑑

0

1

2

3

4

𝑓 (𝑑)

1

4 5

26 35

9 13

49 75

Table 6.9 Values of 𝑓 (𝑑) for small 𝑑 ∈ [4] 0 .

We are now in a position to state the result in [263]. Theorem 6.69 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛, then ∑︁
𝜏(𝐻) ≤ 𝑛 − 𝑓 deg 𝐻 (𝑣) . 𝑣 ∈𝑉 (𝐻 )

We remark that the proof of Theorem 6.69 given in [243] is constructive, and a transversal 𝑇
in a 5-uniform hypergraph of order 𝑛 satisfying |𝑇 | ≤ 𝑛 − Í 𝑣 ∈𝑉 (𝐻 ) 𝑓 deg 𝐻 (𝑣) can be found in polynomial time. As shown in [243], the function 𝑓 defined in Equation (6.12) is convex, assuming we extend the domain of 𝑓 to the nonnegative reals by linear interpolation. This implies that if 𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 are arbitrary nonnegative reals, then  ∑︁  𝑛 𝑛 ∑︁ 1 1 𝑓 (𝑑𝑖 ) ≥ 𝑓 𝑛 𝑑𝑖 . 𝑛 𝑖=1

𝑖=1

Hence, if the 5-uniform hypergraph 𝐻 of order 𝑛 has size 𝑚 = 𝑛, then ∑︁ 𝑛 = 𝑚 = 15 deg 𝐻 (𝑣), 𝑣 ∈𝑉 (𝐻 )

implying that ∑︁




𝑓 deg 𝐻 (𝑣) ≥ 𝑛 · 𝑓

𝑣 ∈𝑉 (𝐻 )



∑︁

1 deg 𝐻 (𝑣) 𝑛 𝑣 ∈𝑉 (𝐻 )

 = 𝑛 · 𝑓 (5).

Therefore, as a consequence of Theorem 6.69, we have the following result. Theorem 6.70 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛 and size 𝑛, then
2453 𝜏(𝐻) ≤ 1 − 𝑓 (5) 𝑛 = 𝑛 ≈ 0.3773 𝑛. 6500 From Observation 6.56 and Theorem 6.70, one can prove the following upper bound on the total domination number of a graph with minimum degree at least 5. Theorem 6.71 ([243]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then   2453 4 11 𝛾t (𝐺) ≤ 𝑛< + 𝑛. 6500 11 800 However, Conjecture 6.66 has yet to be settled. To achieve this goal, a new idea different than those employed in the proofs of Theorems 6.67 and 6.70 seems necessary.

Section 6.3. Bounds on the Total Domination Number

185

6.3.7 Minimum Degree Six It is a natural question to ask whether the Thomassé-Yeo Conjecture 6.66 holds if we relax the minimum degree condition in the statement of the conjecture from minimum degree at least 5 to minimum degree at least 6. This is indeed the case, as shown by Henning and Yeo [493]. In order to state their result, we define the weight w(𝑒) of an edge 𝑒 in a 6-uniform hypergraph 𝐻 as w(𝑒) = 7707. Further, we define the weight of the vertex 𝑣, denoted w(𝑣), in a 6-uniform hypergraph 𝐻 as 7707 if deg 𝐻 (𝑣) ≥ 4, and we define w(𝑣) as 7524, 6861, and 5788 if deg 𝐻 (𝑣) = 𝑖 and 𝑖 equals 3, 2, and 1, respectively. Thus, the weight w(𝑣) of a vertex 𝑣 in the hypergraph 𝐻 is given in Table 6.10. deg 𝐻 (𝑣)

≥4

3

2

1

0

w(𝑣)

7707

7524

6861

5788

0

Table 6.10 The weight w(𝑣) assigned to a vertex 𝑣 in the hypergraph 𝐻 We define the weight of the 6-uniform hypergraph 𝐻 of order 𝑛 and size 𝑚 as ∑︁ ∑︁ w(𝐻) = w(𝑒) + w(𝑣). 𝑒∈𝐸 (𝐻 )

𝑣 ∈𝑉 (𝐻 )

Equivalently, w(𝐻) = 7707𝑛 ≥4 + 7524𝑛3 + 6861𝑛2 + 5788𝑛1 + 7707𝑚,

(6.13)

where 𝑛𝑖 for 𝑖 ∈ [3] 0 denotes the number of vertices of degree 𝑖 in 𝐻, and 𝑛 ≥4 denotes the number of vertices of degree at least 4 in 𝐻. Thus, 𝑛 = 𝑛 ≥4 + 𝑛3 + 𝑛2 + 𝑛1 + 𝑛0 . We are now in a position to state the result given in [493]. Theorem 6.72 ([493]) If 𝐻 is a 6-uniform hypergraph, then 42435𝜏(𝐻) ≤ w(𝐻). As a consequence of Theorem 6.72, we have the following result. Corollary 6.73 ([493]) If 𝐻 is a 6-uniform hypergraph of order 𝑛 and size 𝑚, then 𝜏(𝐻) ≤

2569 (𝑛 + 𝑚). 14145

Proof Let 𝐻 be a 6-uniform hypergraph. By Equation (6.13), w(𝐻) = 5788𝑛1 + 6861𝑛2 + 7524𝑛3 + 7707𝑛 ≥4 + 7707𝑚 ≤ 7707(𝑛0 + 𝑛1 + 𝑛2 + 𝑛3 + 𝑛 ≥4 ) + 7707𝑚 = 7707(𝑛 + 𝑚).

Chapter 6. Upper Bounds in Terms of Minimum Degree

186 Thus, by Theorem 6.72,

7707(𝑛 + 𝑚) ≥ w(𝐻) ≥ 42435𝜏(𝐻), or equivalently, 𝜏(𝐻) ≤

2569 (𝑛 + 𝑚). 14145

As an immediate consequence of Observation 6.56 and Corollary 6.73, we have the following upper bound on the total domination number of a graph with minimum degree at least 6. Theorem 6.74 ([493]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤

5138 𝑛 ≈ 0.3773 𝑛. 14145

Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 6. Let 𝐻 be the open neighborhood hypergraph ONH(𝐺) of 𝐺. Thus, 𝐻 has order 𝑛(𝐻) = 𝑛 and size 𝑚(𝐻) = 𝑛, where all edges have size at least 𝛿(𝐺) ≥ 6. Hence, by Observation 6.56 and Theorem 6.74, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤


2569 2569 5138 𝑛(𝐻) + 𝑚(𝐻) = (𝑛 + 𝑛) = 𝑛, 14145 14145 14145

completing the proof of the theorem. 5138 4 1 We note that 14145 . Hence, as an immediate consequence of Theo< 11 − 2510 rem 6.74, the Thomassé-Yeo Conjecture 6.66 holds if the minimum degree is at least 6. The upper bound in Theorem 6.74 was slightly improved by the same authors in [495] as an application of the so-called Tuza-Vestergaard Theorem, which states that the transversal number of a 6-uniform hypergraph of order 𝑛 is at most 𝑛4 .

Theorem 6.75 ([495]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤ 4549 13299 𝑛 ≈ 0.3420 𝑛. The bound in Theorem 6.75 is currently the best known upper bound on the total domination number of a graph with minimum degree at least 6. However, it is not known if there is a graph with minimum degree at least 6 that achieves this bound. It is unlikely that this bound is achievable. Henning and Yeo [493] pose the following conjecture. Conjecture 6.76 ([493]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤ 4 13 𝑛. If Conjecture 6.76 is true, then the bound is best possible as shown by the authors in [493], who construct a 5-uniform hypergraph 𝐻 as follows. Let 𝐻 be the hypergraph with vertex set 𝑉 (𝐻) = [12] 0 and edge set 𝐸 (𝐻) = {𝑒 0 , 𝑒 1 , . . . , 𝑒 12 }, where the edge 𝑒 𝑖 = 𝑄 + 𝑖 for 𝑖 ∈ [12] 0 and where 𝑄 is the set of non-zero quadratic residues modulo 13, that is, 𝑄 = {1, 3, 4, 9, 10, 12}. The hypergraph 𝐻 is a 6-regular, 6-uniform hypergraph with 𝑛(𝐻) = 13 vertices
and 𝑚(𝐻) = 13 edges, and with 2 transversal number 𝜏(𝐻) = 4 = 13 𝑛(𝐻) + 𝑚(𝐻) .

Section 6.3. Bounds on the Total Domination Number

187

Let 𝐺 26 be the incidence bipartite graph of the hypergraph 𝐻. Thus, 𝐺 26 is the bipartite graph with partite sets (𝑋, 𝑌 ) where 𝑋 = 𝑉 (𝐻) and 𝑌 = 𝐸 (𝐻), where a vertex 𝑣 ∈ 𝑋 is joined to a vertex 𝑒 ∈ 𝑌 in the bipartite graph 𝐺 if the vertex 𝑣 ∈ 𝑉 (𝐻) is contained in the edge 𝑒 ∈ 𝐸 (𝐻) in the hypergraph 𝐻. The graph 𝐺 26 , shown in Figure 6.26, is a 6-regular (bipartite) graph of order 𝑛 = 26 satisfying 4 𝛾t (𝐺 26 ) = 8 = 13 𝑛. Thus, if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then the best 4 upper bound we can hope for is 𝛾t (𝐺) ≤ 13 𝑛. Conjecture 6.76 claims that this is the best possible upper bound.

Figure 6.26 A graph 𝐺 26 of order 𝑛 = 26 satisfying 𝛾t (𝐺 22 ) = 8 =

4 13 𝑛

We summarize the best known upper bounds on the total domination number of a graph 𝐺 in terms of its order 𝑛 and minimum degree 𝛿 ∈ [6] in Table 6.11. The bounds for minimum degree 𝛿 ∈ [3] are tight (in the sense that there are infinitely many connected graphs that achieve the bound in each case). However, although the bound for minimum degree 𝛿 = 4 is best possible, it is not known if the bound is tight. As remarked earlier, the known bounds for minimum degree 𝛿 = 5 and 4 𝛿 = 6 are unlikely best possible, and the conjectured bounds are 𝛾t (𝐺) ≤ 11 𝑛 and 4 𝛾t (𝐺) ≤ 13 𝑛, respectively.

6.3.8

A Heuristic Bound

In this section, we present a heuristic algorithm that yields an upper bound on the total domination of a graph in terms of its order and minimum degree. Moreover, we show that this simple greedy approximation algorithm can be used to efficiently find a TD-set whose cardinality is “close” to the cardinality of a minimum TD-set. We shall prove the following 2007 result due to Henning and Yeo [482]. Theorem 6.77 ([482]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, then   1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛. 𝛿

Chapter 6. Upper Bounds in Terms of Minimum Degree

188 Year

𝛿≥

1980

𝛿(𝐺) ≥ 1

𝛾t (𝐺) ≤

2000

𝛿(𝐺) ≥ 2

𝛾t (𝐺) ≤

2004

𝛿(𝐺) ≥ 3

𝛾t (𝐺) ≤

2007

𝛿(𝐺) ≥ 4

𝛾t (𝐺) ≤

2016

𝛿(𝐺) ≥ 5

𝛾t (𝐺) ≤

2022

𝛿(𝐺) ≥ 6

𝛾t (𝐺) ≤

a b

𝛾t ≤

Citation

2 a 3𝑛 4 b 7𝑛 1 2𝑛 3 7𝑛
2453 4 11 6500 𝑛 < 11 + 800 𝑛
4549 4 17 13299 𝑛 < 13 + 494 𝑛

[182] [453] [35, 176, 720] [709] [263] [495]

If 𝑛 ≥ 3 and 𝐺 is connected. If 𝑛 ≥ 11 and 𝐺 is connected. Table 6.11 Upper bounds on 𝛾t (𝐺) with minimum degree 𝛿 ∈ [6]

Further, using a greedy algorithm we can, in time complexity O (𝑛 + 𝛿 𝑛), find a 𝛿)
TD-set 𝑇 in the graph 𝐺 such that |𝑇 | ≤ 1+ln( 𝑛. 𝛿 Proof Let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. If 𝛿 = 1, 𝛿)
then 𝛾t (𝐺) ≤ 𝑛 = 1+ln( 𝑛. Hence, we may assume that 𝛿 ≥ 2, for otherwise 𝛿 the result is immediate. We now consider the hypergraph 𝐻 obtained from the open neighborhood hypergraph ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 𝛿. By construction, 𝐻 is a 𝛿-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻). We now greedily select vertices of maximum degree at every stage in the algorithm until we obtain a transversal in 𝐻. The resulting set is a TD-set of 𝐺. More precisely, we construct a transversal 𝑇 in 𝐻 as follows. Initially, we select any vertex 𝑣 of maximum degree in 𝐻, delete 𝑣 and all edges incident with 𝑣 from 𝐻, and let 𝑇 = {𝑣}. We note that the resulting hypergraph 𝐻 − 𝑣 is a 𝛿-uniform hypergraph with at most 𝑛 vertices. In this resulting hypergraph, we select any vertex 𝑤 of maximum degree, delete 𝑤 and all edges incident with it, and then add 𝑤 to the set 𝑇. We continue this process until no edges remain. By construction, every deleted edge contains at least one vertex of the resulting set 𝑇, implying that the set 𝑇 is a transversal in 𝐻. Hence, 𝜏(𝐻) ≤ |𝑇 |.
𝛿) We show next that |𝑇 | ≤ 1+ln( 𝑛. Let 𝐻 𝛿 = 𝐻 and let 𝑇 𝛿 denote the set of 𝛿 vertices in 𝑇 of degree 𝛿 or more in the current hypergraph when they were added to the transversal 𝑇. Let 𝑡 𝛿 = |𝑇 𝛿 |. For 𝑗 ∈ [𝛿 − 1], let 𝑇 𝑗 be the set of all vertices that had degree exactly 𝑗 in the current hypergraph when they were added to the transversal 𝑇, and let 𝑡 𝑗 = |𝑇 𝑗 |. By definition, 𝑇=

𝛿 Ø 𝑗=1

𝑇𝑖

and

|𝑇 | =

𝛿 ∑︁ 𝑗=1

𝑡𝑗.

Section 6.3. Bounds on the Total Domination Number

189

For 𝑗 ∈ [𝛿 − 1], let 𝐻 𝑗 = 𝐻 𝑗+1 − 𝑇 𝑗+1 and so, 𝐻 𝑗 is obtained from the hypergraph 𝐻 𝑗+1 by deleting all vertices in 𝑇 𝑗+1 and all edges incident with vertices in 𝑇 𝑗+1 . We note that 𝐻 𝑗 is a 𝛿-uniform hypergraph with at most 𝑛 vertices and with maximum degree Δ(𝐻 𝑗 ) ≤ 𝑗, implying that ∑︁ deg 𝐻 𝑗 (𝑣) ≤ 𝑗𝑛. (6.14) 𝑣 ∈𝑉 (𝐻 𝑗 )

For 𝑗 ∈ [𝛿 − 1] \ {1}, the hypergraph 𝐻 𝑗 −1 is obtained from 𝐻 𝑗 by deleting exactly 𝑗𝑡 𝑗 edges and so, |𝐸 (𝐻 𝑗 )| = |𝐸 (𝐻 𝑗 −1 )| + 𝑗𝑡 𝑗 .

(6.15)

Since Δ(𝐻1 ) ≤ 1, the edges of 𝐻1 have no vertices in common and so, by construction, exactly one vertex from each edge of 𝐻1 is added to the set 𝑇1 . Thus, |𝐸 (𝐻1 )| = 𝑡 1 . For each value of 𝑗 with 𝑗 ∈ [𝛿 − 1], it therefore follows from Inequality (6.14) and Equation (6.15) that 𝑗 ∑︁

𝑖𝑡𝑖 = 𝑡1 +

𝑖=1

𝑗 ∑︁

𝑖𝑡𝑖 = 𝑡1 +

𝑖=2

𝑗 ∑︁

|𝐸 (𝐻𝑖 )| − |𝐸 (𝐻𝑖−1 )|




𝑖=2

= |𝐸 (𝐻 𝑗 )| ∑︁ = 1𝛿 deg 𝐻 𝑗 (𝑣) 𝑣 ∈𝑉 (𝐻 𝑗 )

≤

𝑗𝑛 𝛿 .

The hypergraph 𝐻 𝛿−1 is obtained from 𝐻 𝛿 by deleting at least 𝛿𝑡 𝛿 edges and so, |𝐸 (𝐻 𝛿 )| ≥ |𝐸 (𝐻 𝛿−1 )| + 𝛿𝑡 𝛿 .

(6.16)

Recall that since 𝐻 𝛿 = 𝐻, the hypergraph 𝐻 𝛿 has exactly 𝑛 edges. Thus, |𝐸 (𝐻 𝛿 )| = 𝑛, implying by Inequality (6.16) that 𝛿 ∑︁ 𝑖=1

𝑖𝑡𝑖 = 𝛿𝑡 𝛿 +

𝛿−1 ∑︁

𝑖𝑡𝑖 = 𝛿𝑡 𝛿 + |𝐸 (𝐻 𝛿−1 )|

𝑖=1


≤ |𝐸 (𝐻 𝛿 )| − |𝐸 (𝐻 𝛿−1 )| + |𝐸 (𝐻 𝛿−1 )| = |𝐸 (𝐻 𝛿 )| = 𝑛. Thus, for each value of 𝑗 ∈ [𝛿], the inequality 𝑗 ∑︁ 𝑖=1

𝑖𝑡𝑖 ≤

𝑗𝑛 𝛿

Chapter 6. Upper Bounds in Terms of Minimum Degree

190

holds. Therefore, for each value of 𝑗 ∈ [𝛿] 0 , 𝑗 ∑︁

𝑖𝑡𝑖 =

𝑗𝑛 − 𝜀𝑗 𝛿

(6.17)

𝑖=1

for some real number 𝜀 𝑗 ≥ 0, where 𝜀0 = 0. By Equation (6.17), for each value of 𝑗 ∈ [𝛿], 𝑗 −1 ∑︁

𝑗 ∑︁



   𝑗𝑛 ( 𝑗 − 1)𝑛 𝑛 𝑗𝑡 𝑗 = 𝑖𝑡𝑖 − 𝑖𝑡𝑖 = − 𝜀𝑗 − − 𝜀 𝑗 −1 = − (𝜀 𝑗 − 𝜀 𝑗 −1 ), 𝛿 𝛿 𝛿 𝑖=1 𝑖=1 or equivalently, 𝑡𝑗 =

𝜀 𝑗 − 𝜀 𝑗 −1 𝑛 − . 𝑗𝛿 𝑗

(6.18)

Thus, by Equation (6.18), |𝑇 | =

𝛿 ∑︁

𝑡𝑗 =

𝑗=1

 𝛿  ∑︁ 𝜀 𝑗 − 𝜀 𝑗 −1 𝑛 − 𝑗𝛿 𝑗 𝑗=1

=

   𝛿  𝛿−1 ∑︁ 𝜀 𝛿 ∑︁ 𝑛 1 1 − − 𝜀𝑗 − 𝑗𝛿 𝛿 𝑗 𝑗 +1 𝑗=1 𝑗=1

≤

 𝛿  ∑︁ 𝑛 𝑗𝛿 𝑗=1

=

𝑛 ∑︁ 1 𝛿 𝑗=1 𝑗

≤


𝑛 1 + ln(𝛿) . 𝛿

𝛿

Therefore, by our earlier observations,  𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤ |𝑇 | ≤

 1 + ln(𝛿) 𝑛. 𝛿

This establishes the desired upper bound. We next give a brief discussion of the complexity of the greedy algorithm. With correct implementation, the greedy algorithm to build the set 𝑇 can be made to run in time O (𝑛 + 𝛿 𝑛). It would require us to keep a data structure, such as a hash table, that enables one to find a vertex of maximum degree in constant time. Once such a vertex is deleted from the hypergraph and added to the set 𝑇, the degrees of the vertices in the resulting hypergraph need to be updated. Since the sum of the degrees of vertices in 𝐻 is at most ∑︁ deg 𝐻 (𝑣) = 𝛿 · 𝑚(𝐻) = 𝛿 · 𝑛, 𝑣 ∈𝑉 (𝐻 )

Section 6.4. Bounds on the Independent Domination Number

191

we need to update the degrees at most 𝛿 𝑛 times. The overall complexity of the greedy algorithm can be shown to be O (𝑛 + 𝛿 · 𝑛). It can be deduced from a probabilistic proof due to Alon [17] that the bound in Theorem 6.77 is asymptotically (that is, when 𝛿 → ∞) optimal. We remark that this can also be deduced from the following 2007 result due to Thomassé and Yeo [709]. Theorem 6.78 ([709]) For any 𝜀 > 0 and for sufficiently large 𝛿, there exists a 𝛿-uniform hypergraph 𝐻 of order 𝑛 and size 𝑛 edges satisfying   (1 − 𝜀) ln(𝛿) 𝜏(𝐻) > 𝑛. 𝛿 Thomassé and Yeo [709] also provided the following result showing that the bound on the TD-set produced by the greedy algorithm in the proof of Theorem 6.77 is close to optimal. Theorem 6.79 ([709]) For every integer 𝛿 ≥ 1, there exists a bipartite 𝛿-regular graph 𝐺 of order 𝑛 satisfying   0.1 ln(𝛿) 𝛾t (𝐺) > 𝑛. 𝛿 Although the bound on the total domination number constructed by the heuristic (greedy) algorithm presented in the proof of Theorem 6.77 is asymptotically optimal, the bound is far from optimal when the minimum degree 𝛿 is small. However, for 𝛿 ≥ 8, the upper bound in Theorem 6.77 is better than the upper bound given in Theorem 6.60. In Table 6.12 we compare these two bounds for small 𝛿 ∈ {5, 6, 7, 8, 9}. 𝛿 1 + ln(𝛿) 𝛿   1 + 2𝛿  3𝛿 

5

6

7

8

9

≈ 0.5218

≈ 0.4652

≈ 0.4208

≈ 0.3849

≈ 0.3552

≈ 0.4285

≈ 0.4444

0.4

≈ 0.4166

≈ 0.3846

2

Table 6.12 Comparing the bounds of Theorems 6.60 and 6.77 for 𝛿 ∈ {5, 6, 7, 8, 9} Thus, for example if 𝐺 is a graph of order 𝑛 and 𝛿(𝐺) ≥ 8, then 𝛾t (𝐺) ≤ 0.4166 𝑛 by Theorem 6.60 and 𝛾t (𝐺) ≤ 0.3849 𝑛 by Theorem 6.77.

6.4

Bounds on the Independent Domination Number

In this section, we present a breakthrough result by Sun and Wang that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) = 𝛿 where 𝛿 ≥ 1 is an arbitrary integer, then

192

Chapter 6. Upper Bounds in Terms of Minimum Degree

√ the independent domination number 𝑖(𝐺) is bounded above by 𝑛 + 2𝛿 − 2 𝛿𝑛, thereby proving a conjecture posed by Favaron. We also present a classical result of Rosenfeld that the independence number 𝛼(𝐺) of a regular graph 𝐺 is at most one-half its order. We show that equality in Rosenfeld’s bound for the independent domination number 𝑖(𝐺) is only obtainable for graphs with every component a balanced complete bipartite graph.

6.4.1

Minimum Degree One

As in previous sections, we repeat the upper bounds from Chapter 4 for the case of minimum degree one. Observation 6.80 ([398]) If 𝐺 is a graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺). Theorem 6.81 ([84]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 𝛾(𝐺) −

𝑛 . 𝛾(𝐺)

Theorem 6.82 ([274]) If 𝐺 is an isolate-free graph of order 𝑛, then √ 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛.

6.4.2

Arbitrary Minimum Degree

In this section, we present a tight upper bound on the independent domination number of a graph with arbitrary minimum degree. In 1991 Haviland [398] improved the bound of Favaron in Theorem 6.82 when 𝛿 ≥ 14 𝑛. In order to state this result, we first present the following two lemmas of Haviland [398]. Recall that if 𝑣 is a vertex of 𝐺 and 𝑋 ⊆ 𝑉, then the degree of 𝑣 in 𝑋, denoted deg𝑋 (𝑣), is the number of neighbors of 𝑣 in 𝐺 that belong to the set 𝑋. Lemma 6.83 ([398]) If there exists an 𝑖-set 𝐼 of a graph 𝐺 of order 𝑛 with√𝛿(𝐺) = 𝛿 such that no vertex outside 𝐼 dominates all vertices in 𝐼, then 𝑖(𝐺) ≤ 𝑛 − 𝛿𝑛. Proof Let 𝐼 be an 𝑖-set of 𝐺 such that no vertex in 𝐼 = 𝑉 \ 𝐼 dominates all vertices in 𝐼. Among all vertices in 𝐼, let 𝑥 be chosen to have the maximum number of neighbors in 𝐼, that is, deg𝐼 (𝑥) is a maximum. Let 𝐼 𝑥 = 𝐼 ∩ N𝐺 (𝑥). Let 𝑋 = {𝑣 ∈ 𝐼 : N𝐺 (𝑣) ⊆ 𝐼 𝑥 } and let 𝑊 be a maximal independent set in 𝐺 [𝑋] containing the vertex 𝑥. Since the set 𝑊 ∪ (𝐼 \ 𝐼 𝑥 ) is a maximal independent set, |𝑊 | + |𝐼 | − |𝐼 𝑥 | ≥ 𝑖(𝐺) = |𝐼 |, implying that |𝑊 | ≥ |𝐼 𝑥 |. The number of edges between 𝐼 and 𝐼 is at least 𝛿|𝐼 |, noting that every vertex in 𝐼 has at least 𝛿 neighbors in 𝐼. By the Pigeonhole Principle, there is a vertex in 𝐼 with at least 𝛿|𝐼 |/|𝐼 | = 𝛿|𝐼 |/(𝑛 − |𝐼 |) neighbors in 𝑋. Hence, by our choice of the vertex 𝑥, 𝛿 · |𝐼 | |𝑋 | ≥ |𝑊 | ≥ |𝐼 𝑥 | ≥ . (6.19) 𝑛 − |𝐼 |

Section 6.4. Bounds on the Independent Domination Number

193

By supposition, |𝐼 \ 𝐼 𝑥 | ≥ 1. By definition of the set 𝑋, each vertex in 𝐼 \ 𝐼 𝑥 has all its neighbors in 𝐼 \ 𝑋, implying that |𝐼 \ 𝑋 | ≥ 𝛿.

(6.20)

By Inequalities (6.19) and (6.20), 𝑛 − |𝐼 | = |𝑋 | + |𝐼 \ 𝑋 | ≥

𝛿 · |𝐼 | + 𝛿, 𝑛 − |𝐼 |

or equivalently, |𝐼 | 2 − 2𝑛|𝐼 | − 𝑛(𝑛 − 𝛿) ≥ 0.

(6.21)

√ Solving the quadratic expression in Inequality (6.21) yields 𝑖(𝐺) = |𝐼 | ≤ 𝑛− 𝛿𝑛. Lemma 6.84 ([398]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 14 𝑛, then 𝑖(𝐺) ≤ 12 𝑛. Proof Let 𝐼 be an 𝑖-set of 𝐺. If Δ(𝐺) ≥ 𝑖(𝐺), then by Observation 6.80, 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺) ≤ 𝑛 − 𝑖(𝐺) and so, 𝑖(𝐺) ≤ 12 𝑛. Hence, we may assume that Δ(𝐺) < 𝑖(𝐺), implying that no vertex outside 𝐼 dominates all vertices in 𝐼. Therefore, we can apply Lemma 6.83 to yield √︃ √ 𝑖(𝐺) ≤ 𝑛 − 𝛿𝑛 ≤ 𝑛 − 14 𝑛2 = 12 𝑛. We now present Haviland’s improvement of the bound due to Favaron in Theorem 6.82 when 𝛿 is large relative to the order. Theorem 6.85 ([398]) If 𝐺 is a graph of order 𝑛,
then the following hold: (a) If 14 𝑛 ≤ 𝛿(𝐺) ≤ 25 𝑛, then 𝑖(𝐺) ≤ 23 𝑛 − 𝛿(𝐺) . (b) If 25 𝑛 ≤ 𝛿(𝐺) ≤ 12 𝑛, then 𝑖(𝐺) ≤ 𝛿(𝐺). Proof degree 𝛿(𝐺) = 𝛿. We note that √ Let 𝐺 be a graph of order 𝑛 with minimum √ 𝑛 − 𝛿𝑛 ≤ 23 (𝑛 − 𝛿) if and only if 𝑛 + 2𝛿 ≤ 3 𝛿𝑛 if and only if 4𝛿2 − 5𝛿𝑛 + 𝑛2 ≤ 0 if and only if 14 𝑛 ≤ 𝛿 ≤ 𝑛. Further, we note that 23 (𝑛 − 𝛿) ≤ 𝛿 if and only if 25 𝑛 ≤ 𝛿 ≤ 𝑛. √  Thus, if 𝑖(𝐺) ≤ max 𝛿, 𝑛 − 𝛿𝑛 , then our desired bounds in parts (a) and (b) of the √  theorem hold. Hence, we may assume that 𝑖(𝐺) > max 𝛿, 𝑛 − 𝛿𝑛 . Let 𝐼 be an 𝑖-set √ of 𝐺, and let 𝐼 = 𝑉 \ 𝐼. By Lemma 6.83 and by our assumption that 𝑖(𝐺) > 𝑛 − 𝛿𝑛, there exists a vertex in 𝐼 that dominates all vertices in 𝐼. Let 𝑆 be the set of all such vertices and so,  𝑆 = 𝑣 ∈ 𝐼 : 𝐼 ⊆ N𝐺 (𝑣) . Suppose that 𝑆 = 𝐼, implying that all edges between 𝐼 and 𝐼 are present. Thus, every vertex in 𝐼 has degree at least |𝐼 | = 𝑖(𝐺) > 𝛿, and every vertex in 𝐼 has degree exactly |𝐼 | = 𝑛 − |𝐼 |. Since 𝐺 has minimum degree 𝛿, this implies that 𝑖(𝐺) > 𝛿 = |𝐼 | = 𝑛 − 𝑖(𝐺) and so, 𝑖(𝐺) > 12 𝑛. However, every maximal independent set in 𝐺 [𝑆] is an ID-set of 𝐺 and so, 𝑖(𝐺) ≤ |𝑆| = 𝑛 − 𝑖(𝐺). Therefore, 𝑖(𝐺) ≤ 12 𝑛, a contradiction. Hence, 𝑆 is a proper subset of 𝐼. Let  𝑇 = 𝑣 ∈ 𝐼 \ 𝑆 : 𝑆 ⊆ N𝐺 (𝑣)

Chapter 6. Upper Bounds in Terms of Minimum Degree

194

and so, 𝑇 ⊆ 𝐼 \ 𝑆. Suppose that 𝑇 = 𝐼 \ 𝑆. In this case, every maximal independent set in 𝐺 [𝑆] is an ID-set of 𝐺 and so, |𝑆| ≥ 𝑖(𝐺) > 𝛿, implying that every vertex in 𝐼 ∪ 𝑇 has degree greater than 𝛿 in 𝐺. Further, every vertex in 𝑆 has degree at least |𝐼 | = 𝑖(𝐺) > 𝛿. Hence, all vertices of 𝐺 have degree greater than 𝛿, a contradiction. Thus, 𝑇 is a proper subset of 𝐼 \ 𝑆. Suppose that there exists a vertex 𝑣 ∈ 𝐼 \ (𝑆 ∪ 𝑇) that has more than 𝑛 − 2|𝐼 | neighbors in 𝐼, that is, deg𝐼 (𝑣) = |N𝐺 (𝑣) ∩ 𝐼 | > 𝑛 − 2|𝐼 | = 𝑛 − 2 𝑖(𝐺). By definition, there is a vertex 𝑢 ∈ 𝑆 that is not adjacent to the vertex 𝑣. Let 𝐼𝑢,𝑣 be a maximal independent set in 𝐺 that contains the nonadjacent pair {𝑢, 𝑣}. Since the vertex 𝑢 dominates the set 𝐼, we note that the ID-set 𝐼𝑢,𝑣 contains no vertex from the set 𝐼. Further, the ID-set 𝐼𝑢,𝑣 contains no neighbor of 𝑣, implying that 𝑖(𝐺) ≤ |𝐼𝑢,𝑣 | ≤
𝑛 − |𝐼 | − deg𝐼 (𝑣) < 𝑛 − 𝑖(𝐺) − 𝑛 − 2 𝑖(𝐺) = 𝑖(𝐺), a contradiction. Hence, for every vertex 𝑣 ∈ 𝐼 \ (𝑆 ∪ 𝑇), deg𝐼 (𝑣) ≤ 𝑛 − 2|𝐼 |. (6.22) Let 𝑥 ∈ 𝐼 \ (𝑆 ∪ 𝑇). Further, let 𝐼 𝑥 = 𝐼 ∩ N𝐺 (𝑥). By Inequality (6.22), we have 𝛿 ≤ deg𝐺 (𝑥) = deg𝐼 (𝑥) + deg𝐼 (𝑥) ≤ |𝐼 𝑥 | + 𝑛 − 2|𝐼 | and so, |𝐼 𝑥 | ≥ 𝛿 − 𝑛 + 2|𝐼 |.

(6.23)

Let 𝑋 = {𝑣 ∈ 𝐼 : N𝐺 (𝑣) ∩ 𝐼 ⊆ 𝐼 𝑥 } and let 𝑅 be a maximal independent set of 𝐺 [𝑋] that contains the vertex 𝑣. The set 𝑅 ∪ (𝐼 \ 𝐼 𝑥 ) is a maximal independent set of 𝐺, implying that |𝑅| + |𝐼 | − |𝐼 𝑥 | ≥ 𝑖(𝐺) = |𝐼 | and so, |𝑋 | ≥ |𝑅| ≥ |𝐼 𝑥 |. Thus, by Inequality (6.23), |𝑋 | ≥ 𝛿 − 𝑛 + 2|𝐼 |. (6.24) Since 𝑥 ∉ 𝑆, we know that 𝐼 \ 𝐼 𝑥 ≠ ∅. Further, N𝐺 (𝐼 \ 𝐼 𝑥 ) ⊆ 𝐼 \ 𝑋 and each vertex in 𝐼 \ 𝐼 𝑥 has at least 𝛿 neighbors in 𝐼 \ 𝑋. Thus, 𝑛 − |𝐼 | − |𝑋 | = |𝐼 | − |𝑋 | = |𝐼 \ 𝑋 | ≥ 𝛿.

(6.25)

By Inequalities (6.24) and (6.25), 𝑛 − |𝐼 | ≥ |𝑋 | + 𝛿 ≥ 2𝛿 − 𝑛 + 2|𝐼 | and so, 𝑖(𝐺) = |𝐼 | ≤ 23 (𝑛 − 𝛿). This proves (a). For 𝛿 ≥ 25 𝑛, we note that 23 (𝑛 − 𝛿) ≤ 𝛿, which completes the proof of part (b) and completes the proof of Theorem 6.85. We remark that if 25 𝑛 ≤ 𝛿 ≤ 12 𝑛, then the complete bipartite graph 𝐾 𝛿,𝑛− 𝛿 satisfies 𝑖(𝐺) = 𝛿 and so, the bound in Theorem 6.85(b) is best possible. However, the bound in Theorem 6.85(a) can be improved. Given a connected graph 𝐺 with arbitrary minimum degree 𝛿 ≥ 4, a tight upper bound (that holds for connected graphs of arbitrarily large order) on 𝛾(𝐺) has yet to be determined, even for the special case when 𝛿 = 4. However, this is not the case

Section 6.4. Bounds on the Independent Domination Number

195

for the independent domination number. In 1988 Favaron [274] conjectured such an upper bound on 𝑖(𝐺) as a function of 𝑛 and 𝛿. Conjecture 6.86 ([274]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿, then √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. If 𝛿 = 0, then Conjecture 6.86 simplifies to 𝑖(𝐺) ≤ 𝑛, which is obviously true. If 𝛿 = 1, then Conjecture 6.86 is precisely the statement of Theorem 6.82, which was proven by Favaron. Hence, at the time Favaron posed Conjecture 6.86, it was only of interest for 𝛿 ≥ 2. Favaron [274] showed that for every positive integer 𝛿, the bound in Conjecture 6.86 is attained for infinitely many graphs as follows. For 𝛿 ≥ 1 and ℓ ≥ 2, let 𝐵𝑖 be the complete bipartite graph 𝐾 𝛿, 𝛿 (ℓ −1) with partite sets 𝑋𝑖 and 𝑌𝑖 where |𝑋𝑖 | = 𝛿 and |𝑌𝑖 | = 𝛿(ℓ − 1) for 𝑖 ∈ [ℓ]. Let 𝐺 𝛿,ℓ be the graph obtained from the disjoint union of the graphs 𝐵1 , 𝐵2 , . . . , 𝐵ℓ by adding all edges between the sets 𝑋𝑖 and 𝑋 𝑗 for all 𝑖 and 𝑗, where 𝑖, 𝑗 ∈ [ℓ] and 𝑖 ≠ 𝑗. The resulting graph 𝐺 = 𝐺 𝛿,ℓ has order 𝑛 = 𝛿ℓ 2 and √ 𝑖(𝐺) = 𝛿 + 𝛿(ℓ − 1) 2 = 𝛿ℓ 2 + 2𝛿 − 2𝛿ℓ = 𝑛 + 2𝛿 − 2 𝛿𝑛. As mention earlier, Favaron’s conjecture is true for 𝛿 = 0 and 𝛿 = 1. The conjecture was subsequently proven for 𝛿 = 2 in 1998 by Glebov and Kostochka [338]. The big breakthrough came in 1999 when Sun and Wang [699] proved the conjecture is true for all 𝛿. Before we present their proof, we shall need the following lemma. Consider a graph 𝐺 with minimum degree 𝛿. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. Let 𝑌 = 𝑉 \ 𝑋 and so, |𝑌 | = 𝑛 − 𝑥. Each vertex of 𝑋 has at least 𝛿 neighbors in 𝑌 and so, there are least 𝛿𝑥 edges between 𝑋 and 𝑌 in 𝐺. By the Pigeonhole Principle, there is a vertex in 𝑌 with at least 𝛿𝑥/(𝑛 − 𝑥) neighbors in 𝑋. For subsets 𝑋 and 𝑌 of vertices of a graph 𝐺, we denote the set of edges that join a vertex of 𝑋 and a vertex of 𝑌 in 𝐺 by 𝐺 [𝑋, 𝑌 ], or simply by [𝑋, 𝑌 ] if 𝐺 is clear from context. Thus, [𝑋, 𝑌 ] is the set of edges between 𝑋 and 𝑌 in 𝐺. Lemma 6.87 ([699]) Let 𝐺 be a graph with 𝛿(𝐺) = 𝛿. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. Let 𝑌 = 𝑉 \ 𝑋 and so, |𝑌 | = 𝑛 − 𝑥. Let 𝐵 be the bipartite graph with partite sets 𝑋 and 𝑌 , and whose edge set is 𝐺 [𝑋, 𝑌 ]. Let 𝑦 ′ be a vertex in 𝑌 such that deg 𝐵 (𝑦 ′ ) ≥ 𝛿𝑥/(𝑛 − 𝑥), and let 𝑌 ′ = 𝑦 ∈ 𝑌 : N 𝐵 (𝑦) ⊆ N 𝐵 (𝑦 ′ ) . If 𝑍 is a maximal independent set in 𝐺 [𝑌 ′ ] containing the vertex 𝑦 ′ , then the following hold: √ 𝛿𝑥 (a) If |𝑍 | − 𝛿 ≤ deg 𝐵 (𝑦 ′ ) − 𝑛−𝑥 , then 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. 𝛿𝑥 ′ (b) If |𝑍 | − 𝛿 > deg 𝐵 (𝑦 ′ ) − 𝑛− 𝑥 , then for every proper subset 𝑍 of 𝑍, we have |𝑍 \ 𝑍 ′ | − 𝛿 > deg 𝐵 (𝑦 ′ ) − |N 𝐵 (𝑍 ′ )| − Proof (a) Assume that |𝑍 | − 𝛿 ≤ deg 𝐵 (𝑦 ′ ) − maximal independent set in 𝐺 and so,

𝛿𝑥 𝑛− 𝑥 .

𝛿𝑥 . 𝑛−𝑥


The set 𝑍 ∪ 𝑋 \ N 𝐵 (𝑦 ′ ) is a

Chapter 6. Upper Bounds in Terms of Minimum Degree

196


𝑖(𝐺) ≤ 𝑍 ∪ 𝑋 \ N 𝐵 (𝑦 ′ ) = |𝑍 | + |𝑋 | − |N 𝐵 (𝑦 ′ )| = 𝑥 + |𝑍 | − deg 𝐵 (𝑦 ′ ) 𝛿𝑥 ≤ 𝑥+𝛿− . 𝑛−𝑥 𝛿𝑥 By elementary calculus, the function 𝑓 (𝑥) = 𝑥 − 𝑛− 𝑥 is maximized when √ √
√ √ 𝑥 = 𝑛 − 𝛿𝑛. Further, 𝑓 𝑛 − 𝛿𝑛 = 𝑛 + 2𝛿 − 2 𝛿𝑛. Hence, 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. 𝛿𝑥 ′ (b) Assume that |𝑍
| − 𝛿 > deg 𝐵 (𝑦 ′ ) − 𝑛− 𝑥 and let 𝑍 be a proper subset of 𝑍. The ′ ′ set 𝑍 ∪ 𝑋 \ N 𝐵 (𝑍 ) is a maximal independent set in 𝐺 and so, by the maximality of the set 𝑋,
|𝑍 ′ | + |𝑋 | − |N 𝐵 (𝑍 ′ )| = 𝑍 ′ ∪ 𝑋 \ N 𝐵 (𝑍 ′ ) ≤ |𝑋 |, implying that |𝑍 ′ | ≤ |N 𝐵 (𝑍 ′ )|.

Thus, by assumption,
𝛿𝑥 |𝑍 \ 𝑍 ′ | − 𝛿 = |𝑍 | − 𝛿 − |𝑍 ′ | > deg 𝐵 (𝑦 ′ ) − |N 𝐵 (𝑍 ′ )| − . 𝑛−𝑥 We now present a proof of Conjecture 6.86 due to Sun and Wang [699]. Theorem 6.88 ([699]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿, then √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. Proof If 𝛿 = 0, then the bound simplifies to 𝑖(𝐺) √≤ 𝑛, which is obviously true. Suppose, to the contrary, that 𝑖(𝐺) > 𝑛 + 2𝛿 − 2 𝛿𝑛. Hence, we may assume that 𝛿 ≥ 1.√ An elementary calculus argument shows that the function
𝑓 (𝛿) = 𝑛 + 2𝛿 − 2 𝛿𝑛 is minimized when 𝛿 = 14 𝑛. Further, we note that 𝑓 14 𝑛 = 12 𝑛. √ Therefore, 𝑛 + 2𝛿 − 2 𝛿𝑛 ≥ 12 𝑛, with equality if and only if 𝛿 = 14 𝑛. If 𝛿 = 14 𝑛, then √ by Lemma 6.84, 𝑖(𝐺) ≤ 12 𝑛 = 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝛿 ≠ 14 𝑛, which √ implies by our earlier observations that 𝑛 + 2𝛿 − 2 𝛿𝑛 > 12 𝑛. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. If 𝑥 ≤ 12 𝑛, then √ 𝑖(𝐺) ≤ 𝛼(𝐺) = 𝑥 ≤ 12 𝑛 < 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝑥 > 12 𝑛. If 𝑌 = 𝑉 \ 𝑋, then |𝑌 | = 𝑛 − 𝑥. Let 𝐺 1 be the bipartite graph with partite sets ( 𝐴1 , 𝐵1 ), where 𝐴1 = 𝑋 and 𝐵1 = 𝑌 , and whose edge set consist of all edges of 𝐺 between 𝑋 and 𝑌 . The number of edges between 𝐴1 and 𝐵1 is at least 𝛿|𝑋 | = 𝛿𝑥 and so, by the Pigeonhole Principle, there is a vertex in 𝑦 1 ∈ 𝐵1 with at least 𝛿𝑥/|𝐵1 | = 𝛿𝑥/(𝑛 − 𝑥) neighbors in 𝑋. Thus, deg𝐺1 (𝑦 1 ) ≥ 𝛿𝑥/(𝑛 − 𝑥). Let  𝑌1 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 1 ) , and let 𝑍1 be a maximal independent set in 𝐺 [𝑌1 ] containing the vertex 𝑦 1 . We 𝛿𝑥 note that 𝑍1 ⊆ 𝑌1 . If |𝑍1 | − 𝛿 ≤ deg𝐺1 (𝑦 1 ) − 𝑛− 𝑥 , then by Lemma 6.87(a), we have √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence,

Section 6.4. Bounds on the Independent Domination Number |𝑍1 | − 𝛿 > deg𝐺1 (𝑦 1 ) −

197

𝛿𝑥 . 𝑛−𝑥

(6.26)

Inequality (6.26) implies that |𝑍1 | − 𝛿 > 0. Let 𝐺 2 be the bipartite subgraph of 𝐺 1 induced by the set 𝐴2 ∪ 𝐵2 , where 𝐴2 = 𝑋 \ N𝐺1 (𝑦 1 ) and 𝐵2 = 𝑌 \ 𝑌1 . Thus, 𝐺 2 has partite sets 𝐴2 and 𝐵2 . We note that no vertex of 𝑌1 has a neighbor in 𝐺 that belongs to the set 𝐴2 . Thus, every neighbor in 𝐺 of a vertex in 𝐴2 belongs to the set 𝐵2 . Hence, for every vertex 𝑣 ∈ 𝐴2 , we have deg𝐺2 (𝑣) = deg𝐺 (𝑣) ≥ 𝛿. Claim 6.88.1 There exists a vertex 𝑦 2 ∈ 𝐵2 such that deg𝐺2 (𝑦 2 ) ≥

𝛿𝑥 𝑛− 𝑥 .

𝛿𝑥 Proof Suppose that deg𝐺2 (𝑦) < 𝑛− 𝑥 for all 𝑦 ∈ 𝐴2 . Counting the edges between 𝐴2 and 𝐵2 in 𝐺 2 and noting that |𝐵2 | = |𝑌 | − |𝑌1 | ≤ |𝑌 | − |𝑍1 | = 𝑛 − 𝑥 − |𝑍1 |, we have   ∑︁
𝛿𝑥 𝑛 − 𝑥 − |𝑍1 | > deg𝐺2 (𝑦) 𝑛−𝑥 𝑦 ∈ 𝐵2 ∑︁ = deg𝐺2 (𝑥) 𝑥 ∈ 𝐴2

≥ 𝛿| 𝐴2 |
= 𝛿 𝑥 − deg𝐺1 (𝑦 1 ) , or equivalently,

𝑥 𝑛 − 𝑥 − |𝑍1 | > 𝑥 − deg𝐺1 (𝑦 1 ) (𝑛 − 𝑥), which simplifies to 𝑥|𝑍1 | < (𝑛 − 𝑥) deg𝐺1 (𝑦 1 ).

(6.27)

By Inequalities (6.26) and (6.27), 

 𝛿𝑥 𝑥|𝑍1 | < (𝑛 − 𝑥) |𝑍1 | − 𝛿 + , 𝑛−𝑥 or equivalently,

𝑥 |𝑍1 | − 𝛿 < (𝑛 − 𝑥) |𝑍1 | − 𝛿 .

(6.28)

Since |𝑍1 | − 𝛿 > 0, Inequality (6.28) simplifies to 𝑥 < 𝑛 − 𝑥 and so, 𝑥 < contradicting our earlier assumption.

1 2 𝑛,

By Claim 6.88.1, we may assume there exists a vertex 𝑦 2 ∈ 𝐵2 such that deg𝐺2 (𝑦 2 ) ≥

𝛿𝑥 . 𝑛−𝑥

We now define the sets  𝑌 2 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 2 ) , and  𝑌2 = 𝑦 ∈ 𝐵2 : N𝐺2 (𝑦) ⊆ N𝐺2 (𝑦 2 ) .

(6.29)

Chapter 6. Upper Bounds in Terms of Minimum Degree

198

We note that if 𝑦 ∈ 𝑌 2 , then either 𝑦 ∈ 𝑌1 or 𝑦 ∈ 𝑌2 and so, 𝑌 2 ⊆ 𝑌1 ∪ 𝑌2 . Let 𝑍 2 be a maximal independent set in 𝐺 [𝑌 2 ] containing the vertex 𝑦 2 , and let 𝑍2 be a maximal independent set in 𝐺 [𝑌2 ] containing the set 𝑍 2 ∩ 𝑌2 . We note that 𝑍 2 ⊆ 𝑌1 ∪ 𝑍2 and that the sets 𝑍1 and 𝑍2 are disjoint. By definition, 𝑦 2 ∈ 𝑍√2 . If 𝛿𝑥 |𝑍 2 | − 𝛿 ≤ deg𝐺1 (𝑦 2 ) − 𝑛− 𝑥 , then by Lemma 6.87(a), we have 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝛿𝑥 |𝑍 2 | − 𝛿 > deg𝐺1 (𝑦 2 ) − . (6.30) 𝑛−𝑥 Claim 6.88.2

|𝑍2 | − 𝛿 > deg𝐺2 (𝑦 2 ) −

𝛿𝑥 𝑛− 𝑥 .

Proof Since 𝐺 2 is an induced subgraph of 𝐺 1 , we note that deg𝐺1 (𝑦 2 ) ≥ deg𝐺2 (𝑦 2 ) and so, by Inequality (6.29), 𝛿𝑥 deg𝐺1 (𝑦 2 ) ≥ . 𝑛−𝑥 Let 𝑍 ′ = 𝑍 2 \ 𝑍2 . We note that 𝑍 2 \ 𝑍 ′ ⊆ 𝑍2 and so, |𝑍 2 \ 𝑍 ′ | ≤ |𝑍2 |. Since 𝑦 2 ∈ 𝑍 2 ∩ 𝑍2 , the set 𝑍 ′ is a proper subset of 𝑍 2 . Hence, by Inequality (6.30) and Lemma 6.87(b), 𝛿𝑥 |𝑍2 | − 𝛿 ≥ |𝑍 2 \ 𝑍 ′ | − 𝛿 > deg𝐺1 (𝑦 2 ) − |N𝐺1 (𝑍 ′ )| − . (6.31) 𝑛−𝑥 Since 𝑍 ′ ⊆ 𝑌1 , we note that N𝐺1 (𝑍 ′ ) ⊆ N𝐺1 (𝑦 2 ) \ N𝐺2 (𝑦 2 ) and so, |N𝐺1 (𝑍 ′ )| ≤ deg𝐺1 (𝑦 2 ) − deg𝐺2 (𝑦 2 ), or equivalently, deg𝐺1 (𝑦 2 ) − |N𝐺1 (𝑍 ′ )| ≥ deg𝐺2 (𝑦 2 ). Therefore, by Inequality (6.31), 𝛿𝑥 |𝑍2 | − 𝛿 > deg𝐺2 (𝑦 2 ) − . 𝑛−𝑥 Claim 6.88.2 implies that |𝑍2 | > 𝛿. More generally, for 𝑗 ≥ 2, we define 𝐺 𝑗 recursively to be the bipartite subgraph of 𝐺 𝑗 −1 induced by the set 𝐴 𝑗 ∪ 𝐵 𝑗 , where 𝐴𝑗 = 𝑋 \

𝑗 −1 Ø

N𝐺𝑖 (𝑦 𝑖 )

and

𝐵𝑗 = 𝑌 \

𝑖=1

𝑗 −1 Ø

𝑌𝑖 ,

𝑖=1

and 𝑦 𝑗 ∈ 𝐵 𝑗 such that deg𝐺 𝑗 (𝑦 𝑗 ) ≥

𝛿𝑥 . 𝑛−𝑥

Moreover,  𝑌 𝑗 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 𝑗 ) ,  𝑌 𝑗 = 𝑦 ∈ 𝐵 𝑗 : N𝐺 𝑗 (𝑦) ⊆ N𝐺 𝑗 (𝑦 𝑗 ) , the set 𝑍 𝑗 is a maximal independent set in 𝐺 [𝑌 𝑗 ] containing vertex 𝑦 𝑗 , and the set 𝑍 𝑗 a maximal independent set in 𝐺 [𝑌 𝑗 ] containing the set 𝑍 𝑗 ∩ 𝑌 𝑗 such that |𝑍 𝑗 | − 𝛿 > deg𝐺 𝑗 (𝑦 𝑗 ) −

𝛿𝑥 . 𝑛−𝑥

(6.32)

Section 6.4. Bounds on the Independent Domination Number

199

We note that 𝑍𝑗 ⊆

Ø 𝑗 −1  𝑌𝑖 ∪ 𝑍 𝑗 , 𝑖=1

and that the sets 𝑍1 , 𝑍2 , . . . , 𝑍 𝑗 are pairwise disjoint and |𝑍𝑖 | > 𝛿 for all 𝑖 ∈ [ 𝑗]. Therefore, there exists an integer 𝑡 such that 𝑋=

𝑡 Ø

N𝐺𝑖 (𝑦 𝑖 ).

𝑖=1

We note that 𝑍1 ∪ 𝑍2 ∪ · · · ∪ 𝑍𝑡 ⊆ 𝑌 . Thus, since the sets 𝑍1 , 𝑍2 , . . . , 𝑍𝑡 are pairwise disjoint and |𝑍𝑖 | ≥ 𝛿 for all 𝑖 ∈ [𝑡], 𝑛 − 𝑥 = |𝑌 | ≥

𝑡 ∑︁

|𝑍𝑖 | ≥ 𝑡𝛿

𝑖=1

and so, 𝑡≤

𝑛−𝑥 . 𝛿

(6.33)

Since 𝑡 ∑︁

deg𝐺𝑖 (𝑦 𝑖 ) =

𝑖=1

𝑡 ∑︁

|N𝐺𝑖 (𝑦 𝑖 )| = |𝑋 | = 𝑥,

𝑖=1

by Inequality (6.32), |𝑌 | ≥

𝑡 ∑︁

|𝑍𝑖 | ≥

𝑖=1

∑︁ 𝑡

     𝛿𝑥 𝛿𝑥 deg𝐺𝑖 (𝑦 𝑖 ) + 𝑡 𝛿 − =𝑥+𝑡 𝛿− . 𝑛−𝑥 𝑛−𝑥 𝑖=1

(6.34)

By our earlier assumptions, we have 𝑛 < 2𝑥 and so, 𝛿−

𝛿𝑥 𝛿(𝑛 − 2𝑥) = < 0. 𝑛−𝑥 𝑛−𝑥

Thus, by Inequalities (6.33) and (6.34), 

   𝛿𝑥 𝑛−𝑥 𝛿𝑥 𝑛 = |𝑋 | + |𝑌 | > 2𝑥 + 𝑡 𝛿 − ≥ 2𝑥 + 𝛿− = 𝑛, 𝑛−𝑥 𝛿 𝑛−𝑥 a contradiction. As observed earlier, for every positive integer 𝛿, the bound in Theorem 6.88 is attained for infinitely many graphs. Hence, by Theorem 6.88 and the tightness of this bound, the independent domination number behaves different than the domination number.

200

Chapter 6. Upper Bounds in Terms of Minimum Degree

6.4.3 Regular Graphs As shown in the previous section, for any fixed minimum degree 𝛿 ≥ 1, there are graphs with 𝑖(𝐺) = 𝑛 − O (𝑛). However, the infinite class of graphs constructed earlier by Favaron [274] that achieve equality in the upper bound on the independent domination number given in Theorem 6.88 are far from regular, and the difference between their maximum and minimum degrees is large. If we require the graph to be regular, then the upper bound in Theorem 6.88 can be significantly improved. As first observed in 1964 by Rosenfeld [659], the independence number of a regular graph is at most one-half its order. Theorem 6.89 ([659]) For every integer 𝑟 ≥ 1, if 𝐺 is an 𝑟-regular connected graph of order 𝑛, then 𝛼(𝐺) ≤ 21 𝑛. Proof For 𝑟 ≥ 1, let 𝐺 be an 𝑟-regular connected graph of order 𝑛 and let 𝑋 be an 𝛼set of 𝐺. Let 𝑋 denote the complement of 𝑋 and so, 𝑋 = 𝑉 \ 𝑋. By double counting the
edges joining 𝑋 and its complement 𝑋, we have 𝑟 |𝑋 | = | [𝑋, 𝑋] | ≤ 𝑟 |𝑋 | = 𝑟 𝑛 − |𝑋 | and so, 𝑖(𝐺) ≤ 𝛼(𝐺) = |𝑋 | ≤ 12 𝑛. We note that equality in Theorem 6.89 is only obtainable for graphs with every component a balanced complete bipartite graph, as observed in [358]. Theorem 6.90 ([358]) For every integer 𝑟 ≥ 1, if 𝐺 is an 𝑟-regular connected graph of order 𝑛, then 𝑖(𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 = 𝐾𝑟 ,𝑟 . Proof For 𝑟 ≥ 1, let 𝐺 be an 𝑟-regular connected graph of order 𝑛. Let 𝑋 be an 𝛼-set of 𝐺 and let 𝑋 =
𝑉 \ 𝑋. As shown in the proof of Theorem 6.89, we have 𝑟 |𝑋 | ≤ 𝑟 |𝑋 | = 𝑟 𝑛 − |𝑋 | and therefore 𝑖(𝐺) ≤ |𝑋 | ≤ 12 𝑛. Suppose that 𝑖(𝐺) = 12 𝑛. Hence, we must have equality throughout these two inequality chains, implying that |𝑋 | = |𝑋 | = 12 𝑛 and that each vertex in the complement 𝑋 of 𝑋 has exactly 𝑟 neighbors in 𝑋. Let 𝑋 = 𝑌 and so, 𝐺 is an 𝑟-regular, bipartite graph with partite sets 𝑋 and 𝑌 . Suppose, to the contrary, that 𝐺 ≠ 𝐾𝑟 ,𝑟 . Thus, there exist vertices 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 that are not adjacent in 𝐺. Let 𝑋1 be the set of all vertices in 𝑋 whose neighborhood is N𝐺 (𝑥), and let 𝑌1 be the set of all vertices in 𝑌 whose neighborhood is N𝐺 (𝑦), that is,  𝑋1 = 𝑣 ∈ 𝑋 : N𝐺 (𝑣) = N𝐺 (𝑥) , and  𝑌1 = 𝑣 ∈ 𝑌 : N𝐺 (𝑣) = N𝐺 (𝑦) . We note that 𝑥 ∈ 𝑋1 , 𝑦 ∈ 𝑌1 , and the set 𝑋1 ∪ 𝑌1 is an independent set in 𝐺. Let 𝑋2 = N𝐺 (𝑦) and let 𝑌2 = N𝐺 (𝑥) and so, |𝑋2 | = |𝑌2 | = 𝑟. By the regularity of 𝐺, we have |𝑋1 | ≤ 𝑟 and |𝑌1 | ≤ 𝑟. If |𝑋1 | = 𝑟, then 𝐺 [𝑋1 ∪ 𝑌2 ] = 𝐾𝑟 ,𝑟 , implying that the graph 𝐺 is disconnected, a contradiction. Hence, |𝑋1 | ≤ 𝑟 − 1. Analogously, |𝑌1 | ≤ 𝑟 − 1. Let 𝑆 = 𝑋1 ∪ 𝑌1 and so, |𝑆| = |𝑋1 | + |𝑌1 | ≤ 2(𝑟 − 1). Suppose that 𝑆 is a dominating set of 𝐺. In this case, 𝑆 is an ID-set of 𝐺. Further, 𝑋 = 𝑋1 ∪ 𝑋2 and 𝑌 = 𝑌1 ∪ 𝑌2 and so, 𝑛 = |𝑆| + 2𝑟 ≤ 4𝑟 − 2. Thus, 𝑖(𝐺) ≤ |𝑆| = 𝑛 − 2𝑟 ≤ 12 𝑛 − 1, a contradiction. Hence, 𝑆 is not a dominating set of 𝐺.

Section 6.4. Bounds on the Independent Domination Number

201

Let 𝑋3 = 𝑋 \ (𝑋1 ∪ 𝑋2 ) and let 𝑌3 = 𝑌 \ (𝑌1 ∪ 𝑌2 ). Each vertex in 𝑋3 has no neighbor in 𝑌1 and at most 𝑟 − 1 neighbors in 𝑌2 , and therefore has at least one neighbor in 𝑌3 . Analogously, each vertex in 𝑌3 has at least one neighbor in 𝑋3 . Let 𝐺 ′ be the subgraph of 𝐺 induced by the set 𝑋3 ∪ 𝑌3 . By our earlier observations, 𝐺 ′ is a bipartite
isolate-free graph
with partite sets 𝑋3 and 𝑌3 , implying that 𝑖(𝐺 ′ ) ≤ 21 |𝑋3 | + |𝑌3 | = 12 𝑛 − |𝑆| − 2𝑟 . A minimum ID-set in 𝐺 ′ can be extended to an ID-set of 𝐺 by adding the set 𝑆 to it. Hence, 𝑖(𝐺) ≤ |𝑆| + 𝑖(𝐺 ′ ) ≤ |𝑆| + 12 (𝑛 − |𝑆| − 2𝑟) = 12 𝑛 + 12 |𝑆| − 𝑟 ≤ 12 𝑛 + (𝑟 − 1) − 𝑟 = 12 𝑛 − 1, a contradiction. Hence, 𝐺 = 𝐾𝑟 ,𝑟 . Conversely, if 𝐺 = 𝐾𝑟 ,𝑟 , then 𝑛 = 2𝑟 and 𝑖(𝐺) = 𝑟 = 12 𝑛. We now consider regular graphs of fixed regularity. For 𝑟 ≥ 2 and 𝑛 ≥ 𝑟 +1, let G𝑟𝑛 denote the family of all connected 𝑟-regular graphs of order 𝑛 different from 𝐾𝑟 ,𝑟 . 𝑛 Further, let 𝑐𝑟 denote the supremum of 𝑖 (𝐺) 𝑛 taken over all graphs 𝐺 ∈ G𝑟 , that is, 𝑖(𝐺) . 𝐺 ∈ G𝑟𝑛 𝑛

𝑐𝑟 = sup

As a consequence of Theorem 6.90, we have the following result. Corollary 6.91 For 𝑟 ≥ 2, we have 𝑐𝑟 ≤ 12 . If 𝐺 is a connected 2-regular graph of order 𝑛, then 𝐺 is a cycle 𝐶𝑛 and 𝑖(𝐺) = 13 𝑛 . Hence, if 𝐺 is different from 𝐾2,2 , that is, if 𝑛 = 3 or 𝑛 ≥ 5, then we 3 have 𝑖 (𝐺) 𝑛 ≤ 7 , with equality if and only if 𝑛 = 7. This yields the following result. Proposition 6.92 The supremum 𝑐 2 = 37 . We next consider 3-regular graphs different from 𝐾3,3 . In 1999 Lam et al. [552] presented the best current general upper bound on the independent domination number of a cubic graph. We omit their proof which uses an intricate strong induction argument. Theorem 6.93 ([552]) If 𝐺 ≠ 𝐾3,3 is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 25 𝑛. The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 are shown in Figure 6.27(a) and (b), respectively. As a consequence of Theorem 6.93 and the fact that equality in Theorem 6.93 holds for the 5-prism 𝐶5 □ 𝐾2 , we have the following result. Corollary 6.94 The supremum 𝑐 3 = 25 .

Chapter 6. Upper Bounds in Terms of Minimum Degree

202

(a) 𝐾3,3

(b) 𝐶5 □ 𝐾2

Figure 6.27 The graphs 𝐾3,3 and 𝐶5 □ 𝐾2

In 2013 Goddard and Henning [352] conjectured that the 25 -upper bound on the independent domination number of a 3-regular graph given in Theorem 6.93 can be improved if we forbid the exceptional graphs 𝐾3,3 and 𝐶5 □ 𝐾2 . Conjecture 6.95 ([352]) If 𝐺 ∉ {𝐾3,3 , 𝐶5 □ 𝐾2 } is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. 1 2 Goddard and Henning [352] constructed two infinite families Fcubic and Fcubic of connected cubic graphs achieving the three-eighths upper bound as follows. For 1 𝑘 ≥ 1, a graph in the family Fcubic is constructed by taking two copies of the cycle 𝐶4𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑐 1 𝑑1 . . . 𝑎 𝑘 𝑏 𝑘 𝑐 𝑘 𝑑 𝑘 and 𝑤 1 𝑥1 𝑦 1 𝑧 1 . . . 𝑤 𝑘 𝑥 𝑘 𝑦 𝑘 𝑧 𝑘 , and joining 𝑎 𝑖 to 𝑤 𝑖 , 𝑏 𝑖 to 𝑥𝑖 , 𝑐 𝑖 to 𝑧 𝑖 , and 𝑑𝑖 to 𝑦 𝑖 for each 𝑖 ∈ [𝑘]. For 2 ℓ ≥ 1, a graph in the family Fcubic is constructed by taking a copy of a cycle 𝐶3ℓ with vertex sequence 𝑎 1 𝑏 1 𝑐 1 . . . 𝑎 ℓ 𝑏 ℓ 𝑐 ℓ , and for each 𝑖 ∈ [ℓ], adding the vertices {𝑤 𝑖 , 𝑥𝑖 , 𝑦 𝑖 , 𝑧1𝑖 , 𝑧2𝑖 }, and joining 𝑎 𝑖 to 𝑤 𝑖 , 𝑏 𝑖 to 𝑥𝑖 , and 𝑐 𝑖 to 𝑦 𝑖 , and further for each 𝑗 1 𝑗 ∈ [2], joining 𝑧 𝑖 to each of the vertices 𝑤 𝑖 , 𝑥𝑖 , and 𝑦 𝑖 . Graphs in the families Fcubic 2 and Fcubic are illustrated in Figure 6.28(a) and (b), respectively.

(a) 𝐺

(b) 𝐻

1 2 and 𝐻 ∈ Fcubic Figure 6.28 Graphs 𝐺 ∈ Fcubic

1 2 Proposition 6.96 ([352]) If 𝐺 ∈ Fcubic ∪ Fcubic has order 𝑛, then 𝑖(𝐺) = 38 𝑛.

Section 6.4. Bounds on the Independent Domination Number

203

If Conjecture 6.95 is true, then the bound is tight as shown by Proposition 6.96. A graph operation that occurs frequently in the construction of extremal graphs is the expansion of a graph. If 𝑟 ≥ 1 an integer, the expansion exp(𝐺, 𝑟) of a graph 𝐺 is that graph obtained from 𝐺 by replacing each vertex 𝑣 of 𝐺 with an independent set 𝐼 𝑣 of cardinality 𝑟 and for every vertex 𝑣 in 𝐺, the open neighborhood of every vertex 𝑢 in 𝐼 𝑣 in the expansion of 𝐺 is given by Ø Nexp(𝐺,𝑟 ) (𝑢) = 𝐼𝑤 . 𝑤 ∈N(𝑣)

For example, for 𝑟 ≥ 1 the expansion exp(𝐶4 , 𝑟) of a 4-cycle is the complete bipartite graph 𝐾2𝑟 ,2𝑟 . We note that if 𝑥 and 𝑦 are two open twins in a graph 𝐺, that is, if N𝐺 (𝑥) = N𝐺 (𝑦), then any ID-set of 𝐺 contains either both 𝑥 and 𝑦 or neither of them. It follows that if 𝐷 is an ID-set in exp(𝐺, 𝑟), then for every vertex 𝑣 of 𝐺, the set 𝐷 either contains all of 𝐼 𝑣 or none of 𝐼 𝑣 . Furthermore, the set {𝑣 : 𝐼 𝑣 ⊆ 𝐷} is an ID-set of the graph 𝐺, implying that 𝑖(𝐺) ≤ 𝑟1 · 𝑖(exp(𝐺, 𝑟)), or equivalently, Ð 𝑖(exp(𝐺, 𝑟)) ≥ 𝑟 · 𝑖(𝐺). On the other hand, if 𝐼 is an ID-set of 𝐺, then the set 𝑣 ∈𝐷 𝐼 𝑣 is an ID-set in exp(𝐺, 𝑟), implying that 𝑖(exp(𝐺, 𝑟)) ≤ 𝑟 · 𝑖(𝐺). Consequently, 𝑖(exp(𝐺, 𝑟)) = 𝑟 · 𝑖(𝐺). We state this formally as follows. Lemma 6.97 If 𝐺 is a graph and 𝑟 ≥ 1 an integer, then 𝑖(exp(𝐺, 𝑟)) = 𝑟 · 𝑖(𝐺). To illustrate Lemma 6.97, consider the expansion exp(𝐶7 , 2) of a 7-cycle, illustrated in Figure 6.29. By Lemma 6.97, we have 𝑖(exp(𝐶7 , 2)) = 2 · 𝑖(𝐶7 ) = 2 · 3 = 6. Hence, the expansion 𝐺 = exp(𝐶7 , 2) of a 7-cycle is a connected 4-regular graph of order 𝑛 = 14 satisfying 𝑖(𝐺) = 6 = 37 𝑛, implying that the constant 𝑐 4 ≥ 37 .

Figure 6.29 The expansion exp(𝐶7 , 2) In 2013 Goddard and Henning [352] conjectured that if 𝐺 ≠ 𝐾4,4 is a connected 4-regular graph of order 𝑛, then 𝑖(𝐺) ≤ 37 𝑛. Equivalently, they conjectured that 𝑐 4 = 37 . In 2021 Cho et al. [171] announced they had settled this conjecture in the affirmative. Theorem 6.98 ([171]) The supremum 𝑐 4 = 37 . It would be interesting to determine the exact value of the constant 𝑐𝑟 for all 𝑟 ≥ 5. However, the question of best possible bounds of the independent domination number

Chapter 6. Upper Bounds in Terms of Minimum Degree

204

of connected 𝑟-regular graphs different from 𝐾𝑟 ,𝑟 for 𝑟 ≥ 5 remains unresolved, even for the special case when 𝑟 = 5. The following result was observed in [352]. Lemma 6.99 ([352]) For all positive integers 𝑟 and 𝑠, 𝑐𝑟 𝑠 ≥ 𝑐𝑟 . Proof Let 𝐺 be a connected graph in G𝑟𝑛 that gives the value for 𝑐𝑟 . The expansion 𝐻 = exp(𝐺, 𝑠) of the graph 𝐺 is a connected (𝑟 𝑠)-regular graph of order 𝑛(𝐻) = 𝑛 × 𝑠 satisfying 𝑖(𝐻) 𝑠 × 𝑖(𝐺) 𝑖(𝐺) = = , 𝑛(𝐻) 𝑛×𝑠 𝑛 implying that 𝑐𝑟 𝑠 ≥ 𝑐𝑟 . Goddard and Henning [352] posed the following question. Question 6.100 Is it true that the supremum 𝑐𝑟 tends to

1 2

as 𝑟 → ∞?

Question 6.100 was answered in the affirmative in 2020 by Blumenthal [82] in his PhD thesis. In order to construct connected graphs 𝐺 in the family G𝑟𝑛 such that 𝑖 (𝐺) 1 𝑘 𝑛 ≥ 2 − 𝜀 for every 𝜀 > 0, for 𝑟 > 𝑘 ≥ 2, let 𝐾𝑟 ,𝑟 be the graph obtained from a complete bipartite graph 𝐾𝑟 ,𝑟 by selecting an arbitrary vertex of the graph, which we call the gluing vertex, and removing 𝑘 − 1 edges incident with this vertex. The 4 with gluing vertex 𝑥 is illustrated in Figure 6.30. graph 𝐾5,5

𝑥 4 with gluing vertex 𝑥 Figure 6.30 The graph 𝐾5,5

For integers 𝑟 > 𝑘 ≥ 2 with 𝑘 even, let G𝑘,𝑟 be the family of graphs 𝐺 𝑘,𝑟 constructed as follows. Let 𝐺 𝑘,𝑟 be obtained from 𝑘 vertex-disjoint copies of 𝐾𝑟𝑘,𝑟 by adding all edges between the 𝑘 gluing vertices, so that the gluing vertices form a clique 𝐾 𝑘 . Let 𝑀 be an arbitrary perfect matching in this complete graph of (even) order 𝑘 consisting of the gluing vertices. For each gluing vertex 𝑣, let 𝐺 𝑣 be the copy of 𝐾𝑟𝑘,𝑟 that contains 𝑣, and let 𝑁 𝑣 be the set of vertices of degree 𝑟 − 1 in 𝐺 𝑣 different from 𝑣. We note that |𝑁 𝑣 | = 𝑘 − 1 and that the 𝑘 − 1 edges joining 𝑣 to vertices in 𝑁 𝑣 were deleted when constructing 𝐺 𝑣 . Further, we note that in 𝐺 𝑣 every vertex has degree 𝑟, except for the vertex 𝑣 which has degree 𝑟 − 𝑘 + 1 and the vertices in 𝑁 𝑣 which have degree 𝑟 − 1. For each edge 𝑢𝑣 ∈ 𝑀, we add a perfect matching between the vertices of 𝑁𝑢 and 𝑁 𝑣 . Let 𝐺 𝑘,𝑟 be the resulting 𝑟-regular graph of order 2𝑟 𝑘 and let G𝑘,𝑟 be the family of all such graphs 𝐺 𝑘,𝑟 . A graph in the family G4,5 is illustrated in Figure 6.31. Blumenthal [82] proved if 𝐺 ∈ G𝑘,𝑟 , then 𝑖(𝐺) ≥ 𝑘 + (𝑟 − 𝑘 − 1) (𝑘 − 1) = 𝑟 (𝑘 − 1) − 𝑘 2 + 𝑘 + 1. With a more detailed analysis, we can determine precisely the independent domination number of a graph in the family G𝑘,𝑟 .

Section 6.4. Bounds on the Independent Domination Number

205

Figure 6.31 A graph in the family G4,5

Proposition 6.101 For 𝑟 > 𝑘 ≥ 2 with 𝑘 even, if 𝐺 ∈ G𝑘,𝑟 , then 𝑖(𝐺) = 𝑟 (𝑘 − 1) − 1 2 𝑘 (𝑘 − 3). Proof Let 𝐺 ∈ G𝑘,𝑟 and let 𝐴 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be the set of 𝑘 gluing vertices used in the construction of 𝐺. Let 𝐺 𝑖 be the copy of 𝐾𝑟𝑘,𝑟 that contains the gluing vertex 𝑣 𝑖 for 𝑖 ∈ [𝑘]. Renaming vertices if necessary, we may assume that the perfect matching 𝑀 in the complete graph 𝐺 [ 𝐴] = 𝐾 𝑘 used in the construction of 𝐺 is 𝑀=

𝑘/2 Ø

{𝑣 2𝑖−1 , 𝑣 2𝑖 }.

𝑖=1

Let   𝐺 2𝑖−1,2𝑖 be the subgraph of 𝐺 induced by the set 𝑉 (𝐺 2𝑖−1 ) ∪ 𝑉 (𝐺 2𝑖 ) for 𝑖 ∈ 𝑘2 . We note that the subgraphs 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘/2 are vertex-disjoint. Further, the only edges of 𝐺 joining a vertex in one such subgraph to a vertex in another such subgraph are edges joining two link vertices. Let 𝐼 be an 𝑖-set of 𝐺. We note that 𝐺 [ 𝐴] is a complete graph 𝐾 𝑘 and so, the independent set 𝐼 contains at most one vertex from the set 𝐴, that   is, the set 𝐼 contains at most one gluing vertex. Let 𝐼𝑖 = 𝐼 ∩ 𝑉 (𝐺 2𝑖−1,2𝑖 ) for 𝑖 ∈ 𝑘2 . If (exactly) one of 𝑣 2𝑖−1 and 𝑣 2𝑖 belongs to the set 𝐼, then |𝐼𝑖 | ≥ 𝑘 + (𝑟 − 𝑘 + 1) = 𝑟 + 1. If neither 𝑣 2𝑖−1 nor 𝑣 2𝑖 belongs to the set 𝐼, then |𝐼𝑖 | ≥ 𝑟 + (𝑟 − 𝑘 + 1) = 2𝑟 − 𝑘 + 1. Since the set 𝐼 contains at most one gluing vertex, this implies that 𝑖(𝐺) = |𝐼 | =

𝑘/2 ∑︁

|𝐼𝑖 | ≥ 𝑟 + 1 + (2𝑟 − 𝑘 + 1)

1 2𝑘


− 1 = 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). (6.35)

𝑖=1

We construct next an independent dominating set 𝐷 of 𝐺 satisfying |𝐷| ≤ 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). For 𝑗 ∈ [𝑘], let 𝑋 𝑗 be the set of neighbors of 𝑣 𝑗 in 𝐺 𝑗 and let 𝑌 𝑗 be the set of vertices at distance 3 from 𝑣 𝑗 in 𝐺 𝑗 . We note that 𝑌 𝑗 is the set of vertices in 𝐺 𝑗 of degree 𝑟 − 1 that are different from 𝑣 𝑗 , and that |𝑋 𝑗 | = 𝑟 − 𝑘 + 1 and |𝑌 𝑗 | = 𝑘 − 1. Let Ø  Ø  𝑘/2 𝑘 𝐷 = {𝑣 1 } ∪ 𝑌1 ∪ 𝑋𝑖 ∪ 𝑌2𝑖 . 𝑖=2

𝑖=2

Chapter 6. Upper Bounds in Terms of Minimum Degree

206

The set 𝐷 is an independent dominating set of 𝐺 satisfying 𝑖(𝐺) ≤ |𝐷 | = 1 + |𝑌1 | +

𝑘 ∑︁ 𝑖=2

|𝑋𝑖 | +

𝑘/2 ∑︁

|𝑌2𝑖 |

𝑖=2

= 1 + (𝑘 − 1) + (𝑘 − 1) (𝑟 − 𝑘 + 1) + (𝑘 − 1)

𝑘 2

−1




= 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). Consequently, by Inequality (6.35), we have 𝑖(𝐺) = 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). To illustrate Proposition 6.101, if 𝐺 ∈ G4,5 , then 𝑖(𝐺) = 13. An example of an 𝑖-set in the graph 𝐺 ∈ G4,5 shown in Figure 6.31 is given by the set of 13 highlighted vertices. We are now in a position to state the following result due to Blumenthal [82]. However, the proof we present follows from Proposition 6.101. Theorem 6.102 ([82]) The values of 𝑐𝑟 tend to

1 2

as 𝑟 → ∞.

Proof For integers 𝑟 and 𝑘, where 𝑟 > 𝑘 ≥ 2 with 𝑘 even, let 𝐺 ∈ G𝑘,𝑟 be a graph of order 𝑛. Let 𝑓𝑟 (𝑘) be the function defined by 𝑓𝑟 (𝑘) =

1 1 𝑘 3 − − + . 2 2𝑘 4𝑟 4𝑟

By Proposition 6.101 and by the definition of the constant 𝑐𝑟 , 𝑐𝑟 ≥

1 𝑖(𝐺) 𝑟 (𝑘 − 1) − 2 𝑘 (𝑘 − 3) = 𝑓𝑟 (𝑘). = 𝑛 2𝑟 𝑘

Thus, 𝑐𝑟 ≥ 𝑓𝑟 (𝑘) for all integer values of 𝑘, where 2 ≤ 𝑘 ≤ 𝑟 − 1 and 𝑘 is even. For real optimization with 𝑟 ≥ 3 a fixed integer and 𝑘√a real number √ (where 2 ≤ 𝑘 ≤ 𝑟 − 1), the function 𝑓𝑟 (𝑘) is maximized when 𝑘 = 2𝑟. Thus, if 2𝑟 is an even integer, then this yields 𝑐 𝑟 ≥ 𝑓𝑟

√
1 3 1 2𝑟 = + −√ . 2 4𝑟 2𝑟

(6.36)

For integer optimization with 𝑟 ≥ 3 a fixed integer and 𝑘 an even integer satisfying 2 ≤ 𝑘 ≤ 𝑟 − 1, the maximum value of the √ function 𝑓𝑟 (𝑘) is max{𝑘 1 , 𝑘 2 }, where 𝑘 1 and 𝑘 2 are even integers such that 𝑘 1 ≤ 2𝑟 ≤ 𝑘 2 and 𝑘 2 = 𝑘 1 + 2. We remark that sometimes the maximum value of 𝑓𝑟 (𝑘) is attained at 𝑘 = 𝑘 1 and sometimes the maximum value of 𝑓𝑟 (𝑘) is attained at 𝑘 = 𝑘 2 . In any event, this yields a lower bound of 𝑐𝑟 of approximately 12 + 4𝑟3 − √1 , which tends to 12 as 𝑟 → ∞. By Corollary 6.91, 2𝑟 we have 𝑐𝑟 ≤ 12 . Consequently, 𝑐𝑟 tends to 12 as 𝑟 → ∞. We next consider the independent domination number in regular graphs of larger degree. Favaron [274] was the first to improve the upper bound of Theorem 6.89 for 𝛿 ≥ 12 𝑛.

Section 6.4. Bounds on the Independent Domination Number

207

Theorem 6.103 ([274]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 ≥ 21 𝑛, then 𝑖(𝐺) ≤ 𝑛 − 𝛿, with equality only for complete multipartite graphs with all partite sets of the same order. Haviland [398–400] improved the upper bound √ of
Theorem 6.89 for values of 𝛿 in the range 14 𝑛 ≤ 𝛿 ≤ 12 𝑛. We remark that 12 3 − 5 ≈ 0.3820. Theorem 6.104 ([398–400]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 ≤ then ( √
√ 𝑛 − 𝑛𝛿 if 14 𝑛 ≤ 𝛿 ≤ 12 3 − 5 𝑛 𝑖(𝐺) ≤ √
𝛿 if 12 3 − 5 𝑛 ≤ 𝛿 ≤ 12 𝑛.

1 2 𝑛,

√ We remark that in the statement of Theorem 6.104, the two values 𝑛 − 𝑛𝛿 and 𝛿 √
are the same when 𝛿 = 12 3 − 5 𝑛. This bound in Theorem 6.104 was subsequently improved for 𝛿 ≥ 25 𝑛 by Goddard et al. [358]. Theorem 6.105 ([358]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 25 𝑛 ≤ 𝛿 < 12 𝑛, then 𝑖(𝐺) ≤ 23 (𝑛 − 𝛿). For small regularity, namely 𝛿 < 14 𝑛, Haviland [402] established the following bounds. Theorem 6.106 ([402]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 < 14 𝑛, then 1   𝑛    2 3 𝑖(𝐺) ≤ 5 (𝑛 − 𝛿)     2𝛿 

if 1 ≤ 𝛿 ≤ 16 𝑛 if if

1 3 6 𝑛 ≤ 𝛿 ≤ 13 𝑛 3 1 13 𝑛 ≤ 𝛿 ≤ 4 𝑛.

We remark that in the statement of Theorem 6.106, the two values 12 𝑛 and 35 (𝑛 − 𝛿) are the same when 𝛿 = 16 𝑛. Moreover, the two values 35 (𝑛 − 𝛿) and 2𝛿 are the same 3 when 𝛿 = 13 𝑛. 1 For 𝛿 < 4 𝑛, Haviland [402] established the following upper bounds on 𝑖(𝐺) for 𝛿-regular graphs. Theorem 6.107 ([402]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 14 𝑛 ≤ 𝛿 ≤ 25 𝑛, then √ (1 √

2 2 if 𝛿 < 12 33 − 5 𝑛 2 3𝑛 + 2𝛿 − 𝑛 + 4𝛿 + 20𝑛𝛿 𝑖(𝐺) ≤ √
𝛿 if 𝛿 ≥ 12 33 − 5 𝑛. We close this section with the following result of Lyle [576] who studied a structural approach for independent domination in graphs. If 𝐺 is a 𝛿-regular graph on 𝑛 vertices with 𝛿 < 12 𝑛, then its complement 𝐺 is always connected.

Chapter 6. Upper Bounds in Terms of Minimum Degree

208

Theorem 6.108 ([576]) For every integer 𝑘 ≥ 4, if 𝐺 is a 𝛿-regular graph of order 𝑛 such that 𝐺 is connected, then  3   5 (𝑛 − 𝛿)      1𝑛 𝑖(𝐺) ≤ 2 5    8 (𝑛 − 𝛿)    2𝑛 𝑘

6.5

if

1 6𝑛

< 𝛿 < 14 𝑛

if 𝛿 = 14 𝑛 1 4𝑛

< 𝛿 < 𝑛 − 8 and 𝛿 ≠
if 𝛿 = 𝑘−3 𝑘 𝑛. if

𝑘−3 𝑘




𝑛

Summary

In this chapter, we have presented upper bounds on the domination number, the total domination number, and the independent domination number, in terms of the order 𝑛 and minimum degree 𝛿 of the graph. For a connected graph of order 𝑛 ≥ 3, we compare these best known upper bounds for small minimum degree 𝛿 ∈ [6] in Table 6.13, where Bdom and Btdom are families of seven and six, respectively, exceptional graphs of small orders.

Bound on domination parameter 𝛿≥

𝛾(𝐺) ≤

𝛾t (𝐺) ≤

1

1 2𝑛

2 3𝑛

2 3 4 5 6

2 5𝑛

if 𝐺 ∉ Bdom 3 8𝑛 4 11 𝑛 1 3𝑛 127 418 𝑛

4 7𝑛

if 𝐺 ∉ Btdom

4 11 4 13

1 2𝑛 3 7𝑛
11 + 800 𝑛
17 + 494 𝑛

𝑖(𝐺) ≤ √ 𝑛+2−2 𝑛 √ 𝑛 + 4 − 2 2𝑛 √ 𝑛 + 6 − 2 3𝑛 √ 𝑛 + 8 − 2 4𝑛 √ 𝑛 + 10 − 2 5𝑛 √ 𝑛 + 12 − 2 6𝑛

Table 6.13 A summary of upper bounds for small 𝛿 ∈ [6]

Chapter 7

Probabilistic Bounds and Domination in Random Graphs 7.1 Introduction In Chapter 6, we presented upper bounds on the domination number of a graph in terms of its order 𝑛 and minimum degree 𝛿. For small 𝛿, the best known bounds to date are summarized in Table 6.5 in Chapter 6. Recall that for 𝛿 ∈ [3], the bounds given in the table are tight, while for all values of 𝛿 ≥ 4, no tight bound on the domination number is yet known. When 𝛿 is sufficiently large, optimal bounds on the domination number can be found using the Probabilistic Method. We present several such probabilistic bounds, including one for the total domination number, in this chapter. We also show that if we carefully choose the probability 𝑝 that an edge is chosen in a random graph of order 𝑛, then the domination numbers √︁ and total domination √︁ enjoy a tight concentration, roughly between √1 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). We give 2 2 2 similar results for the independent domination number.

7.2

Probabilistic Bounds

One of the earliest bounds on the domination number was due to Arnautov [36] in 1974 and Payan [632] in 1975. Theorem 7.1 ([36, 632]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then 𝛿+1 𝑛 ∑︁ 1 𝛾(𝐺) ≤ . 𝛿 + 1 𝑗=1 𝑗 The proof of Theorem 7.1 presented in [36, 632] appears to follow a careful © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_7

209

210

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

analysis of a greedy algorithm to find a dominating set. Since the 𝑘 th harmonic number, 𝑘 ∑︁ 1 𝐻𝑘 = , 𝑗 𝑗=1 1 is approximately Φ + ln(𝑘) + 2𝑘 , where Φ = 0.57721 . . . is the Euler-Mascheroni constant, as an immediate consequence of Theorem 7.1 we have the following upper bound on the domination number of a graph in terms of its order and minimum degree.

Theorem 7.2 ([36, 632]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   1 + ln(𝛿 + 1) 𝛾(𝐺) ≤ 𝑛. 𝛿+1 We present here a proof of the upper bound in Theorem 7.2 using the probabilistic method. The idea of the proof is to generate a random set of vertices in a given graph where each vertex is chosen with a certain probability. Since the random choice of our set is not necessarily a dominating set, the set of vertices not yet dominated is added to this random set to form a dominating set. By carefully choosing the probability that each vertex is included in the initial random set, we obtain an upper bound on the domination number. Probabilistic Proof of Theorem 7.2 Let 𝑅 be a random subset of vertices of a graph 𝐺 with minimum degree 𝛿 ≥ 1, where a vertex is chosen to be in 𝑅 with probability 𝑝 and independently of the choice for any other vertex, where 𝑝=

ln(𝛿 + 1) . 𝛿+1

Let 𝑆 be the set of vertices in 𝐺 outside 𝑅 that have no neighbor in 𝑅, that is,  𝑆 = 𝑣 ∈ 𝑉 \ 𝑅 : N(𝑣) ∩ 𝑅 = ∅ . The set 𝑅 ∪ 𝑆 is a dominating set of 𝐺. The expected value of |𝑅| is E(|𝑅|) = 𝑛𝑝. The random variable |𝑆| can be written as the sum of 𝑛 indicator random variables 𝑋𝑣 (𝑆) for each 𝑣 ∈ 𝑉, where 𝑋𝑣 (𝑆) = 1 if 𝑣 ∈ 𝑆 and 𝑋𝑣 (𝑆) = 0 otherwise. For each vertex 𝑣 ∈ 𝑉, the expected value of 𝑋𝑣 (𝑆) is the probability that 𝑣 and its neighbors are not in 𝑅; that is,
E 𝑋𝑣 (𝑆) = (1 − 𝑝) deg(𝑣)+1 ≤ (1 − 𝑝) 𝛿+1 since deg(𝑣) ≥ 𝛿 and 0 ≤ 1 − 𝑝 ≤ 1. Using the inequality 1 − 𝑥 ≤ 𝑒 −𝑥 for 𝑥 a real number,
E 𝑋𝑣 (𝑆) ≤ (1 − 𝑝) 𝛿+1 ≤ 𝑒 − 𝑝 ( 𝛿+1)

Section 7.2. Probabilistic Bounds

211

for each vertex 𝑣 in 𝐺. Thus, by linearity of expectation,


E |𝑅 ∪ 𝑆| = E |𝑅| + E |𝑆| ∑︁
E 𝑋𝑣 (𝑆) ≤ 𝑛𝑝 + 𝑣 ∈𝑉

≤ 𝑛𝑝 + 𝑛𝑒 − 𝑝 ( 𝛿+1)   1 + ln(𝛿 + 1) = 𝑛. 𝛿+1 Since expectation is an average value, there is a set 𝑅 and an associated set 𝑆 such that 𝑅 ∪ 𝑆 is a dominating set in 𝐺 and   1 + ln(𝛿 + 1) |𝑅| + |𝑆| ≤ 𝑛. 𝛿+1 Since 𝛾(𝐺) ≤ |𝑅| + |𝑆|, this completes the proof of Theorem 7.2. The bound on the domination number in Theorem 7.2 is not very good for small values of 𝛿. However, as 𝛿 increases, the bound gets increasingly tight. Indeed, in 1990 Alon [17] provided a probabilistic proof that shows that the bound in Theorem 7.2 is asymptotically optimal, that is, when 𝛿 → ∞. Many probabilistic bounds on the domination number of a graph have been established over the past few decades. Using probabilistic arguments, in 1985 Caro and Roditty [134] (see also [135]) established the following upper bound on the domination number of a graph that is valid for all 𝛿 ≥ 1. Theorem 7.3 ([134]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then 

1 𝛾(𝐺) ≤ 1 − 𝛿 𝛿+1

 1+ 𝛿1 ! 𝑛.

In 1998 Clark et al. [180] established the following probabilistic upper bound on the domination number. Theorem 7.4 ([180]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then  𝛿+1 Ö 𝛾(𝐺) ≤ 1 − 𝑗=1

 𝑗𝛿 𝑛. 𝑗𝛿 + 1

The bound given in Theorem 7.4 is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 5. Recall that the results presented in Chapter 6 showed that Conjecture 6.28 holds, that is, if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1

212

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

unless 𝛿 = 2 and 𝐺 is one of the seven graphs in the family Bdom shown in Figure 6.1. We note that the bound given in Theorem 7.4 is better than the bound in Conjecture 6.28 for all 𝛿 ≥ 7. In 2012 Biró et al. [78] further improved the upper bound on the domination number and proved the following result. Theorem 7.5 ([78]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then ! 𝛿2 − 𝛿 + 1 𝛾(𝐺) ≤ 1 −
𝑛. Î 𝛿+1 1 + 𝛿 𝛿−1 𝑗=1 1 + 𝑗 𝛿 The bound given in Theorem 7.5 is better than the bound given in Theorem 7.4 for all 𝛿 ≥ 1 and is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 6. In 1999 Harant et al. [380] used probabilistic arguments to obtain upper bounds on the domination number of a graph. Theorem 7.6 ([380]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then
  𝛿1 ! ∑︁  1  𝛿1 deg(𝑣)+1 1 𝛾(𝐺) ≤ 1 − 𝛿 . 𝑛+ 𝛿 𝛿+1 𝑣 ∈𝑉 Further, a dominating set of cardinality at most the expression on the right hand side
of the above inequality can be constructed in O Δ2 𝑛 time. Theorem 7.7 ([380]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   𝛿1   𝛿1 Ö   𝛿1 ! ∑︁ 1 1 1 𝛾(𝐺) ≤ 1− + . deg(𝑣) + 1 deg(𝑣) + 1 deg(𝑢) + 1 𝑣 ∈𝑉 𝑢∈N(𝑣)

In the special case when 𝐺 is a regular graph, the bounds in Theorems 7.6 and 7.7 simplify as follows. Theorem 7.8 ([380]) If 𝐺 is a 𝛿-regular graph of order 𝑛, then ! 𝛿 𝛾(𝐺) ≤ 1 − 𝑛. 1 (𝛿 + 1) 1+ 𝛿 In 2019 Jafari Rad [509] gave a new probabilistic upper bound on the domination number that improves the previous bounds. Before stating and proving the result due to Jafari Rad, we present two key preliminary lemmas. Lemma 7.9 ([509]) Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 0 < 𝑝 < 1. Let 𝐴 be a subset of vertices of 𝐺, where a vertex is chosen to be in 𝐴 with probability 𝑝 and independently of the choice for any other vertex. Let 𝐴′′ ⊆ 𝐴′ ⊆ 𝐴 be defined by   𝐴′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴 and 𝐴′′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴′ .

Section 7.2. Probabilistic Bounds

213

For any integer 𝑠 ≥ 1, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠 − 1)| 𝐴′ |, where 𝑓 (0) = 𝑝 + (1 − 𝑝) 1+ 𝛿 , and 𝑓 (0)

1− 𝑓 (0)

{ ∑︁ }| }| z { z 𝑗
𝑓 ( 𝑗) = 𝑝 + (1 − 𝑝) 1+ 𝛿 − 1 − 𝑝 − (1 − 𝑝) 1+ 𝛿 𝑝 𝑖 (Δ+1) 𝑖=1

for every integer 𝑗 ≥ 1. Proof We proceed by induction on 𝑠 ≥ 1. To prove the base case, we show that there is a subset 𝑆 ⊆ 𝐴 such that 𝑆 dominates 𝐴′′ and
|𝑆| ≤ 𝑓 (0)| 𝐴′ | = 𝑝 + (1 − 𝑝) 1+ 𝛿 | 𝐴′ |. Let 𝐴1 be a subset of vertices of 𝐴′ , where a vertex is chosen to be in 𝐴1 with probability 𝑝 and independently of the choice for any other vertex. Let 𝐵1 ⊆ 𝐴′′ be the set of vertices in 𝐴′′ that are not dominated by 𝐴1 . We now let 𝑆1 = 𝐴1 ∪ 𝐵1 and note that the set 𝑆1 dominates 𝐴′′ . The expected value of | 𝐴1 | is
E | 𝐴1 | = | 𝐴′ | 𝑝. By definition, we note that if 𝑣 ∈ 𝐴′′ , then 𝑣 and its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Thus, Pr(𝑣 ∈ 𝐵1 ) = (1 − 𝑝) 1+deg𝐺 [ 𝐴′ ] (𝑣) = (1 − 𝑝) 1+deg𝐺 (𝑣) ≤ (1 − 𝑝) 1+ 𝛿 , implying that


E |𝐵1 | ≤ | 𝐴′ |(1 − 𝑝) 1+ 𝛿 .

Thus, by linearity of expectation,



E |𝑆1 | = E | 𝐴1 ∪ 𝐵1 | = E | 𝐴1 | + E |𝐵1 | ≤ | 𝐴′ | 𝑝 + | 𝐴′ | (1 − 𝑝) 1+ 𝛿
= 𝑝 + (1 − 𝑝) 1+ 𝛿 | 𝐴′ | = 𝑓 (0)| 𝐴′ |. Since expectation is an average value, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (0)| 𝐴′ |. This establishes the base case. For the inductive hypothesis, let 𝑠 ≥ 1 and assume the result holds for all positive integers 𝑠′ , where 𝑠′ ≤ 𝑠. We show that the result holds for 𝑠 + 1. As before, let 𝐴1 be a subset of vertices of 𝐴′ , where a vertex is chosen to be in 𝐴1 with probability 𝑝 and independently of the choice for any other vertex, and let 𝐵1 ⊆ 𝐴′′ be the set of vertices in 𝐴′′ that are not dominated by 𝐴1 . Let 𝐴1′′ ⊆ 𝐴1′ ⊆ 𝐴1 be defined by  𝐴1′ = 𝑣 ∈ 𝐴1 : N𝐺 [ 𝐴′ ] [𝑣] ⊆ 𝐴1

and

 𝐴1′′ = 𝑣 ∈ 𝐴1 : N𝐺 [ 𝐴′ ] [𝑣] ⊆ 𝐴1′ .

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

214

We note that 𝐴1′ ⊆ 𝐴′′ and 𝐴1 ⊆ 𝐴′ . If 𝑣 ∈ 𝐴1′ , then 𝑣 and all its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Thus, 𝐺 [ 𝐴′ ] is a graph with minimum degree 𝛿 > 1. Applying the inductive hypothesis to the graph 𝐺 [ 𝐴′ ], there is a subset 𝑆 𝑠 ⊆ 𝐴1′ such that 𝑆 𝑠 dominates 𝐴1′′ and |𝑆 𝑠 | ≤ 𝑓 (𝑠 − 1)| 𝐴1′ |. We now let 𝑆 𝑠+1 = ( 𝐴1 \ 𝐴1′ ) ∪ 𝑆 𝑠 ∪ 𝐵1 , and note that 𝑆 𝑠+1 ⊆ 𝐴′ and the set 𝑆 𝑠+1 dominates 𝐴′′ . Moreover,
E | 𝐴1 | = | 𝐴′ | 𝑝,
E |𝐵1 | ≤ | 𝐴′ |(1 − 𝑝) 1+ 𝛿 , and
E | 𝐴1′ | ≥ | 𝐴′ | 𝑝 1+Δ . Thus, by linearity of expectation,

E |𝑆 𝑠+1 | = E |( 𝐴1 \ 𝐴1′ ) ∪ 𝑆 𝑠 ∪ 𝐵1 |



= E | 𝐴1 | − E | 𝐴1′ | + E |𝑆 𝑠 | + E |𝐵1 |

≤ | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − E | 𝐴1′ | + 𝑓 (𝑠 − 1)E | 𝐴1′ |

= | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − 1 − 𝑓 (𝑠 − 1) E | 𝐴1′ |
≤ | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − 1 − 𝑓 (𝑠 − 1) | 𝐴′ | 𝑝 1+Δ . By definition of the function 𝑓 ,   𝑠−1 ∑︁
1+Δ 1+ 𝛿 𝑖 (Δ+1) 1+ 𝛿
𝑝 1+Δ 𝑝 = 1 − 𝑝 − (1 − 𝑝) + 1 − 𝑝 − (1 − 𝑝) 1 − 𝑓 (𝑠 − 1) 𝑝 𝑖=1

= 1 − 𝑝 − (1 − 𝑝)

1+ 𝛿


𝑠 ∑︁

𝑝

𝑖 (Δ+1)

.

𝑖=1 (1− 𝑓 (𝑠−1) ) | 𝐴′ | 𝑝 1+Δ

Thus, z


′

′

E |𝑆 𝑠+1 | ≤ | 𝐴 | 𝑝 + | 𝐴 |(1 − 𝑝)

1+ 𝛿

′

}|

− | 𝐴 | 1 − 𝑝 − (1 − 𝑝)

1+ 𝛿


𝑠 ∑︁

{ 𝑝

𝑖 (Δ+1)

𝑖=1

  𝑠
∑︁ = | 𝐴′ | 𝑝 + (1 − 𝑝) 1+ 𝛿 − 1 − 𝑝 − (1 − 𝑝) 1+ 𝛿 𝑝 𝑖 (Δ+1) 𝑖=1

= 𝑓 (𝑠)| 𝐴′ |. Since expectation is an average value, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠)| 𝐴′ |. Arguments similar to those used in the proof of Lemma 7.9, together with the well-known inequality 1 − 𝑥 ≤ 𝑒 − 𝑥 for 0 ≤ 𝑥 ≤ 1, yield the following result. Lemma 7.10 ([509]) Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 0 < 𝑝 < 1. Let the sets 𝐴, 𝐴′ , and 𝐴′′ be defined as in

Section 7.2. Probabilistic Bounds

215

the statement of Lemma 7.9. For any integer 𝑠 ≥ 1, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠 − 1)| 𝐴′ |, where 𝑓 (0) = 𝑝 + 𝑒 − 𝑝 (1+ 𝛿 ) , and 𝑓 (0)

1− 𝑓 (0)

}| { ∑︁ z }| { z 𝑗 − 𝑝 (1+ 𝛿 ) − 𝑝 (1+ 𝛿 )
𝑓 ( 𝑗) = 𝑝 + 𝑒 𝑝 𝑖 (Δ+1) − 1− 𝑝−𝑒 𝑖=1

for every integer 𝑗 ≥ 1. We are now in a position to state and prove the result by Jafari Rad [509]. Theorem 7.11 ([509]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, then for every integer 𝑘 ≥ 1,  𝑖 (1+Δ) ! 𝑘 
∑︁ ln(𝛿 + 1) 𝑛 𝛾(𝐺) ≤ ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) . 𝛿 + 1 𝛿 + 1 𝑖=1 Proof Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 𝑘 ≥ 1 be an integer. Let 𝐴 be a subset of vertices of 𝐺, where a vertex is chosen to be in 𝐴 with probability 𝑝 and independently of the choice for any other vertex, where ln(𝛿 + 1) 𝑝= . 𝛿+1 Let 𝐵 = 𝑉 \ N[ 𝐴] and let 𝐴′′ ⊆ 𝐴′ ⊆ 𝐴 be defined by   𝐴′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴 and 𝐴′′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴′ . By definition, if 𝑣 ∈ 𝐴′′ , then 𝑣 and its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Further, by definition, every vertex in 𝐴′ \ 𝐴′′ has a neighbor
in 𝐴 \ 𝐴′ . For 𝑝 = ln(𝛿 + 1) /(𝛿 + 1), Lemma 7.9 shows that there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑘 − 1)| 𝐴′ |, where 𝑓 (0) =

ln(𝛿 + 1) + 1 , and 𝛿+1

!  𝑗 
∑︁ ln(𝛿 + 1) 𝑖 (1+Δ) 1 𝑓 ( 𝑗) = ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) 𝛿+1 𝛿+1 𝑖=1 for every integer 𝑗 ≥ 1. We now let 𝐷 = ( 𝐴 \ 𝐴′ ) ∪ 𝐵 ∪ 𝑆 and note that 𝐷 is a dominating set of 𝐺 satisfying |𝐷| = | 𝐴 \ 𝐴′ | + |𝐵| + |𝑆| ≤ | 𝐴| + |𝐵| − | 𝐴′ | + 𝑓 (𝑘 − 1)| 𝐴′ |
= | 𝐴| + |𝐵| − 1 − 𝑓 (𝑘 − 1) | 𝐴′ |.

216

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

By linearity of expectation,




E |𝐷 | ≤ E | 𝐴| + E |𝐵| − 1 − 𝑓 (𝑘 − 1) E | 𝐴′ | . Moreover,
E | 𝐴| = 𝑛𝑝,
E |𝐵| ≤ 𝑛(1 − 𝑝) 1+ 𝛿 , and
E | 𝐴′ | ≥ 𝑛𝑝 1+Δ . Thus, by linearity of expectation,




E |𝐷| ≤ E | 𝐴| + E |𝐵| − 1 − 𝑓 (𝑘 − 1) E | 𝐴′ |
≤ 𝑛𝑝 + 𝑛(1 − 𝑝) 1+ 𝛿 − 𝑛 1 − 𝑓 (𝑘 − 1) 𝑝 1+Δ
≤ 𝑛𝑝 + 𝑛𝑒 − 𝑝 (1+ 𝛿 ) − 1 − 𝑓 (𝑘 − 1) 𝑛𝑝 1+Δ

𝑛 ≤ ln(𝛿 + 1) + 1 − 𝑛 1 − 𝑓 (𝑘 − 1) 𝑝 1+Δ 𝛿+1  𝑖 (1+Δ) ! 𝑘 
∑︁ ln(𝛿 + 1) 𝑛 = ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) . 𝛿 + 1 𝛿 + 1 𝑖=1 Since expectation is an average value, there is a dominating set 𝐷 ★ of 𝐺 of the desired cardinality. We note that the bound in Theorem 7.11 is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 2, since 𝛿 − ln(𝛿 + 1) > 0. Theorem 7.12 ([509]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, then for every integer 𝑘 ≥ 1,  𝑖 (1+Δ) ! 𝑘  ∑︁ 𝛿 𝛿 1 𝛾(𝐺) ≤ 1 − − 1− 𝑛. 1 1 1 (𝛿 + 1) 1+ 𝛿 (𝛿 + 1) 1+ 𝛿 𝑖=1 (𝛿 + 1) 𝛿 Proof The proof is analogous to the proof given for Theorem 7.11, except that in this case we apply Lemma 7.10 with 𝑝 =1−

1

, 1

(𝛿 + 1) 𝛿

and define 𝑓 (0) = 1 −

𝛿 1

(𝛿 + 1) 1+ 𝛿

, and 1− 𝑓 (0)

𝑓 (0)

}| 𝛿

z 𝑓 ( 𝑗) = 1 −

{ z 1

(𝛿 + 1) 1+ 𝛿

for every integer 𝑗 ≥ 1.

−

}| 𝛿

{

𝑗  ∑︁

1

(𝛿 + 1) 1+ 𝛿

𝑖=1

1−

 𝑖 (1+Δ)

1 1

(𝛿 + 1) 𝛿

Section 7.3. Domination in Random Graphs

217

We note that the bound in Theorem 7.12 is better than the bound given in Theorem 7.3 for all 𝛿 ≥ 2. We also note that if 𝐺 is a 𝛿-regular graph, then the bound in Theorem 7.12 is better than the bound given in Theorem 7.8 for all 𝛿 ≥ 2. Similar probabilistic bounds also exist for the total domination number. Recall that in Theorem 6.77 in Chapter 6, we proved that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛. 𝛿 This bound can also be proved using probabilistic arguments analogous to the probabilistic proof of Theorem 7.2. In 2019 Henning and Jafari Rad [462] gave the following improved probabilistic upper bound on the total domination number of a graph in terms of its minimum degree. Theorem 7.13 ([462]) If 𝐺 is a graph with minimum degree 𝛿 ≥ 2 and maximum degree Δ, then     1   1+Δ(Δ−1) 1 + ln(𝛿) 1 1 𝛿−1 ln(𝛿) 𝛾t (𝐺) ≤ 𝑛 −𝑛 1− . 𝛿 𝛿 Δ 𝛿 

7.3

Domination in Random Graphs

In this section, we study the behavior of the domination, total domination, and independent domination numbers using the Erdős-Rényi random graph model, which they introduced in [258] in 1960. For a positive integer 𝑛 and real number 𝑝 with 0 < 𝑝 < 1, we denote by G(𝑛, 𝑝) the probability space whose elements 𝐺 are all possible graphs of order 𝑛 with vertex set 𝑉𝑛 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }, where an edge is chosen to be in 𝐺 with probability 𝑝 and independently of the choice for any other edge. We call an element 𝐺 ∈ G(𝑛, 𝑝) a random graph. A graph property or graph invariant is a property of graphs that is preserved under all possible isomorphisms of a graph, and therefore depends only on the abstract structure of graphs and not on the way the graph is drawn or represented. Two examples of graph properties are connectedness and having diameter 2. Let P be the set of graphs of order 𝑛 having Property 𝑃, and let 𝐺 ∈ G(𝑛, 𝑝) be a random graph, where 0 < 𝑝 < 1 and where 𝑝 may depend on 𝑛. If lim Pr(𝐺 ∈ P) = 1,

𝑛→∞

then we say that almost every random graph 𝐺 ∈ G(𝑛, 𝑝) has property 𝑃, or that the property 𝑃 holds asymptotically almost surely (abbreviated, a.a.s.). On the other hand, if lim Pr(𝐺 ∈ P) = 0, 𝑛→∞

then we say that almost no random graph 𝐺 ∈ G(𝑛, 𝑝) has property 𝑃. In 1981 Weber [747] was the first to study the domination and independent domination numbers of random graphs.

218

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Theorem 7.14 ([747]) If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with 𝑝 = 21 , and 
 𝜉1 (𝑛) = log2 (𝑛) − log2 log2 (𝑛) ln(𝑛) , then a.a.s. 𝛾(𝐺) = 𝜉1 (𝑛) + 1 or 𝛾(𝐺) = 𝜉1 (𝑛) + 2, and 𝑖(𝐺) = 𝜉1 (𝑛) + 2 or 𝑖(𝐺) = 𝜉1 (𝑛) + 3. In 2001 Wieland and Godbole [751] extended Weber’s result and showed that the domination number of a random graph has the following two-point concentration. Theorem 7.15 ([751]) Let 𝐺 ∈ G(𝑛, 𝑝) be a random graph of order 𝑛, where either 𝑝 is a constant or 𝑝 = 𝑝(𝑛) such that  2  ln (𝑛) 2 𝑝 ln(𝑛) ≥ 40 ln . 𝑝 If 
 𝜉2 (𝑛) = log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) , where 𝑞 =

1 1− 𝑝

and log𝑞 denotes the logarithm with base 𝑞, then a.a.s. 𝛾(𝐺) = 𝜉2 (𝑛) + 1 or

𝛾(𝐺) = 𝜉2 (𝑛) + 2.

We remark that the probability function 𝑝 = 𝑝(𝑛) in the statement of Theorem 7.15 tends to 0 as 𝑛 tends to infinity and is equivalent (see [339]) to √︄
10 ln ln(𝑛) 𝑝 = 𝑝(𝑛) ≥ . ln(𝑛) The proof of Theorem 7.15, which we omit, uses first and second moment methods to prove the two-point concentration of the domination number. Wieland and Godbole [751] raised the problem whether the validity of this two-point concentration in Theorem 7.15 can be extended to a wider range of 𝑝. This question was answered in 2015 by Glebov et al. [339]. Theorem 7.16 ([339])

If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with ln(𝑛) √ ≪ 𝑝 < 1, 𝑛

and  𝜉3 (𝑛, 𝑝) = log𝑞 where 𝑞 =

1 1− 𝑝 ,



𝑛 ln(𝑞) ln2 (𝑛𝑝)

 (1 + O (1) ,

then a.a.s. 𝛾(𝐺) = 𝜉3 (𝑛, 𝑝) + 1 or

𝛾(𝐺) = 𝜉3 (𝑛, 𝑝) + 2.

In 2014 Henning and Yeo [491] showed that if we choose the probability 𝑝 carefully, then the domination and total domination numbers of the random graph G(𝑛, 𝑝)

Section 7.3. Domination in Random Graphs

219

√︁ √︁ are concentrated between roughly √1 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). Before we present 2 2 2 a precise statement and proof of this result, we first give some preliminary results. We shall need the following well-known lemma. Lemma 7.17 For all 𝑥 > 1, 1−


1 𝑥 𝑥

< 𝑒 −1 < 1 −


1 𝑥−1 . 𝑥

In what follows, we define √︄ 𝑐𝑛 =


ln ln(𝑛) 1 1 . + + 2 ln(𝑛) 2 ln(𝑛)

The function 1/ln(𝑛) is a decreasing function. For 𝑛 ≥ 24, we therefore have 1/ln(𝑛) ≤ 1/ln(24). Using the well-known calculus result that for 𝑥 > 0 the maximum
value of ln(𝑥)/𝑥 is 𝑒 −1 , we have that ln ln(𝑛) /ln(𝑛) ≤ 𝑒 −1 . Hence, √︄ 1 1 1 𝑐𝑛 ≤ + + < 1. 2 ln(24) 2𝑒 We state this formally as follows. Observation 7.18 If 𝑛 ≥ 24, then 𝑐 𝑛 < 1. We shall need the following upper bound on the total domination number of a graph of diameter 2. Lemma 7.19 ([491]) If 𝐺 is a diameter-2 graph of order 𝑛 ≥ 24, then 𝛾t (𝐺) < √︁ 𝑐 𝑛 𝑛 ln(𝑛) + 1. Proof Let 𝐺 be a diameter-2 graph of order 𝑛 ≥ 24. Since the closed neighborhood of any vertex in a diameter-2 graph is a TD-set in the graph, we can choose a vertex of minimum degree 𝛿(𝐺) =√︁𝛿 and form a TD-set of cardinality 𝛿 + 1. Hence, 𝛾t (𝐺) ≤ 𝛿 + 1. Thus, if 𝛿 < 𝑐 𝑛 𝑛 ln(𝑛), then the desired result is immediate. √︁ Therefore, we may assume that 𝛿 ≥ 𝑐 𝑛 𝑛 ln(𝑛). Since 𝑛 ≥ 24, by Observation 7.18, we have 𝑐 𝑛 < 1,
implying that ln(𝑐 𝑛 ) < 0. Thus, by Theorem 6.77 in Chapter 6 and since 1 + ln(𝛿) /𝛿 is a decreasing function for all 𝛿 ≥ 1, the following holds: √︁
    1 + ln 𝑐 𝑛 𝑛 ln(𝑛) 1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛≤ 𝑛 √︁ 𝛿 𝑐 𝑛 𝑛 ln(𝑛)
 √︁  𝑛 ln(𝑛) ln(𝑛) ln ln(𝑛) = 1 + ln(𝑐 𝑛 ) + + 2 2 𝑐 𝑛 ln(𝑛)
  √︁ ln ln(𝑛) 1 1 1 < + + 𝑛 ln(𝑛) 𝑐 𝑛 ln(𝑛) 2 2 ln(𝑛) √︁ 1 = × 𝑐2𝑛 × 𝑛 ln(𝑛) 𝑐𝑛 √︁ = 𝑐 𝑛 𝑛 ln(𝑛).

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

220

As a consequence of Lemma 7.19, we have the following result. Theorem 7.20 ([491]) Given any 𝜀 > 0, if 𝐺 is a diameter-2 graph of sufficiently large order 𝑛, then   √︁ 1 𝛾t (𝐺) < √ + 𝜀 𝑛 ln(𝑛). 2 Proof The function √︄

tends to holds:

√1 2


ln ln(𝑛) 1 1 1 + √︁ + + , 2 ln(𝑛) 2 ln(𝑛) 𝑛 ln(𝑛)

when 𝑛 tends to infinity. Therefore, for 𝑛 sufficiently large, the following √︄


ln ln(𝑛) 1 1 1 + √︁ + + < 2 ln(𝑛) 2 ln(𝑛) 𝑛 ln(𝑛)
√︄  √︁ ln ln(𝑛) 1 1 × 𝑛 ln(𝑛) + 1 < + + 2 ln(𝑛) 2 ln(𝑛) √︁ 𝑐 𝑛 𝑛 ln(𝑛) + 1
2 2 𝜀 > 𝜀 > 𝜀 ′ . Let 𝜀1 be defined such 2 2 that 0 < 𝜀 ′ < 𝜀1 < 𝜀★, and let 𝑁 be large enough so that

Section 7.3. Domination in Random Graphs 

1 1 + 𝜀1

 √︂

221

  √︂ 𝑛 ln(𝑛) 1 𝑛 ln(𝑛) − >1 8 1 + 𝜀★ 8

holds for all 𝑛 > 𝑁. Let  𝑡=

1 1 + 𝜀★

 √︂

𝑛 ln(𝑛) 8



and define 𝜀★ 1 such that 

1 𝑡= 1 + 𝜀★ 1

 √︂

𝑛 ln(𝑛) . 8

★ When 𝑛 > 𝑁, we note that 0 < 𝜀 ′ < 𝜀 1 < 𝜀★ 1 ≤ 𝜀 . By definition of 𝑝 and 𝑡, and ★ since 𝜀1 < 𝜀1 , (1 + 𝜀 ′ ) ln(𝑛) (1 + 𝜀 ′ ) ln(𝑛) 𝑝𝑡 = < . 2(1 + 𝜀1 ) 2(1 + 𝜀★ 1)

For any 𝜀2 such that 0 < 𝜀2 < 1, let 𝑁 be large enough so that the following holds when 𝑛 > 𝑁. Then   √︂    √︂  1 𝑛 ln(𝑛) 1 ln(𝑛) 𝑛−𝑡 =𝑛− ≥ 𝑛 1 − ≥ 𝑛(1 − 𝜀2 ). 8 1 + 𝜀1 8𝑛 1 + 𝜀★ 1 Let 𝜀3 be defined such that 0 < 𝜀3 < 1 −

1 + 𝜀′ , 1 + 𝜀1

which is possible as 𝜀 ′ < 𝜀 1 . Let 𝑁 be large enough so that 1 − 𝑝 ≥ 1 − 𝜀3 when 𝑛 > 𝑁, which is possible as 𝑝 tends to zero when 𝑛 goes to infinity. Let 𝜀4 = 1 −

1 + 𝜀′ (1 + 𝜀1 ) (1 − 𝜀3 )

and note that by the definition of 𝜀3 , we have 𝜀4 > 0. Further, we note that none ★ ′ ★ of 𝜀★, 𝜀 ′ , 𝜀1 , 𝜀2 , 𝜀3 , and 𝜀4 depends on 𝑛 (but 𝜀★ 1 does). The properties of 𝜀 , 𝜀 , 𝜀 1 , 𝜀1 , 𝜀2 , 𝜀3 , and 𝜀4 when 𝑛 > 𝑁 are summarized in the following claim. Claim 7.21.1 The following hold: ★ (a) 0 < 𝜀 ′ < 𝜀1 < 𝜀★ 1 ≤ 𝜀 . (b) 𝑛 − 𝑡 ≥ 𝑛(1 − 𝜀2 ) and 0 < 𝜀2 < 1. (c) 1 − 𝑝 ≥ 1 − 𝜀3 . (d) 𝜀4 > 0. ′ ) ln(𝑛) (e) 𝑝 × 𝑡 < (1+𝜀 2(1+𝜀1 ) . We are now in a position to establish the lower bound in the statement of the theorem.
√︁ Claim 7.21.2 If 𝐺 ∈ G(𝑛, 𝑝), then a.a.s. 𝛾(𝐺) > √1 − 𝜀 𝑛 ln(𝑛). 2 2

222

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Proof Let 𝐺 ∈ G(𝑛, 𝑝) and let 𝑉 = 𝑉 (𝐻) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }. Let 𝑆 be an arbitrary subset of vertices of 𝐺 with |𝑆| = 𝑡. Let 𝑥 ∈ 𝑉 \ 𝑆 be arbitrary. The probability that 𝑆 does not dominate 𝑥 is (1 − 𝑝) 𝑡 , since there are 𝑡 vertices in 𝑆 and each vertex has probability of 1 − 𝑝 of not being adjacent to 𝑥. Therefore, vertex 𝑥 has probability 1 − (1 − 𝑝) 𝑡 of being dominated by 𝑆. The probability of one vertex in 𝑉 \ 𝑆 being dominated by 𝑆 is independent of any other vertex in 𝑉 \ 𝑆 being dominated by 𝑆. Thus, the probability that 𝑆 dominates all of 𝑉 \ 𝑆 is given by
𝑛−𝑡 (7.1) Pr(𝑆 is a dominating set) = 1 − (1 − 𝑝) 𝑡
𝑛−𝑡 since |𝑉 \𝑆| = 𝑛−𝑡. We now provide an upper bound on the expression 1−(1− 𝑝) 𝑡 as follows. We first establish the following lower bound on the term (1 − 𝑝) 𝑡 :
( 1 −1) 𝑝𝑡 (1 − 𝑝) 𝑡 = 1 − 1/1𝑝 𝑝 1− 𝑝
𝑝𝑡 > 𝑒 −1 1− 𝑝 (by Lemma 7.17) 𝑝𝑡
≥ 𝑒 −1 1− 𝜀3 (by Claim 7.21.1(c)) (1+𝜀 ′ ) ln(𝑛)
≥ 𝑒 −1 2(1+𝜀1 ) (1− 𝜀3 ) (by Claim 7.21.1(d))
(1− 𝜀4 ) ln(𝑛)/2 = 𝑒 −1 (by definition of 𝜀4 ), and so, (1 − 𝑝) 𝑡 ≥ 𝑛 − (1− 𝜀4 )/2 . (7.2) Hence, the probability that 𝑆 is a dominating set is bounded as follows:
𝑛−𝑡 Pr(𝑆 is a dominating set) = 1 − (1 − 𝑝) 𝑡 (by Equation (7.1))   𝑛 [ (1− 𝜀4 ) /2] × [ (1−𝑛−𝑡 𝜀4 ) /2] 𝑛 1 ≤ 1 − (1− 𝜀 )/2 (by Inequality (7.2)) 4 𝑛
𝑛−𝑡 < 𝑒 −1 𝑛 [ (1− 𝜀4 ) /2] (by Lemma 7.17) 𝑛(1− 𝜀2 )
≤ 𝑒 −1 𝑛 [ (1− 𝜀4 ) /2] (by Claim 7.21.1(b)) [ (1+𝜀4 ) /2]
(1− 𝜀 )𝑛 2 = 𝑒 −1 . 𝑛
As there are 𝑡 possible choices for choosing 𝑆, the probability that 𝛾(𝐺) ≤ √ 𝑛 ln(𝑛) √ is bounded as follows: ★

2 2(1+𝜀 )



√︁





√︁

 𝑛 ln(𝑛) Pr 𝛾(𝐺) ≤ √ ≤ Pr 𝛾(𝐺) ≤ √ 2 2(1 + 𝜀★) 2 2(1 + 𝜀★ 1)   𝑛 ≤ × Pr(𝑆 is a dominating set) 𝑡
(1− 𝜀2 )𝑛 [ (1+𝜀4 ) /2] ≤ 𝑛𝑡 × 𝑒 −1 𝑛 ln(𝑛)

= 𝑒𝑡

ln(𝑛) − (1− 𝜀2 )𝑛 [ (1+𝜀4 ) /2]

.

Section 7.3. Domination in Random Graphs

223

By the definition of 𝑡, it follows that 𝑡 ln(𝑛) − (1 − 𝜀2 )𝑛

[ (1+𝜀4 )/2]

√︂  1 𝑛 ln(𝑛) = ln(𝑛) − (1 − 𝜀2 )𝑛 [ (1+𝜀4 )/2] 8 1 + 𝜀★ 1 √︁   √ ln(𝑛) ln(𝑛) ≤ √ − (1 − 𝜀2 )𝑛 𝜀4 /2 𝑛. (1 + 𝜀1 ) 8



By Claim 7.21.1, we have that 𝜀1 > 0, 0 < 𝜀2 < 1, and 𝜀4 > 0. Since 𝑛 𝜀4 /2 grows faster than any poly-log function in 𝑛, we have that 𝑡 ln(𝑛) − (1 − 𝜀2 )𝑛 (1+𝜀4 )/2 → −∞ as 𝑛 → ∞, implying that √︁



𝑛 ln(𝑛)



Pr 𝛾(𝐺) ≤ √ →0 2 2(1 + 𝜀★) as 𝑛 → ∞. Hence, a.a.s. we have √︁

  √︁ 𝑛 ln(𝑛) 1 𝛾(𝐺) > √ = √ − 𝜀 𝑛 ln(𝑛), 2 2(1 + 𝜀★) 2 2 which completes the proof of Claim 7.21.2. We prove next that a.a.s. if 𝐺 ∈ G(𝑛, 𝑝), then diam(𝐺) = 2. For this purpose, we define a bad pair of vertices in a graph 𝐺 to be a pair of vertices 𝑢 and 𝑣 at distance at least 3 in 𝐺. In particular, if 𝑢 and 𝑣 is a bad pair, then 𝑢 and 𝑣 are neither adjacent nor do they have a common neighbor. We note that diam(𝐺) > 2 if and only if 𝐺 has a bad pair of vertices. Claim 7.21.3 If 𝐺 ∈ G(𝑛, 𝑝), then a.a.s. 𝐺 is a diameter-2 graph. Proof Let 𝐺 ∈ G(𝑛, 𝑝) with 𝑉 = 𝑉 (𝐺) and let 𝑢 and 𝑣 be an arbitrary pair of bad vertices in the graph 𝐺 and let 𝑤 ∈ 𝑉 \ {𝑢, 𝑣}. The probability that 𝑤 is adjacent to both 𝑢 and 𝑣 is 𝑝 2 and hence the probability that 𝑤 is not a common neighbor of 𝑢 and 𝑣 is 1 − 𝑝 2 . This is true for all 𝑛 − 2 vertices in 𝑉 \ {𝑢, 𝑣}. Hence, the
𝑛−2 probability that no vertex is adjacent to both 𝑢 and 𝑣 is 1 − 𝑝 2 . Further, the probability that 𝑢 and 𝑣 are not adjacent is 1 − 𝑝. Hence, the probability that 𝑢
𝑛−2 and 𝑣 is a bad pair is (1 − 𝑝) 1 − 𝑝 2 . When 𝑛 is sufficiently large, we note that √

2 𝑝 ≤ −1 + 5 /2 < 0.618 and therefore that (1 − 𝑝)/ 1 − 𝑝 2 < 1. Hence, for 𝑛 large enough, the following holds by Lemma 7.17,

224

Chapter 7. Probabilistic Bounds and Domination in Random Graphs
𝑛−2
Pr 𝑑 (𝑢, 𝑣) > 2 = (1 − 𝑝) 1 − 𝑝 2   12 × 𝑝2 𝑛 𝑝 1− 𝑝 1 = 1 − 2 2 2 (1 − 𝑝 ) 1/𝑝   12 × 𝑝2 𝑛 𝑝 1 < 1− 2 1/𝑝
𝑝2 𝑛 < 𝑒 −1
(1+𝜀 ′ ) 2 × ( 2 ln(𝑛) 𝑛 )𝑛 = 𝑒 −1 1 = . ′ )2 2(1+𝜀 𝑛

Let 𝑋 be
the random variable that counts the number of bad pairs in 𝐺. Since there are 𝑛2 pairs of vertices, the expected number of bad pairs is E(𝑋) =

∑︁

Pr 𝑑 (𝑢, 𝑣, >)2




{𝑢,𝑣 } ⊆𝑉

   𝑛 1 2 𝑛2(1+𝜀 ′ ) 2   1 ≤ 𝑛2 𝑛2(1+2𝜀 ′ +( 𝜀 ′ ) 2 ) 1 = . ′ +2( 𝜀 ′ ) 2 4𝜀 𝑛 ≤

Thus, lim E(𝑋) → 0,

𝑛→∞

implying that a.a.s. the graph 𝐺 has no bad pairs and √︁ hence has diameter at most 2. By Claim 7.21.2, a.a.s. we have 𝛾(𝐺) > √ 1 ★ 𝑛 ln(𝑛) > 1, implying that 𝐺 2 2(1+𝜀 ) does not have a vertex adjacent to every other vertex. Therefore, a.a.s. we have diam(𝐺) = 2. The upper bound now follows from Claim 7.21.3 and Theorem 7.20. In 2021 Dubickas [245] posed the following problem. Problem 7.22 ([245]) Find 𝑐 min = lim inf 𝑛→∞

min𝐺 ∈ G𝑛 𝛾t (𝐺) √︁ 𝑛 ln(𝑛)

and

𝑐 max = lim sup 𝑛→∞

min𝐺 ∈ G𝑛 𝛾t (𝐺) , √︁ 𝑛 ln(𝑛)

where G𝑛 is the class of diameter-2 graphs of order 𝑛. In 2014 Desormeaux et al. [230] showed that 14 ≤ 𝑐 min . As a consequence of Theorem 7.21, we have the improved result that √1 ≤ 𝑐 min and 𝑐 max ≤ √1 . 2 2

2

Section 7.3. Domination in Random Graphs

225

√︃ Dubickas [245] further improved the value of 𝑐 min to 38 ≤ 𝑐 min . Thus, the best results to date are given by √︃ 0.612372 ≈ 38 ≤ 𝑐 min ≤ 𝑐 max ≤ √1 ≈ 0.707107. 2

The exact values of 𝑐 min and 𝑐 max have yet to be determined. Dubickas [245] √︃ 3 wrote: “It is quite tempting to conjecture that 𝑐 min = 𝑐 max = 8 .” Results similar to those for the domination number and total domination number also hold for the independent domination number of a random graph. In 2008 Bonato and Wang [85] proved the following two-point concentration result. Theorem 7.23 ([85]) If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with 𝑝 a constant, then a.a.s. 
 log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) + 1 ≤ 𝑖(𝐺) ≤ ⌊log𝑞 (𝑛)⌋, where 𝑞 =

1 1− 𝑝 .

The result of Theorem 7.23 was improved in 2011 by Clark and Johnson [179], who proved the following two-point concentration for the independent domination number of a random graph. Theorem 7.24 ([179]) Let 𝐺 ∈ G(𝑛, 𝑝) be a random graph of order 𝑛, where either 𝑝 is a constant or 𝑝 = 𝑝(𝑛) such that   ln(𝑛) 2 𝑝 ln(𝑛) ≥ 64 ln . 𝑝 If  
𝜉4 (𝑛) = log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) + log𝑞 (2) , where 𝑞 =

1 1− 𝑝

then a.a.s. 𝑖(𝐺) = 𝜉4 (𝑛) + 1 or 𝑖(𝐺) = 𝜉4 (𝑛) + 2.

Theorem 7.24 extends the result of Weber in Theorem 7.23 in the case when 𝑝 = 12 , and the result of Bonato and Wang in Theorem 7.23 in the case when 𝑝 is constant. We close this section with results on the domination and independent domination numbers of a random cubic graph. In 1995 Molloy and Reed [599] established the following bounds on the domination number of a random cubic graph. Theorem 7.25 ([599]) If 𝐺 is a random cubic graph of order 𝑛, then a.a.s. 0.2636𝑛 ≤ 𝛾(𝐺) ≤ 0.3126𝑛. In 2002 Duckworth and Wormald [246] proved a better upper bound for the independent domination number 𝑖(𝐺) of a random cubic graph than the upper bound for 𝛾(𝐺) in Theorem 7.25.

226

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Theorem 7.26 ([246]) If 𝐺 is a random cubic graph of order 𝑛, then a.a.s. 0.2641𝑛 ≤ 𝑖(𝐺) ≤ 0.27942𝑛. The lower bound in Theorem 7.26 was calculated by means of a direct expectation argument. The upper bound can be calculated by using differential equations to analyze the performance of a randomized greedy algorithm used in [246]. A simple randomized greedy algorithm for finding an independent dominating set 𝑆 in an arbitrary graph 𝐺 can be constructed by iteratively selecting a vertex 𝑣 to be in 𝑆 and then removing 𝑣 and all its neighbors N[𝑣] from the graph 𝐺. In the remaining graph, select a second vertex for the set 𝑆 and then remove it and all of its neighbors. Repeat this process until there are no remaining vertices. Removing a selected vertex and its neighbors guarantees that the final set is independent and all vertices removed are either in the set 𝑆 or are dominated by a vertex in 𝑆. Thus, when no vertices remain, 𝑆 is guaranteed to be an independent dominating set. This algorithm is highlighted in Algorithm 4. The choice of the next vertex to be selected for the set 𝑆 can be made using a variety of heuristics, such as selecting a vertex of maximum or minimum degree in the remaining subgraph. Such a greedy algorithm becomes randomized when the vertex is selected uniformly at random from the set of vertices meeting the selection criteria. In [246] the authors selected a vertex of minimum degree. Algorithm 4 Randomized Independent Dominating Set Input : A graph 𝐺 = (𝑉, 𝐸) Output : An independent dominating set 𝑆 of 𝐺 1 2

3 4

5 6 7

[Initialize 𝑆 and the set 𝑊 from which vertices are to be selected] Set 𝑆 = ∅ Set 𝑊 = 𝑉 [Iteratively select vertices for 𝑆 and remove them and their neighbors] while 𝑊 ≠ ∅ do Select a vertex 𝑣 uniformly at random from 𝑊 of minimum degree in 𝐺 [𝑊] Set 𝑆 = 𝑆 ∪ {𝑣} Set 𝑊 = 𝑊 − N[𝑣] od

7.4

Summary

In this chapter, we presented probabilistic bounds for the domination and total domination numbers of a graph in terms of its order and minimum degree. We showed that the bounds are optimal when the minimum degree 𝛿 is sufficiently large. We also showed that if we carefully choose the probability 𝑝 that an each edge is chosen in a random graph G(𝑛, 𝑝) of order 𝑛, then the domination, total domination, and independent domination numbers enjoy a tight concentration.

Chapter 8

Bounds in Terms of Size 8.1 Introduction The order |𝑉 | = 𝑛 and size |𝐸 | = 𝑚 of a graph 𝐺 = (𝑉, 𝐸) are the two most fundamental graph parameters. In this chapter, we study how the size of a graph relates to its domination, total domination, and independent domination numbers.

8.2 Domination and Size One of the earliest results relating the domination number and the size of a graph is the following 1962 result due to Berge [67]. Theorem 8.1 ([67]) If 𝐺 is a graph of order 𝑛 and size 𝑚, then 𝛾(𝐺) ≥ 𝑛 − 𝑚, with equality if and only if each component of 𝐺 is a star. Proof Let 𝐺 have order 𝑛 and size 𝑚. Since 𝛾(𝐺) ≥ 1, the result is immediate for 𝑚 ≥ 𝑛. Hence, we may assume that 𝑚 ≤ 𝑛−1. Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 be the components of 𝐺 where 𝑘 ≥ 1, and let 𝐺 𝑖 have order 𝑛𝑖 and size 𝑚 𝑖 for 𝑖 ∈ [𝑘]. We note that 𝑚=

𝑘 ∑︁ 𝑖=1

𝑚𝑖 ≥

𝑘 ∑︁

(𝑛𝑖 − 1) = 𝑛 − 𝑘,

(8.1)

𝑖=1

with equality if and only if 𝐺 𝑖 is a tree for each 𝑖 ∈ [𝑘]. In particular, 𝑘 ≥ 𝑛 − 𝑚. By linearity, 𝑘 𝑘 ∑︁ ∑︁ 𝛾(𝐺) = 𝛾(𝐺 𝑖 ) ≥ 1 = 𝑘 ≥ 𝑛 − 𝑚. (8.2) 𝑖=1

𝑖=1

This establishes the lower bound. Further, if 𝛾(𝐺) = 𝑛 − 𝑚, then we have equality throughout Inequality (8.2), implying that 𝑘 = 𝑛 − 𝑚 and 𝛾(𝐺 𝑖 ) = 1 for all 𝑖 ∈ [𝑘]. This in turn implies that we have equality throughout Inequality (8.1), and so 𝐺 𝑖 is a tree with a vertex adjacent to every other vertex in 𝐺 𝑖 for all 𝑖 ∈ [𝑘], that is, either © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_8

227

Chapter 8. Bounds in Terms of Size

228

𝑛𝑖 = 1 in which case 𝐺 𝑖 = 𝐾1 , or 𝑛𝑖 ≥ 2 in which case 𝐺 𝑖 = 𝐾1,𝑛𝑖 −1 for all 𝑖 ∈ [𝑘]. Thus, every component of 𝐺 is a star. Conversely, if every component of 𝐺 is a star, then 𝛾(𝐺) = 𝑛 − 𝑚. In 1965 Vizing [733] proved a classic result bounding the size of a graph in terms of its order and domination number. Theorem 8.2 (Vizing’s Theorem [733]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾, then 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). Proof We proceed by induction on the order 𝑛 of a graph of size 𝑚 with domination number 𝛾. The result is immediate if 𝑛 = 1, since in this case 𝑚 = 0 and 𝛾 = 1. This establishes the base case. Assume that 𝑛 ≥ 2 and that if 𝐺 ′ is a graph of order 𝑛′ and size 𝑚 ′ with domination number 𝛾 ′ , where 1 ≤ 𝑛′ < 𝑛, then 𝑚 ′ ≤ 12 (𝑛′ − 𝛾 ′ ) (𝑛′ − 𝛾 ′ + 2). Let 𝐺 be a graph of order 𝑛 and size 𝑚 with domination number 𝛾. Let Δ = Δ(𝐺).
If 𝛾 = 1, then 𝑚 ≤ 𝑛2 < 12 (𝑛 − 1)(𝑛 + 1) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). Hence, we may assume that 𝛾 ≥ 2. If 𝐺 is the empty graph 𝐾 𝑛 , then 𝛾 = 𝑛 and 𝑚 = 0 = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2)/2. Hence, we may further assume that 𝑚 ≥ 1, and so Δ ≥ 1. By Theorem 4.3, we have 𝛾 ≤ 𝑛 − Δ. Let 𝑣 be a vertex of 𝐺 of maximum degree Δ, and so deg(𝑣) = |N(𝑣)| = Δ ≤ 𝑛 − 𝛾. Thus, |N(𝑣)| = 𝑛 − 𝛾 − 𝑘 for some integer 𝑘, where 0 ≤ 𝑘 ≤ 𝑛 − 𝛾 −1. Let 𝑆 = 𝑉 \N[𝑣], and so
|𝑆| = 𝛾 + 𝑘 −1. Let 𝑢 be an arbitrary neighbor of 𝑣, and consider the set 𝑆𝑢 = 𝑆 \N(𝑢) ∪ {𝑢, 𝑣}. The set 𝑆𝑢 is a dominating set of 𝐺 and so, 𝛾 ≤ |𝑆𝑢 | = |𝑆| − |𝑆 ∩ N(𝑢)| + 2 = (𝛾 + 𝑘 − 1) − |𝑆 ∩ N(𝑢)| + 2, or equivalently, |𝑆 ∩ N(𝑢)| ≤ 𝑘 + 1

for all 𝑢 ∈ N(𝑣).

(8.3)

Let 𝑚 1 be the number of edges between N(𝑣) and 𝑆, and so, 𝑚 1 = | [N(𝑣)]𝑆|. By Inequality (8.3), 𝑚 1 ≤ Δ(𝑘 + 1).

(8.4)

Let 𝐺 ′ = 𝐺 [𝑆] and let 𝐷 ′ be a 𝛾-set of 𝐺 ′ . Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and domination number 𝛾 ′ . Thus, 𝑛′ = |𝑆| = 𝛾 + 𝑘 − 1. The set 𝐷 ′ ∪ {𝑣} is a dominating set of 𝐺 and so, 𝛾 ≤ |𝐷 ′ | + 1. Thus, 𝛾 ′ = |𝐷 ′ | ≥ 𝛾 − 1 ≥ 1. We note that 𝑛′ − 𝛾 ′ ≤ (𝛾 + 𝑘 − 1) − (𝛾 − 1) = 𝑘. Applying the inductive hypothesis to the graph 𝐺 ′ , we have 𝑚 ′ ≤ 12 (𝑛′ − 𝛾 ′ ) (𝑛′ − 𝛾 ′ + 2) ≤ 12 𝑘 (𝑘 + 2).

(8.5)

Recall that deg(𝑣) = Δ and 𝑘 = 𝑛 − 𝛾 − |N(𝑣)| = 𝑛 − 𝛾 − Δ. We note that |N[𝑣] | = Δ+1 and each vertex in N[𝑣] has degree at most Δ in 𝐺. By Inequalities (8.4) and (8.5),

Section 8.2. Domination and Size 2𝑚 =

∑︁

deg(𝑢) +

𝑢∈N[𝑣 ]

∑︁

229 deg(𝑢)

𝑢∈𝑆 ′

≤ Δ(Δ + 1) + (2𝑚 + 𝑚 1 ) ≤ Δ(Δ + 1) + 𝑘 (𝑘 + 2) + Δ(𝑘 + 1) = Δ(Δ + 1) + (𝑛 − 𝛾 − Δ) (𝑛 − 𝛾 − Δ + 2) + Δ(𝑛 − 𝛾 − Δ + 1) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − Δ(𝑛 − 𝛾 − Δ) ≤ (𝑛 − 𝛾) (𝑛 − 𝛾 + 2), or equivalently, 𝑚 ≤ 21 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). As shown in the proof of Theorem 8.2, if 𝐺 is a graph of order 𝑛 and size 𝑚 with
domination number 𝛾 = 1, then 𝑛2 < 12 (𝑛 − 1)(𝑛 + 1) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). However, if 𝛾 ≥ 2, then Vizing [733] showed that the bound is achievable and constructed the following family ofgraphs 𝐻𝑛,𝛾 of order𝑛 and size 𝑚 with domination number 𝛾 ≥ 2 that satisfy 𝑚 = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) . If 𝛾 = 2, then let 𝐻𝑛,2 be the graph obtained as follows. For 𝑛 even, let 𝐻𝑛,2 be obtained from a complete graph 𝐾𝑛 by removing the edges of a perfect matching. For 𝑛 ≥ 3 odd, let 𝐻𝑛,2 be obtained from a complete graph 𝐾𝑛 by deleting a maximum matching (of size (𝑛 − 1)/2) and an additional edge incident to the remaining vertex, which is not incident to an edge of the removed matching, that is, 𝐻𝑛,2 is obtained from 𝐾𝑛 by
removing from it the edges of a spanning subgraph isomorphic to 𝑃3 ∪ 𝑛−1 2 𝑃2 . Equivalently, for 𝑛 ≥ 2 the graph 𝐻 𝑛,2 is obtained from a complete graph 𝐾𝑛 by removing a minimum edge cover. The graphs 𝐻8,2 and 𝐻7,2 , for example, are shown in Figure 8.1(a) and (b), respectively.

(a) 𝐻8,2

(b) 𝐻7,2

Figure 8.1 The graphs 𝐻8,2 and 𝐻7,2 For all 𝑛 ≥ 2, the graph 𝐻𝑛,2 has size         𝑛 𝑛 𝑛(𝑛 − 2) (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) 𝑚(𝐻𝑛,2 ) = − = = . 2 2 2 2 For 𝛾 > 2, let 𝐻𝑛,𝛾 = 𝐻𝑛−𝛾+2,2 ∪ (𝛾 − 2)𝐾1 , that is, 𝐻𝑛,𝛾 is obtained from the disjoint union of 𝐻𝑛−𝛾+2,2 and 𝛾 − 2 copies of 𝐾1 . The graphs 𝐻11,5 = 𝐻8,2 ∪ 3𝐾1 and 𝐻10,5 = 𝐻7,2 ∪ 3𝐾1 , for example, are shown in Figure 8.2(a) and (b), respectively.

Chapter 8. Bounds in Terms of Size

230

(a) 𝐻11,5

(b) 𝐻10,5

Figure 8.2 The graphs 𝐻11,5 and 𝐻10,5

For 𝛾 > 2, the graph 𝐻𝑛,𝛾 has order 𝑛, domination number 𝛾, and size     𝑚(𝐻𝑛,𝛾 ) = 𝑚(𝐻𝑛−𝛾+2,2 ) = 12 (𝑛 − 𝛾 + 2 − 2)(𝑛 − 𝛾 + 2) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) . As a corollary of Vizing’s Theorem, we have the following upper bound for the domination number of a graph in terms of its order and size. Theorem 8.3 ([663]) If 𝐺 is a graph of order 𝑛 and size 𝑚, then √ 𝛾(𝐺) ≤ 𝑛 + 1 − 1 + 2𝑚. Proof Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾. By Theorem 8.2, we have (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − 2𝑚 ≥ 0, or equivalently, (𝑛 − 𝛾) 2 + 2(𝑛 − 𝛾) √ √ − 2𝑚 ≥ 0. Thus, since 𝑛 − 𝛾 ≥ 0, we have 𝑛 − 𝛾 ≥ −1 + 1 + 2𝑚, or 𝛾 ≤ 𝑛 + 1 − 1 + 2𝑚. For 𝛾 ≥ 2, the family of graphs 𝐻𝑛,𝛾 constructed by Vizing are disconnected and all have maximum degree Δ = 𝑛 − 𝛾, showing that the upper bound of Theorem 8.2 is tight when Δ = 𝑛 − 𝛾. However, for smaller maximum degrees, in 1991 Sanchis [663] showed that the bound of Theorem 8.2 can be improved slightly. Theorem 8.4 ([663]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 ≥ 2 and Δ(𝐺) = Δ satisfying Δ ≤ 𝑛 − 𝛾 − 1, then 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 1). Proof For 1 ≤ Δ ≤ 𝑛 − 𝛾 − 1, let 𝑓 (Δ) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − Δ(𝑛 − 𝛾 − Δ). As shown in the proof of Theorem 8.2, we have 2𝑚 ≤ 𝑓 (Δ). The parabolic function 𝑓 (Δ) = Δ2 − (𝑛 − 𝛾)Δ + (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) achieves its maximum value at one of its endpoints, namely Δ = 1 or Δ = 𝑛 − 𝛾 − 1. Since 𝑓 (1) = 𝑓 (𝑛 − 𝛾 − 1) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 1) + 1, which is odd, and since 𝑚 ≤ 12 𝑓 (Δ) , it follows that 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 1). Vizing’s result in Theorem 8.2 was generalized in 1994 by Fulman [316] and others. For 𝛾 ≥ 2, the family of graphs 𝐻𝑛,𝛾 constructed by Vizing that achieve equality in the upper bound of Theorem 8.2 have two unusual properties. The first property is that they are disconnected, and the second property is that their edges are unevenly distributed in the sense that if 𝐺 is a graph in the family, then Δ(𝐺) = 0 while Δ(𝐺) = 𝑛 − 𝛾(𝐺). In 1991 Sanchis [663, 665] showed that if we restrict the

Section 8.2. Domination and Size

231

graph to being connected, then the bounds can be improved. However, the family of graphs that achieve the improved bound of Sanchis still have the second property that edges are unevenly distributed. In 1999 Rautenbach [649] showed that the square dependence on 𝑛 and 𝛾(𝐺) in Vizing’s result in Theorem 8.2 can be improved to a linear dependence on 𝑛, 𝛾(𝐺), and Δ(𝐺). For the remainder of this section, let Δ ≥ 3 be a given fixed integer, where Δ is not necessarily equal to the maximum degree Δ(𝐺). Theorem 8.5 ([649]) If 𝐺 is an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 and Δ(𝐺) ≤ Δ, where Δ ≥ 3, then 𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾.

(8.6)

Proof We proceed by induction on the order 𝑛 ≥ 2. If 𝑛 = 2, then 𝐺 = 𝐾2 and 𝑚 = 𝛾 = 1, and Δ𝑛 − (Δ + 1)𝛾 = 2Δ − (Δ + 1) = Δ − 1 ≥ 2 > 𝑚. This establishes the base case. Let 𝑛 ≥ 3 and suppose that the result is true for all isolate-free graphs of order less than 𝑛. Let 𝐺 be an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾, 𝛿(𝐺) = 𝛿, and Δ(𝐺) ≤ Δ, where Δ ≥ 3. If 𝐺 is disconnected with components 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 , where 𝑘 ≥ 2 and 𝐺 𝑖 has order 𝑛𝑖 , size 𝑚 𝑖 , and domination number 𝛾𝑖 for 𝑖 ∈ [𝑘], then applying the inductive hypothesis to each component, we have 𝑚=

𝑘 ∑︁ 𝑖=1

𝑚𝑖 ≤

𝑘 ∑︁


Δ𝑛𝑖 − (Δ + 1)𝛾𝑖 = Δ𝑛 − (Δ + 1)𝛾.

𝑖=1

Hence, since Inequality (8.6) is linear in 𝑚, 𝑛, and 𝛾, the result is immediate if 𝐺 is disconnected. Thus, we may assume that 𝐺 is connected. If Δ(𝐺) ≤ 2, then 𝐺 is a cycle or a path of order   at least 2. Thus, 𝑚 = 𝑛 if 𝐺 is a cycle or 𝑚 = 𝑛 − 1 is 𝐺 is a path. Further, 𝛾 = 31 𝑛 . Therefore,     𝑛  2𝑛   𝑛  Δ𝑛 − (Δ + 1)𝛾 = Δ𝑛 − (Δ + 1) 𝑛3 = Δ 2𝑛 3 − 3 ≥ 3 3 − 3 ≥ 𝑛 ≥ 𝑚, with strict inequality if 𝑛 ≠ 4 or Δ ≠ 3. Hence, if Δ(𝐺) ≤ 2, then Inequality (8.6) holds, and we have equality if and only if 𝐺 = 𝐶4 and Δ = 3. Hence, we may assume that Δ(𝐺) ≥ 3. Since Δ𝑛 − (Δ + 1)𝛾 is monotonically increasing in Δ, we can assume that Δ(𝐺) = Δ ≥ 3. (8.7) Let 𝑤 be a vertex of 𝐺 of minimum degree 𝛿 in 𝐺 and let 𝑣 be an arbitrary neighbor of 𝑤. Let 𝑉0 be the set of isolated vertices in 𝐺 − N[𝑣], let 𝑉1 be the set of vertices that are not isolated in 𝐺 − N[𝑣], and let 𝑉2 = N[𝑣] ∪ 𝑉0 . For 𝑖 ∈ [2], let 𝐺 𝑖 = 𝐺 [𝑉𝑖 ], where 𝐺 𝑖 has order 𝑛𝑖 , size 𝑚 𝑖 , and domination number 𝛾𝑖 . By construction, both 𝐺 1 and 𝐺 2 are isolate-free graphs. We note that Δ(𝐺 𝑖 ) ≤ Δ(𝐺) = Δ for 𝑖 ∈ [2]. Further, 𝑉 = 𝑉1 ∪𝑉2 , 𝑛 = 𝑛1 + 𝑛2 , and 𝛾 ≤ 𝛾1 + 𝛾2 . Let |𝑉0 | = 𝑛0 , and so 𝑛2 = deg𝐺 (𝑣) + 𝑛0 + 1. Thus, 𝑛1 = 𝑛 − 𝑛2 = 𝑛 − deg𝐺 (𝑣) − 𝑛0 − 1 and 𝛾1 ≥ 𝛾 − 𝛾2 . Applying the inductive hypothesis to 𝐺 1 , we have
𝑚 1 ≤ Δ𝑛1 − (Δ + 1)𝛾1 ≤ Δ 𝑛 − deg𝐺 (𝑣) − 𝑛0 − 1 − (Δ + 1) (𝛾 − 𝛾2 ). (8.8)

Chapter 8. Bounds in Terms of Size

232

Let 𝑚★ be the number of edges incident with a vertex in N(𝑣). By construction, every edge of 𝐺 belongs to 𝐺 1 or is incident with a vertex in N(𝑣), and so 𝑚 = 𝑚 1 +𝑚★. Thus, by Inequality (8.8),
𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾 − Δ deg𝐺 (𝑣) + 𝑛0 + 1 + (Δ + 1)𝛾2 + 𝑚★ .

(8.9)

Therefore, if
𝑚★ + (Δ + 1)𝛾2 ≤ Δ deg𝐺 (𝑣) + 𝑛0 + 1 ,

(8.10)

then Inequality (8.6) follows from Inequality (8.9). Since the neighbor 𝑤 of the vertex 𝑣 has degree 𝛿 and each of the remaining deg𝐺 (𝑣) − 1 neighbors of 𝑣 has degree at most Δ,
(8.11) 𝑚★ ≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 . Thus, by Inequalities (8.9) and (8.11), 𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾 − 2Δ + 𝛿 − Δ𝑛0 + (Δ + 1)𝛾2 .

(8.12)

𝛿 + (Δ + 1)𝛾2 ≤ Δ(𝑛0 + 2),

(8.13)

Therefore, if

then Inequality (8.6) follows from Inequality (8.12). The set 𝑉0 ∪ {𝑣} is a dominating set of 𝐺 2 , implying that 𝛾2 ≤ 𝑛0 + 1 and therefore 𝛿 + (Δ + 1)𝛾2 ≤ 𝛿 + (Δ + 1) (𝑛0 + 1) = Δ(𝑛0 + 2) + (𝑛0 + 1 − Δ + 𝛿). Hence, if 𝑛0 ≤ Δ − 𝛿 − 1, then Inequality (8.13) holds, yielding Inequality (8.6). Therefore, we may assume that 𝑛0 ≥ Δ − 𝛿. Moreover, the set N(𝑣) is a dominating set of 𝐺 2 , implying that 𝛾2 ≤ deg𝐺 (𝑣) and therefore 𝛿 + (Δ + 1)𝛾2 ≤ Δ(𝑛0 + 2) + 𝛿 + (Δ + 1) deg𝐺 (𝑣) − 2Δ − Δ𝑛0 . Hence, if Δ𝑛0 ≥ (Δ+1) deg𝐺 (𝑣) + 𝛿 −2Δ, then Inequality (8.13) holds, once again yielding Inequality (8.6). Therefore, we may assume that Δ𝑛0 < (Δ + 1) deg𝐺 (𝑣) + 𝛿 − 2Δ. With these assumptions, and noting that deg𝐺 (𝑣) + 𝛿 − 2Δ ≤ 𝛿 − Δ ≤ 0, we have Δ − 𝛿 ≤ 𝑛0
1.15 × 1026 by proving the following result. Theorem 8.8 ([764]) For every Δ ≥ 3, there exists a bipartite Δ-regular graph 𝐺 of order 𝑛 with 𝛾(𝐺) = 𝛾 satisfying   0.05 ln(Δ) 𝛾> (8.16) 𝑛. Δ We note that if 𝐺 is a bipartite Δ-regular graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 satisfying Inequality (8.16), then     Δ + 0.05 ln(Δ) Δ 0.05 ln(Δ) 1 𝑚 = 2 Δ𝑛 = 𝑛− 𝑛 2 2 Δ   Δ + 0.05 ln(Δ) Δ > 𝑛− 𝛾 2 2     Δ + 0.05 ln(Δ) Δ+5 > 𝑛− 𝛾. 2 2 Hence, as an immediate consequence of Theorem 8.8, we have the following result. Corollary 8.9 ([764]) For every Δ ≥ 3, there exists a bipartite Δ-regular graph 𝐺 of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 satisfying     Δ + 0.05 ln(Δ) Δ+5 𝑚> 𝑛− 𝛾. 2 2 Corollary 8.9 disproves Conjecture 8.7 when 0.05 ln(Δ) > 3, which happens when Δ > 1.15 × 1026 . This gives rise to the following open problem. Problem 8.10 For each Δ ≥ 3, find the smallest value 𝑐 Δ such that for every isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 and Δ(𝐺) ≤ Δ,     Δ + 𝑐Δ Δ + 𝑐Δ + 2 𝑚≤ 𝑛− 𝛾. 2 2

Chapter 8. Bounds in Terms of Size

238

Theorem 8.5, due to Rautenbach, implies that 𝑐 Δ ≤ Δ. Theorem 8.8, due to Yeo, implies that 𝑐 Δ > 0.05 ln(Δ). We state this formally as follows. Theorem 8.11 ([649, 764]) If Δ ≥ 3, then 0.05 ln(Δ) < 𝑐 Δ ≤ Δ. However, it remains an open problem to determine if 𝑐 Δ grows proportionally with ln(Δ) or some completely different function.

8.3

Total Domination and Size

In this section, we relate the size and the total domination number of a graph of given order 𝑛. To present our first such result, we shall need the following definition and lemma. For integers 𝑛 and 𝑘, where 𝑛 ≥ 𝑘 ≥ 1, let ( 𝑓 (𝑛, 𝑘) =

𝑛−𝑘+2
2 𝑛−𝑘+1
2

+ +

𝑘 2 𝑘 2

−1

if 𝑘 is even

1 2

if 𝑘 is odd.

+

The following properties of 𝑓 (𝑛, 𝑘) are readily determined from elementary arithmetic. Lemma 8.12 ([215]) The following properties of 𝑓 (𝑛, 𝑘) hold: (a) If 1 ≤ 𝑘 1 ≤ 𝑛1 and 1 ≤ 𝑘 2 ≤ 𝑛2 , then 𝑓 (𝑛1 + 𝑛2 , 𝑘 1 + 𝑘 2 ) ≥ 𝑓 (𝑛1 , 𝑘 1 ) + 𝑓 (𝑛2 , 𝑘 2 ), unless 𝑘 1 = 𝑛1 and 𝑘 1 is odd, or 𝑘 2 = 𝑛2 and 𝑘 2 is odd. (b) If 1 ≤ 𝑘 1 ≤ 𝑘 2 ≤ 𝑛, then 𝑓 (𝑛, 𝑘 2 ) ≤ 𝑓 (𝑛, 𝑘 1 ). (c) If 2 ≤ 𝑘 ≤ 𝑛 − 1, then 𝑓 (𝑛 − 1, 𝑘) < 𝑓 (𝑛, 𝑘) < 𝑓 (𝑛, 𝑘 − 1). (d) If 1 ≤ 𝑘 ≤ 𝑛 and 0 ≤ ℓ ≤ 𝑘 − 1, then 𝑓 (𝑛 − ℓ, 𝑘 − ℓ) ≤ 𝑓 (𝑛, 𝑘 − 1). (e) If 3 ≤ 𝑘 ≤ 𝑛 − 2 and 𝑘 is odd, then 𝑓 (𝑛 − 1, 𝑘) + 𝑛 − 𝑘 = 𝑓 (𝑛, 𝑘) = 𝑓 (𝑘 − 1, 𝑘 − 2) + (𝑛 − 𝑘) (𝑛 − 𝑘 + 1)/2. In 2004 Dankelmann et al. [215] established the following Vizing-like relation between the size and the total domination number of a graph of given order. Theorem 8.13 ([215]) If 𝐺 is an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t ≥ 2, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof We proceed by induction on the order 𝑛 of an isolate-free graph of size 𝑚 with 𝛾t (𝐺) = 𝛾t ≥ 2. The result is immediate if 𝑛 = 2 since in this case 𝛾t = 2 and 𝑚 = 1 = 𝑓 (2, 2) = 𝑓 (𝑛, 𝛾t ). This establishes the base case. Let 𝑛 ≥ 3 and assume that if 𝐺 ′ is an isolate-free graph having order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ , where 2 ≤ 𝑛′ < 𝑛, then 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ). Let 𝐺 be an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t and Δ(𝐺) = Δ. If
𝑛 is even, then 𝑛 ≥ 𝛾t ≥ 2, while if 𝑛 is odd, then 𝑛 > 𝛾t ≥ 2. If 𝛾t = 2, then 𝑚 ≤ 𝑛2 = 𝑓 (𝑛, 2) = 𝑓 (𝑛, 𝛾t ). Hence, we may assume that 𝛾t ≥ 3, for otherwise the desired result follows. We proceed with the following series of claims. Claim 8.13.1 If 𝐺 is disconnected, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ).

Section 8.3. Total Domination and Size

239

Proof Suppose that 𝐺 is disconnected. Let 𝐺 1 be a component of 𝐺 and let 𝐺 2 = 𝐺 −𝑉 (𝐺 1 ). Although 𝐺 2 may be disconnected, both 𝐺 1 and 𝐺 2 are isolate-free graphs. Let 𝐺 𝑖 have order 𝑛𝑖 and size 𝑚 𝑖 with total domination number 𝛾t𝑖 for 𝑖 ∈ [2]. Applying the inductive hypothesis to 𝐺 𝑖 , we have 𝑚 𝑖 ≤ 𝑓 (𝑛𝑖 , 𝛾t𝑖 ) for 𝑖 ∈ [2]. We note that a graph of odd order has total domination number strictly less than its order. Thus, if 𝑛𝑖 is odd, then 𝛾t𝑖 < 𝑛𝑖 for 𝑖 ∈ [2]. Hence, by Lemma 8.12(a), 𝑚 = 𝑚 1 + 𝑚 2 ≤ 𝑓 (𝑛1 , 𝛾t1 ) + 𝑓 (𝑛2 , 𝛾t2 ) ≤ 𝑓 (𝑛1 + 𝑛2 , 𝛾t1 + 𝛾t2 ) = 𝑓 (𝑛, 𝛾t ). By Claim 8.13.1, we may assume that 𝐺 is connected, for otherwise Theorem 8.13 holds. Recall that 𝛾t ≥ 3, implying that 𝑛 ≥ 4. By Theorem 6.40, we have 𝛾t ≤ 23 𝑛. In particular, we note that 𝛾t ≤ 𝑛 − 2. Let 𝑣 be a vertex of 𝐺 of maximum degree Δ and let 𝐴 = N(𝑣) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 Δ } be the set of neighbors of 𝑣. Let 𝐵 = 𝑉 \ N[𝑣]. Since 𝐺 is connected and 𝛾t ≥ 3, 𝐵 must be nonempty. Claim 8.13.2 deg 𝐵 (𝑣 𝑖 ) ≤ 𝑛 − Δ − 𝛾t + 1 for all 𝑖 ∈ [Δ]. Proof Let 𝑖 ∈ [Δ], and let 𝐵𝑖 be the set of all vertices in 𝐵 that are not dominated by the vertex 𝑣 𝑖 , that is, 𝐵𝑖 = 𝐵 \ N 𝐵 (𝑣 𝑖 ). Let 𝐷 𝑖 = {𝑣, 𝑣 𝑖 } ∪ 𝐵𝑖 . If the graph 𝐺 [𝐷 𝑖 ] has no isolated vertices, then let 𝑆𝑖 = 𝐷 𝑖 . If the graph 𝐺 [𝐷 𝑖 ] contains an isolated vertex,
then the neighbors of such an isolated vertex belong to the set 𝐴 \ {𝑣 𝑖 } ∪ N 𝐵 (𝑣 𝑖 ). In this case, we replace each such isolated vertex in 𝐷 𝑖 with one of its neighbors, and let 𝑆𝑖 denote the resulting set. In both cases, the set 𝑆𝑖 is a TD-set of 𝐺 and 𝛾t ≤ |𝑆𝑖 | ≤ |𝐷 𝑖 | = 𝑛 − Δ + 1 − deg 𝐵 (𝑣 𝑖 ). Claim 8.13.3 Δ ≤ 𝑛 − 𝛾t . Proof By our earlier assumptions, 𝐺 is connected and 𝛾t ≥ 3, which implies that 𝐵 ≠ ∅ and at least one vertex in 𝐴 has a neighbor in 𝐵. Renaming vertices if necessary, we may assume that deg 𝐵 (𝑣 1 ) ≥ 1. By Claim 8.13.2, we have 𝛾t ≤ 𝑛 − Δ + 1 − deg 𝐵 (𝑣 1 ) ≤ 𝑛 − Δ.
Our next claim gives an upper bound on the size 𝑚 𝐺 [𝐵] of the graph 𝐺 [𝐵].
Claim 8.13.4 𝑚 𝐺 [𝐵] ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2). Proof Let 𝐼 𝐵 be the set of isolated vertices in 𝐺 [𝐵] and let |𝐼 𝐵 | = ℓ. If 𝐼 𝐵 is a proper subset of 𝐵, then let 𝐺 ′ = 𝐺 [𝐵\𝐼 𝐵 ]. Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Let 𝑆 ′ be a 𝛾t -set of 𝐺 ′ and so 𝛾t′ = |𝑆 ′ |. Applying the inductive hypothesis to 𝐺 ′ , we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ). If ℓ = 0, then 𝑛′ = 𝑛 − Δ − 1 and the set 𝑆 ′ ∪ {𝑣, 𝑣 1 } is a TD-set of 𝐺, implying that 𝛾t ≤ |𝑆 ′ | + 2 = 𝛾t′ + 2. Therefore, 1 ≤ 𝛾t − 2 ≤ 𝛾t′ ≤ 𝑛′ . Thus, by Lemma 8.12(b), we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2). Hence, we may assume that ℓ ≥ 1. Let 𝑢 ∈ 𝐼 𝐵 be an isolated vertex in 𝐺 [𝐵] having at least one neighbor in 𝐴. Renaming vertices if necessary, we may assume that 𝑢𝑣 1 is an edge. Let 𝑆 =
𝑆 ′ ∪ {𝑣, 𝑣 1 } ∪ 𝐼 𝐵 \ {𝑢} . If the graph 𝐺 [𝑆] has no isolated vertices, then let 𝐷 = 𝑆. If the graph 𝐺 [𝑆] has an isolated vertex, then such a vertex belongs to 𝐼 𝐵 and its neighbors belong to the set 𝐴 \ {𝑣 1 }. In this case, we replace each such isolated vertex

Chapter 8. Bounds in Terms of Size

240

in 𝑆 with one of its neighbors and let 𝐷 denote the resulting set. In both cases, the set 𝐷 is a TD-set of 𝐺 and 𝛾t ≤ |𝐷| ≤ |𝑆| = |𝑆 ′ | + 2 + (ℓ − 1) = 𝛾t′ + ℓ + 1 and so 𝛾t − ℓ − 1 ≤ 𝛾t′ . We note that 𝑛′ = 𝑛 − Δ − 1 − ℓ. Thus, by Lemma 8.12(b) and (d),
𝑚 𝐺 [𝐵] = 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛 − Δ − 1 − ℓ, 𝛾t − 1 − ℓ) ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2), which yields the desired inequality, noting that 𝛾t − 2 ≥ 1 and the definition of 𝑓 (𝑛, 𝑘) also applies for 𝑘 = 1. By Claim 8.13.2, 2𝑚 = deg𝐺 (𝑣) +

Δ ∑︁

deg𝐺 (𝑣 𝑖 ) + | [ 𝐴, 𝐵] | + 2𝑚 𝐺 [𝐵]




𝑖=1

≤ Δ + Δ2 +

Δ ∑︁

deg 𝐵 (𝑣 𝑖 ) + 2𝑚 𝐺 [𝐵]




𝑖=1


≤ Δ + Δ + Δ(𝑛 − Δ − 𝛾t + 1) + 2𝑚 𝐺 [𝐵] , 2

or equivalently,
2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 2𝑚 𝐺 [𝐵] .

(8.17)

By Claim 8.13.4 and Inequality (8.17), 2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 2 𝑓 (𝑛 − Δ − 1, 𝛾t − 2).

(8.18)

By Claim 8.13.3 and our earlier observations, 2 ≤ Δ ≤ 𝑛 − 𝛾t . Claim 8.13.5 If 𝛾t is even, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that 𝛾t is even. The right-hand side of Inequality (8.18) is a parabola as a function of Δ. Since the second derivative of this function is positive, the function is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t . The extremum is 2 𝑓 (𝑛, 𝛾t ), which occurs at Δ = 2, unless 𝛾t ≥ 𝑛 − 1. However, by our earlier observations, we have 𝛾t ≤ 𝑛 − 2, and so 𝛾t ≥ 𝑛 − 1 is not possible. Hence, by Inequality (8.18), we have 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ), or equivalently, 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.5, we may assume that 𝛾t is odd, for otherwise Theorem 8.13 holds. Claim 8.13.6 If Δ ≤ 𝑛 − 𝛾t − 1, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that Δ ≤ 𝑛−𝛾t −1. As in the proof of Claim 8.13.5, the right-hand side of Inequality (8.18) is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t − 1. The extremum at these two values is the same value, namely 2 𝑓 (𝑛, 𝛾t ) + 2. Hence, by Inequality (8.18), we have 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ) + 2, that is, 𝑚 ≤ 𝑓 (𝑛, 𝛾t ) + 1. Suppose that 𝑚 = 𝑓 (𝑛, 𝛾t ) + 1. With this assumption, we must have equality in Inequality (8.18), which in turn implies that we have equality in Claim 8.13.2, that is, deg 𝐵 (𝑣 𝑖 ) = 𝑛 − Δ − 𝛾t + 1 for all 𝑖 ∈ [Δ]. Thus, the proof of Claim 8.13.2 implies that the number of vertices needed to totally dominate the set 𝐵1 = 𝐵 \ N(𝑣 1 )

Section 8.3. Total Domination and Size

241

equals |𝐵1 |. Therefore, 𝐺 [𝐵1 ] is the disjoint union of copies of 𝐾2 . Further, every vertex in N
𝐵 (𝑣 1 ) has at most one neighbor in 𝐵1 . Hence, the graph 𝐺 [𝐵1 ] has size 𝑚 𝐺 [𝐵1 ] = 21 |𝐵1 | and | [𝐵1 , N 𝐵 (𝑣 1 )] | ≤ |N𝐺 (𝑣 1 )| = deg 𝐵 (𝑣 1 ). Therefore,  
deg 𝐵 (𝑣 1 ) 2𝑚 𝐺 [𝐵] ≤ |𝐵1 | + 2 deg 𝐵 (𝑣 1 ) + 2 2
= 𝛾t − 2 + deg 𝐵 (𝑣 1 ) deg 𝐵 (𝑣 1 ) + 1 = 𝛾t − 2 + (𝑛 − Δ − 𝛾t + 1) (𝑛 − Δ − 𝛾t + 2). By Inequality (8.17), 2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 𝛾t − 2 + (𝑛 − Δ − 𝛾t + 1) (𝑛 − Δ − 𝛾t + 2).

(8.19)

Calculus arguments show that the right-hand side of Inequality (8.19) is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t − 1. The extremum at these two values is the same value, namely 2 𝑓 (𝑛, 𝛾t ) + 1. Thus, 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ) + 1, implying that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ), a contradiction. Hence, our assumption that 𝑚 = 𝑓 (𝑛, 𝛾t ) + 1 is incorrect, implying that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.6, we may assume that Δ = 𝑛 − 𝛾t , for otherwise Theorem 8.13 holds. Claim 8.13.7 If there exists a pair of vertices 𝑥 and 𝑦 with N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥} such that the graph 𝐺 − 𝑦 has no isolated vertices, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Let 𝑥 and 𝑦 be vertices of 𝐺 such that N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥}. Let 𝐺 ′ = 𝐺 − 𝑦 and suppose, to the contrary, that 𝐺 ′ has no isolated vertices. Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Since every TD-set of 𝐺 ′ is a TD-set of 𝐺, 3 ≤ 𝛾t ≤ 𝛾t′ ≤ 𝑛′ ≤ 𝑛 − 1. By Lemma 8.12(b) and (c), we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛′ , 𝛾t ) ≤ 𝑓 (𝑛 − 1, 𝛾t ). Recall that 𝛾t is odd. By Lemma 8.12(d), 𝑚 = 𝑚 ′ + deg𝐺 (𝑦) ≤ 𝑓 (𝑛 − 1, 𝛾t ) + Δ = 𝑓 (𝑛 − 1, 𝛾t ) + 𝑛 − 𝛾t = 𝑓 (𝑛, 𝛾t ), which yields the desired inequality. By Claim 8.13.7, if 𝑥 and 𝑦 are distinct vertices of 𝐺 with N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥}, then we may assume that the graph 𝐺 − 𝑦 contains isolated vertices, for otherwise the desired result holds. Claim 8.13.8 If the graph 𝐺 − 𝑣 has an isolated vertex, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that 𝐺 − 𝑣 has an isolated vertex. Renaming vertices in 𝐴 if necessary, we may assume that 𝑣 1 is isolated in 𝐺 − 𝑣 and deg𝐺 (𝑣 1 ) = 1. Since we may
assume by Claim 8.13.6 that Δ = 𝑛 − 𝛾t , by Claim 8.13.4, we have 𝑚 𝐺 [𝐵] ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2) = 𝑓 (𝛾t − 1, 𝛾t − 2). Returning to our derivation of Inequality (8.17), we now have by Lemma 8.12(e) that

Chapter 8. Bounds in Terms of Size

242 2𝑚 = deg𝐺 (𝑣) + deg𝐺 (𝑣 1 ) +

Δ ∑︁

deg𝐺 (𝑣 𝑖 ) + | [ 𝐴, 𝐵] | + 2𝑚 𝐺 [𝐵]




𝑖=2 Δ ∑︁

deg 𝐵 (𝑣 𝑖 ) + 2𝑚 𝐺 [𝐵]




≤ Δ + 1 + (Δ − 1) (𝑛 − Δ − 𝛾t + 1) + 2𝑚 𝐺 [𝐵]
≤ (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) + 1 + 2𝑚 𝐺 [𝐵] ≤ (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) + 1 + 2 𝑓 (𝛾t − 1, 𝛾t − 2) = 2 𝑓 (𝑛, 𝛾t ) + 1.




≤ Δ + 1 + (Δ − 1)Δ +

𝑖=2 2

Since 𝑓 (𝑛, 𝛾t ) is an integer, it follows that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.8, we may assume that the graph 𝐺 − 𝑣 has no isolated vertices. Claim 8.13.9 The following hold: (a) Each vertex in 𝐴 has exactly one neighbor in 𝐵. (b) | [ 𝐴, 𝐵] | = Δ. Proof By our earlier assumptions, the graph 𝐺 − 𝑣 has no isolated vertices. If there is a vertex 𝑣 𝑖 ∈ 𝐴 such that N(𝑣 𝑖 )\{𝑣} ⊆ 𝐴, then by Claim 8.13.7, we have 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Hence, we may assume that no neighbor of 𝑣 has all its other neighbors in 𝐴. With this assumption, every neighbor of 𝑣 has a neighbor in 𝐵, and so deg 𝐵 (𝑣 𝑖 ) ≥ 1 for all 𝑖 ∈ [Δ]. By Claim 8.13.2 and our earlier assumption that Δ = 𝑛 − 𝛾t , we have deg 𝐵 (𝑣 𝑖 ) ≤ 𝑛 − Δ − 𝛾t + 1 = 1 for all 𝑖 ∈ [Δ]. Consequently, deg 𝐵 (𝑣 𝑖 ) = 1 for all 𝑖 ∈ [Δ]. This proves part (a). Part (b) is an immediate consequence of part (a). Let 𝐺 ′ = 𝐺 [𝐵] and let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . We note that 𝑛′ = |𝐵| = 𝑛 − Δ − 1 = 𝛾t − 1. By our earlier assumption, 𝛾t is odd and so 𝑛′ is even. Claim 8.13.10 The following hold: (a) Every component of 𝐺 ′ is a 𝐾2 . (b) 𝑚 ′ = 12 𝑛′ = 12 (𝛾t − 1). (c) Every vertex in 𝐵 has at least one neighbor in 𝐴. Proof (a) Since the graph 𝐺 − 𝑣 is isolate-free, no vertex in 𝐵 has all of its neighbors in 𝐴. Thus, every vertex in 𝐵 has at least one neighbor in 𝐵, and so the graph 𝐺 [𝐵] is isolate-free. If 𝛾t′ ≤ 𝑛′ − 2, then since every 𝛾t -set of 𝐺 ′ can be extended to a TD-set of 𝐺 by adding the vertices 𝑣 and 𝑣 1 to it, this would imply that 𝑛′ + 1 = 𝛾t ≤ 𝛾t′ + 2 ≤ 𝑛′ , a contradiction. Hence, 𝛾t′ ≥ 𝑛′ − 1, implying that every component of 𝐺 [𝐵] is a 𝐾2 , except possibly for one component which is either a 𝑃3 or a 𝐾3 . However, 𝑛′ is even, implying that every component of 𝐺 [𝐵] is a 𝐾2 . This proves (a). (b) By part (a), we have 𝑚 ′ = 12 𝑛′ = 12 (𝛾t − 1). (c) Suppose, to the contrary, that there is a vertex 𝑥 ∈ 𝐵 with no neighbor in 𝐴. By part (a), every component of 𝐺 [𝐵] is a 𝐾2 . Let 𝑥 ′ be the neighbor of 𝑥 in 𝐵. Since 𝐺 is connected, the vertex 𝑥 ′ has a neighbor 𝑦 ∈ 𝐴. Thus, 𝑦 = 𝑣 𝑖 for some 𝑖 ∈ [Δ].

Section 8.3. Total Domination and Size

243

We note that N(𝑥) = {𝑥 ′ } ⊂ N(𝑦). However, the graph 𝐺 − 𝑦 does not contain an isolated vertex, contradicting our earlier assumptions. Hence, every vertex in 𝐵 has a neighbor in 𝐴.

Claim 8.13.11 𝑚 𝐺 [ 𝐴] ≤ Δ2 − (Δ − 1). Proof By Claim 8.13.9(a), every vertex in 𝐴 has exactly one neighbor in 𝐵, and by Claim 8.13.10(c), every vertex in 𝐵 has at least one neighbor in 𝐴. Let 𝐵 = {𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑛′ }, where we note that 𝑛′ ≥ 2. Let 𝑁𝑖 = N 𝐴 (𝑤 𝑖 ) for 𝑖 ∈ [𝑛′ ]. By our earlier observations, {𝑁1 , 𝑁2 , . . . , 𝑁 𝑛′ } is a partition of the set 𝐴. Let 𝐷 be a set of vertices in 𝐴 that contains exactly one vertex from each of the sets 𝑁𝑖 , where 𝑖 ∈ [𝑛′ ]. We note that |𝐷| = |𝐵| = 𝑛′ = 𝛾t − 1, implying that the set 𝐷 is not a TD-set of 𝐺. By construction of the set 𝐷, every vertex in 𝐵 ∪ {𝑣} is totally dominated by the set 𝐷. Hence, there exists a vertex 𝑧 ∈ 𝐴 that is not totally dominated by 𝐷, where possibly, 𝑧 ∈ 𝐷. Let 𝑦 𝑖 and 𝑦 𝑗 be an arbitrary pair of vertices in 𝑁𝑖 and 𝑁 𝑗 , respectively, where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛′ . Consider the subgraph 𝐺 [ 𝐴] of the complement 𝐺 of 𝐺 induced by the set 𝐴. If 𝑦 𝑖 and 𝑦 𝑗 are not adjacent in 𝐺, then 𝑦 𝑖 and 𝑦 𝑗 are adjacent in the complement 𝐺 [ 𝐴]. If 𝑦 𝑖 and 𝑦 𝑗 are adjacent in 𝐺, then 𝑧 is distinct from 𝑦 𝑖 and 𝑦 𝑗 , implying that 𝑦 𝑖 𝑧 𝑦 𝑗 is a path in the complement 𝐺 [ 𝐴]. In both cases, 𝑦 𝑖 and 𝑦 𝑗 are connected in 𝐺 [ 𝐴]. Since 𝑦 𝑖 and 𝑦 𝑗 are arbitrary vertices in 𝑁𝑖 and 𝑁 𝑗 , respectively, where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛′ , and since 𝑛′ ≥ 2, this implies that the subgraph 𝐺 [ 𝐴] is connected. Hence, 𝐺 [ 𝐴] contains at least | 𝐴| − 1 =
Δ −
1 edges, and therefore 𝐺 [ 𝐴] has at least Δ − 1 missing edges, that is, 𝑚 𝐺 [ 𝐴] ≤ Δ2 − (Δ − 1).
By Claim 8.13.10(b), we have 𝑚 𝐺 [𝐵] = 𝑚 ′ = 12 (𝛾t − 1). By our earlier assumption that Δ = 𝑛 − 𝛾t and by Claims 8.13.4, 8.13.9(b), and 8.13.11,

𝑚 = deg𝐺 (𝑣) + 𝑚 𝐺 [ 𝐴] + | [ 𝐴, 𝐵] | + 𝑚 𝐺 [𝐵]
≤ 𝛿 + 2𝛿 − (𝛿 − 1) + 𝛿 + 21 (𝛾t − 1) 𝛿(𝛿 + 1) 𝛾t 1 = + + 2 2 2 (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) 𝛾t 1 = + + 2 2 2 = 𝑓 (𝑛, 𝛾t ). The bound in Theorem 8.13 is tight, as may be seen by considering the graph 𝐺 (𝑛, 𝛾t ) defined as follows. For 𝛾t ≥ 2 even and 𝑛 ≥ 𝛾t , let 𝐺 (𝑛, 𝛾t ) be the disjoint union of 𝐾𝑛−𝛾t +2 and 12 (𝛾t − 2) copies of 𝐾2 . For 𝛾t ≥ 3 odd and 𝑛 ≥ 𝛾t + 1, let 𝐺 (𝑛, 𝛾t ) be the graph obtained from 𝐺 (𝑛 − 2, 𝛾t − 1) by subdividing one edge of the component isomorphic to 𝐾𝑛−𝛾t +1 twice. The graphs 𝐺 (11, 8) and 𝐺 (13, 9), for example, are illustrated in Figure 8.4(a) and (b), respectively. In the case where 𝛾t ≥ 2 is even, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size   𝑛 − 𝛾t + 2 𝛾t 𝑚= + − 1 = 𝑓 (𝑛, 𝛾t ). 2 2

Chapter 8. Bounds in Terms of Size

244

(a) 𝐺 (11, 8)

(b) 𝐺 (13, 9)

Figure 8.4 The graphs 𝐺 (11, 8) and 𝐺 (13, 9)

In the case where 𝛾t ≥ 3 is even, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size     𝑛 − 𝛾t + 1 𝛾t − 3 𝑛 − 𝛾t + 1 𝛾t 1 𝑚= +2+ = + + = 𝑓 (𝑛, 𝛾t ). 2 2 2 2 2 Thus, in both cases, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size 𝑚 = 𝑓 (𝑛, 𝛾t ). The bound in Theorem 8.13 is therefore tight. For 𝛾t ≥ 2, the extremal graphs 𝐺 (𝑛, 𝛾t ) that achieve equality in the bound have two unusual properties, which is similar to the situation that arose with the extremal graphs 𝐻𝑛,𝛾 that achieve equality in the upper bound of Theorem 8.2. The first property is that the extremal graphs 𝐺 (𝑛, 𝛾t ) are disconnected, and the second property is that their edges are very unevenly distributed in the sense that 𝛿(𝐺) and Δ(𝐺) differ greatly. Indeed, for 𝛾t ≥ 3, the graph 𝐺 = 𝐺 (𝑛, 𝛾t ) has minimum degree 𝛿(𝐺) = 1 and maximum degree Δ(𝐺) = 𝑛 − 𝛾t + 1. In 2004 Sanchis [666] improved the bound of Theorem 8.13 slightly in the case when the graph 𝐺 is connected and 𝛾t (𝐺) ≥
5. To state her result, for 𝑛 ≥ 1, let 𝐹𝑛 = 𝑛2 𝐾2 if 𝑛 is even and let 𝐹𝑛 = 𝐾1 ∪ 𝑛−1 2 𝐾2 if 𝑛 is odd. Theorem 8.14 ([666]) If 𝐺 is a connected graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t , then     𝑛 − 𝛾t + 1 𝛾t 𝑚≤ + . 2 2 Further, if 𝐺 achieves equality in this bound, then it has one of the following forms: (a) The graph 𝐺 is obtained from 𝐾𝑛−𝛾t ∪ 𝐹𝛾t by adding edges between the clique and the graph 𝐹𝛾t in such a way that each vertex in the clique is adjacent to exactly one vertex in 𝐹𝛾t and each component of 𝐹𝛾t has at least one vertex adjacent to a vertex in the clique. (b) For 𝛾t = 5 and 𝑛 ≥ 9, the graph 𝐺 is obtained from 𝐾𝑛−7 ∪ 𝑃3 ∪ 𝑃4 by joining every vertex in the clique to both ends of the 𝑃4 and to exactly one end of the 𝑃3 in such a way that each end of the 𝑃3 is adjacent to at least one vertex in the clique. (c) For 𝛾t = 6 and 𝑛 ≥ 9, 𝐺 is obtained from 𝐾𝑛−6 ∪ 𝐹6 by letting 𝑆 be a maximum independent set in 𝐹6 and joining every vertex in the clique to exactly two vertices of 𝑆 in such a way that each vertex in 𝑆 is adjacent to at least one vertex in the clique.

Section 8.3. Total Domination and Size

245

Examples of graphs constructed in the statement of Theorem 8.14(a), (b), and (c) are shown in Figure 8.5(a), (b), and (c), respectively, where some of the edges of the complete subgraphs are omitted for clarity. 𝐾8

𝐾6

𝐾5

(a)

(b)

(c)

Figure 8.5 Examples of graphs in the statement of Theorem 8.14 The graphs that achieve equality in the improved bound of Sanchis in Theorem 8.14 are also very unevenly distributed in the sense that the extremal graphs 𝐺 have small minimum degree and large maximum degree, namely Δ(𝐺) = 𝑛 − 𝛾t 𝐺. In 2005 Henning [458] and in 2007 Shan et al. [672] established a linear Vizinglike theorem relating the size of a graph and its order, total domination number, and maximum degree. Their results showed that the square dependence on 𝑛 and 𝛾t in Theorem 8.13 and Theorem 8.14 can be improved to a linear dependence on 𝑛, 𝛾t (𝐺), and the maximum degree Δ(𝐺). For the remainder of this section, let Δ ≥ 3 be a given fixed integer, where Δ is not necessarily equal to the maximum degree Δ(𝐺). Theorem 8.15 ([458, 672]) Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t and Δ(𝐺) ≤ Δ, where Δ ≥ 3. If every component of 𝐺 has order at least 3, then 𝑚 ≤ Δ(𝑛 − 𝛾t ).

(8.20)

Proof We proceed by induction on the order 𝑛 ≥ 3. If 𝑛 = 3, then 𝐺 = 𝑃3 or 𝐺 = 𝐾3 . In both cases, 𝛾t = Δ(𝐺) = 2, and so Δ(𝑛 − 𝛾t ) = Δ ≥ 3 ≥ 𝑚. This establishes the base case. Let 𝑛 ≥ 4 and suppose that the result is true for all graphs in which every component has order at least 3. Let 𝐺 be a graph, each component of which has order at least 3, of order 𝑛 and size 𝑚 with Δ(𝐺) ≤ Δ, where Δ ≥ 3. Since Inequality (8.20) is linear in 𝑚, 𝑛, and 𝛾t , the result is immediate if 𝐺 is disconnected. Hence, we may assume that 𝐺 is connected. IfΔ(𝐺)  ≤  2,then 𝐺 is a cycle or a path of order at least 4 and 𝑚 ≤ 𝑛. In this case, 𝛾t = 12 𝑛 + 14 𝑛 − 14 𝑛 ≤ 23 𝑛, and so Δ(𝑛 − 𝛾t ) ≥ 3(𝑛 − 𝛾t ) ≥ 𝑛 ≥ 𝑚. Let 𝐺 have minimum degree 𝛿(𝐺) = 𝛿. If 𝛿 ≥ 3, then by Theorem 6.52, we have 𝛾t ≤ 12 𝑛 and so Δ(𝑛 − 𝛾t ) ≥ 3(𝑛 − 𝛾t ) ≥ 32 𝑛 ≥ 𝑚. Hence, we may assume that 𝛿 ∈ [2] and Δ(𝐺) ≥ 3, for otherwise Inequality (8.20) holds as desired. Since Δ(𝑛 − 𝛾t ) is monotonically increasing in Δ, we can assume that Δ(𝐺) = Δ ≥ 3.

(8.21)

Let 𝑢 be a vertex of 𝐺 of minimum degree 𝛿 in 𝐺 and let 𝑣 be an arbitrary neighbor of 𝑢. Let 𝑉1 be the set of isolated vertices in 𝐺 − N[𝑣], and let 𝑉2 be the set of vertices

Chapter 8. Bounds in Terms of Size

246

in 𝐺 − N[𝑣]. Further, let 𝑛𝑖 = |𝑉𝑖 | for 𝑖 ∈ [2] and let that belong to 𝑃2 -components
𝑉 ′ = 𝑉 \ N[𝑣] ∪ 𝑉1 ∪ 𝑉2 . If 𝑉 ′ ≠ ∅, then let 𝐺 ′ = 𝐺 [𝑉 ′ ] and let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Let 𝐻 = 𝐺 − 𝑉 (𝐺 ′ ) = 𝐺 [N[𝑣] ∪ 𝑉1 ∪ 𝑉2 ]. We note that 𝑛 = 𝑛(𝐻) + 𝑛′ = deg𝐺 (𝑣) + 1 + 𝑛1 + 𝑛2 + 𝑛′ , 𝛾t ≤ 𝛾t (𝐻) + 𝛾t′ , and each of 𝐺 ′ and 𝐻 has maximum degree at most Δ. Applying the inductive hypothesis to 𝐺 ′ , we have 𝑚 ′ ≤ Δ(𝑛′ − 𝛾t′ ). Claim 8.15.1 If 𝑛1 + 𝑛2 ≥ 1, then the following hold: (a) 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 + 1. (b) If a vertex in N(𝑣) has two or more neighbors in 𝑉1 ∪ 𝑉2 , then 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . (c) 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 21 𝑛2 + 1. (d) If 𝑛1 = 0, then 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 . Proof (a) By supposition, 𝐺 is connected and has order 𝑛 ≥ 4. Thus, each vertex in 𝑉1 has at least one neighbor in N(𝑣), while each 𝑃2 -component of 𝐺 [𝑉2 ] contains a vertex that has at least one neighbor in N(𝑣). Let 𝑈2 ⊂ 𝑉2 consist of one vertex from every 𝑃2 -component of 𝐺 [𝑉2 ] that has a neighbor in N(𝑣). Let 𝑆 be a minimum set in N(𝑣) that dominates the set 𝑉1 ∪ 𝑈2 . We note that |𝑆| ≤ |𝑉1 | + |𝑈2 |. Since 𝑆 ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | + 1 ≤ |𝑉1 | + 2|𝑈2 | + 1 = 𝑛1 + 𝑛2 + 1. (b) Suppose that a neighbor 𝑤 of 𝑣 has two or more neighbors in 𝑉1 ∪ 𝑉2 . We show that 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . If 𝑤 is adjacent to at most one vertex from each 𝑃2 -component of 𝐺 [𝑉2 ], then we can choose the sets 𝑆 and 𝑈2 so that 𝑤 ∈ 𝑆 and |𝑆| ≤ |𝑉1 | + |𝑈2 | − 1, implying that 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . Hence, we may assume that 𝑤 is adjacent to both vertices from the same 𝑃2 -component of 𝐺 [𝑉2 ]. If 𝑥 is the
vertex in such a 𝑃2 -component that belongs to the set 𝑈2 , then the set 𝑆 ∪ 𝑈2 \ {𝑥} ∪ {𝑣} is a TD-set of 𝐻. Therefore, 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | ≤ 𝑛1 + 𝑛2 . (c) Since N(𝑣) ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + |𝑈2 | + 1 = deg𝐺 (𝑣) + 12 𝑛2 + 1. (d) Suppose that 𝑛1 = 0. As before, let 𝑆 be a minimum set in N(𝑣) that dominates the set 𝑈2 . If 𝑆 ≠ N(𝑣), then |𝑆| ≤ deg𝐺 (𝑣) − 1 and the set 𝑆 ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, implying that 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | + 1 ≤ deg𝐺 (𝑣) + 12 𝑛2 . If 𝑆 = N(𝑣), then every vertex in N(𝑣) uniquely dominates a vertex in 𝑈2 that is not dominated by any other vertex in N(𝑣), implying that the set N(𝑣) ∪ 𝑈2 is a TD-set of 𝐻. Therefore, 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 . Claim 8.15.2 If 𝑛1 + 𝑛2 ≥ 1 and 𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ Δ(𝑛1 + 𝑛2 + 2),

(8.22)

then Inequality (8.20) holds. Proof Suppose that 𝑛1 + 𝑛2 ≥ 1 and that Inequality (8.22) holds. Let 𝑚 1 be the number of edges in 𝐺 incident with vertices in N(𝑣). Since the vertex 𝑢 ∈ N(𝑣) has
degree 𝛿 in 𝐺, 𝑚 1 ≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 . By construction, every edge of 𝐺 is incident with a vertex in N(𝑣) or belongs to 𝐺 ′ or to a 𝑃2 -component in 𝐺 [𝑉2 ]. Recall that 𝑛 = deg𝐺 (𝑣) + 1 + 𝑛1 + 𝑛2 + 𝑛′ and 𝛾t ≤ 𝛾t (𝐻) + 𝛾t′ . These observations, together with our supposition that Inequality (8.22) holds, imply that the number of edges 𝑚 of 𝐺 satisfies

Section 8.3. Total Domination and Size

247


𝑚 = 𝑚 1 + 𝑚 𝐺 [𝑉2 ] + 𝑚 ′ ≤ 𝑚 1 + 21 𝑛2 + Δ(𝑛′ − 𝛾𝑡′ )

≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 + 12 𝑛2 + Δ 𝑛 − deg𝐺 (𝑣) − 1 − 𝑛1 − 𝑛2 − 𝛾t + 𝛾t (𝐻) ≤ Δ(𝑛 − 𝛾𝑡 ) + 𝛿 + 12 𝑛2 + Δ𝛾t (𝐻) − Δ(𝑛1 + 𝑛2 + 2) ≤ Δ(𝑛 − 𝛾𝑡 ), that is, Inequality (8.20) holds. By Claim 8.15.2, we may assume that if 𝑛1 + 𝑛2 ≥ 1, then Inequality (8.22) does not hold, for otherwise Inequality (8.20) follows. With this assumption, we have the following range of possible values for 𝑛2 . Claim 8.15.3 If 𝑛1 + 𝑛2 ≥ 1, then 2(Δ − 𝛿 + 1) ≤ 𝑛2 < 2 deg𝐺 (𝑣). Proof Suppose that 𝑛1 + 𝑛2 ≥ 1. By Claim 8.15.1(a), we have 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 + 1. If 𝑛2 ≤ 2(Δ − 𝛿), then 𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ 𝛿 + (Δ − 𝛿) + Δ(𝑛1 + 𝑛2 + 1) = Δ(𝑛1 + 𝑛2 + 2), that is, Inequality (8.22) holds, a contradiction. Hence, since 𝑛2 is even, we have 𝑛2 ≥ 2(Δ − 𝛿 + 1). This establishes the lower bound on 𝑛2 . We prove next the upper bound on 𝑛2 . Assume that 𝑛1 ≥ 1. By Claim 8.15.1(c), we have 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 + 1. Thus, ⇑ ⇕ ⇕

𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ Δ(𝑛1 + 𝑛2 + 2)
𝛿 + 12 𝑛2 + Δ deg𝐺 (𝑣) + 12 𝑛2 + 1 ≤ Δ(𝑛1 + 𝑛2 + 2)   Δ+1 − Δ 𝑛2 ≤ Δ(𝑛1 + 1) − 𝛿 − Δ deg𝐺 (𝑣) 2
2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 𝑛2 ≥ . Δ−1


Thus, if 𝑛2 ≥ 2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 /(Δ − 1), then Inequality (8.22) holds, a contradiction. Hence, by our supposition that 𝑛1 ≥ 1, we have
2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 Δ−1
2 = 2 deg𝐺 (𝑣) + deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 Δ−1
2 ≤ 2 deg𝐺 (𝑣) + Δ+𝛿−Δ−Δ Δ−1 ≤ 2 deg𝐺 (𝑣).

𝑛2
(8.24) 𝑛. Δ As a consequence of Theorem 8.21, we have the following result.

Section 8.4. Independent Domination and Size

255

Corollary 8.22 ([764]) For every Δ ≥ 3, 𝑟Δ >

0.1 ln(Δ) 1−

(8.25)

.

0.1 ln(Δ) Δ

Proof Suppose, to the contrary, that Inequality (8.25) is not true. Let 𝐺 be a bipartite Δ-regular graph 𝐺 with order 𝑛 and size 𝑚 satisfying Inequality (8.24). Thus,   0.1 ln(Δ) 0.1 ln(Δ) 𝑟Δ ≤ and 𝛾t (𝐺) > 𝑛. Δ 1 − 0.1 ln(Δ) Δ By the definition of 𝑟 Δ , 𝛾t ≤ 𝑛 −

2𝑚 Δ𝑛 ≤ 𝑛− Δ + 𝑟Δ Δ + 0.10.1ln(Δ) ln(Δ) 1−

= 1−

Δ

Δ Δ+

!

Δ0.1 ln(Δ) Δ−0.1 ln(Δ)

𝑛

  Δ − 0.1 ln(Δ) = 1− 𝑛 Δ   0.1 ln(Δ) = 𝑛, Δ a contradiction. Therefore, Inequality (8.25) holds. As an immediate consequence of Corollary 8.22, we have the slightly weaker, but simpler, bound on 𝑟 Δ . Corollary 8.23 ([764]) For every Δ ≥ 3, 𝑟 Δ > 0.1 ln(Δ). Corollary 8.23 disproves Conjecture 8.20 when 0.1 ln(Δ) > 3, which happens when Δ > 1.07 × 1013 . We summarize the results in this section as follows.  √ Theorem 8.24 ([458, 764]) For all Δ ≥ 3, 0.1 ln(Δ) < 𝑟 Δ ≤ max 3, 2 Δ . √It remains an open problem to determine if 𝑟 Δ grows proportionally with ln(Δ) or Δ, or some completely different function.

8.4

Independent Domination and Size

In this section, we relate the size and the independent domination number of a graph of given order. Recall that by Observation 6.80 in Section 6.4.1, if 𝐺 is a graph of order 𝑛 with maximum degree Δ, then we have the trivial bound 𝑖(𝐺) ≤ 𝑛 − Δ. Hence, if 𝐺 has size 𝑚, then 𝑚 ≤ 12 Δ𝑛, implying that 𝑖(𝐺) ≤ 𝑛 − 2𝑚 𝑛 , or equivalently,

𝑛
1 1 𝑚 ≤ 2 𝑛 𝑛 − 𝑖(𝐺) = 2 − 2 𝑛 𝑖(𝐺) − 1 . We state this formally as follows.

256

Chapter 8. Bounds in Terms of Size

Theorem 8.25 If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, then
𝑚 ≤ 𝑛2 − 21 𝑛(𝑘 − 1). If 𝐺 is a complete multipartite graph of order 𝑛 and size 𝑚 with 𝑘 vertices in each partite set, then 𝐺 has independent domination number 𝑘 and size 𝑚 = 12 𝑛(𝑛 − 𝑘) = 𝑛
1 2 − 2 𝑛(𝑘 − 1). Hence, the trivial upper bound given in Theorem 8.25 is tight if 𝑛 is a multiple of 𝑘. In 2004 Dankelmann et al. [215] extended the result in Theorem 8.25 to handle the case when 𝑛 is not a multiple of 𝑘. Their key result is the following lemma. Lemma 8.26 ([215]) If 𝐺 is a graph of order 𝑛 and size 𝑚 such that every vertex is in a 𝑘-clique, then 𝑚 ≥ 12 𝑛(𝑘 − 1) + 12 𝑟 (𝑘 − 𝑟), where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘. Proof Suppose, to the contrary, that the lemma is false, and let 𝐺 = (𝑉, 𝐸) be a counterexample of minimum size. The graph 𝐺 is edge-minimal with respect to the property that every vertex is in a 𝑘-clique. Every vertex of 𝐺 has degree at least 𝑘 − 1. Let 𝐴 be the set of vertices of 𝐺 of degree exactly 𝑘 − 1. Since every vertex of 𝐺 is in a 𝑘-clique, there is a collection 𝑉1 , 𝑉2 , . . . , 𝑉 𝑝 of 𝑘-element distinct subsets of 𝑉 each of which induces a clique in 𝐺 and whose union is 𝑉. Let 𝐴𝑖 = 𝐴 ∩ 𝑉𝑖 for 𝑖 ∈ [ 𝑝] and let 𝑎 𝑖 = | 𝐴𝑖 |. We note that 𝐴𝑖 is the set of those vertices in 𝑉𝑖 which are not contained in any
other set 𝑉 𝑗 . If 𝑝 = 1, then 𝐺 = 𝐾 𝑘 , and so 𝑛 = 𝑘 and 𝑟 = 0, implying that 𝑚 = 𝑛2 = 12 𝑛(𝑛 − 1) = 12 𝑛(𝑘 − 1) + 12 𝑟 (𝑘 − 𝑟), contradicting the supposition that 𝐺 is a counterexample. Hence, 𝑝 ≥ 2. We now sequentially consider the sets 𝑉𝑖 for 𝑖 ≥ 2. By the minimality of 𝐺, the set 𝐴𝑖 is nonempty. Suppose that 𝐴𝑖 ≠ 𝑉𝑖 . Let 𝐴1,𝑖 be an arbitrary set of 𝑘 − 𝑎 𝑖 vertices that belong to the set 𝑉1 . Further, let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 and any edge in 𝐺 [𝑉𝑖 \ 𝐴𝑖 ] joining two vertices no longer in a common 𝑉 𝑗 , and adding the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝐴1,𝑖 . Note that in the resulting graph 𝐺 ′ , 𝑚(𝐺 ′ ) ≤ 𝑚(𝐺). Furthermore, every vertex of 𝐺 ′ is in a 𝑘-clique. Let 𝑉𝑖′ = 𝐴𝑖 ∪ 𝐴1,𝑖 for 𝑖 ≥ 2 and let 𝑉1′ = 𝑉1 . Thus, 𝑉1′ , 𝑉2′ , . . . , 𝑉 𝑝′ is a collection of 𝑘-element distinct subsets of 𝑉 each of which induces a clique in 𝐺 and whose union is 𝑉, and such that for every 𝑖 ≥ 2, the set 𝑉𝑖′ is either disjoint from the other sets 𝑉 𝑗′ or it overlaps with the 𝑘 − 𝑎 𝑖 vertices in 𝐴1,𝑖 of 𝑉1′ , but is otherwise disjoint in the sense that 𝑉𝑖′ ∩ 𝑉 𝑗′ ⊆ 𝑉1′ for all 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Hence, we may assume for the collection {𝑉1 , 𝑉2 , . . . , 𝑉 𝑝 } of 𝑘-element subsets in the original graph 𝐺, that the clique 𝐺 [𝑉𝑖 ], for 𝑖 ≥ 2, consists of 𝐴𝑖 and the first 𝑘 − 𝑎 𝑖 vertices of 𝑉1 . Thus, for every 𝑖 ≥ 2, the set 𝑉𝑖 is either disjoint from the other sets 𝑉 𝑗 , or it overlaps the first 𝑘 − 𝑎 𝑖 vertices of 𝑉1 , but is otherwise disjoint in the sense that 𝑉𝑖 ∩ 𝑉 𝑗 ⊆ 𝑉1 for all 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Suppose that two sets 𝑉𝑖 and 𝑉 𝑗 both intersect 𝑉1 for some 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Suppose that 𝑎 𝑖 + 𝑎 𝑗 ≤ 𝑘. In this case, we replace both sets 𝑉𝑖 and 𝑉 𝑗 with a single set 𝑉𝑖′ which consists of 𝐴𝑖 ∪ 𝐴 𝑗 and the first 𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In other words, we let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 )

Section 8.4. Independent Domination and Size

257

edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 , deleting the 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) edges between 𝐴 𝑗 and 𝑉𝑖 \ 𝐴 𝑗 , adding the 𝑎 𝑖 𝑎 𝑗 edges between 𝐴𝑖 and 𝐴 𝑗 , and adding the (𝑎 𝑖 + 𝑎 𝑗 ) (𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) edges between 𝐴𝑖 ∪ 𝐴 𝑗 and the first 𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In this case, we retain the property that every vertex of 𝐺 ′ is in a 𝑘-clique. However, the size of the resulting graph 𝐺 ′ is strictly less than the size of 𝐺, noting that 𝑚(𝐺 ′ ) = 𝑚 − 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) − 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) + 𝑎 𝑖 𝑎 𝑗 + (𝑎 𝑖 + 𝑎 𝑗 ) (𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) = 𝑚 − 𝑎𝑖 𝑎 𝑗 < 𝑚, contradicting the minimality of 𝐺. Hence, 𝑎 𝑖 + 𝑎 𝑗 > 𝑘. We now replace both sets 𝑉𝑖 and 𝑉 𝑗 with a set 𝑉𝑖′ that consists of 𝑘 vertices of 𝐴𝑖 ∪ 𝐴 𝑗 and a set 𝑉 𝑗′ that consists of the remaining 𝑎 𝑖 + 𝑎 𝑗 − 𝑘 vertices of 𝐴𝑖 ∪ 𝐴 𝑗 , together with the first 2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In other words, we let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 , deleting the 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) edges between 𝐴 𝑗 and 𝑉 𝑗 \ 𝐴 𝑗 , adding the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and a selected set 𝐴 𝑗,1 of 𝑘 − 𝑎 𝑖 vertices in 𝐴 𝑗 , adding the (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) edges between 𝑎 𝑖 + 𝑎 𝑗 − 𝑘 vertices in 𝐴 𝑗 \ 𝐴 𝑗,1 and the first 2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 , and deleting the (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (𝑘 − 𝑎 𝑖 ) edges between 𝐴 𝑗,1 and 𝐴 𝑗 \ 𝐴 𝑗,1 . In this case, we retain the property that every vertex of 𝐺 ′ is in a 𝑘-clique. However, the size of the resulting graph 𝐺 ′ is strictly less than the size of 𝐺, noting that 𝑚(𝐺 ′ ) = 𝑚 − 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) − 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) − (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (𝑘 − 𝑎 𝑖 ) + 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) + (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) = 𝑚 − (𝑘 − 𝑎 𝑖 ) (𝑘 − 𝑎 𝑗 ) < 𝑚, once again, contradicting the minimality of 𝐺. As a consequence of Lemma 8.26, we have the following result. Theorem 8.27 ([215]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘, then
𝑚 ≤ 𝑛2 − 12 𝑛(𝑘 − 1) − 12 𝑟 (𝑘 − 𝑟). Proof Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘. If a vertex 𝑣 of 𝐺 belongs to a maximal independent set 𝐼 𝑣 of cardinality less than 𝑘, then 𝑖(𝐺) ≤ |𝐼 𝑣 | < 𝑘, a contradiction. Hence, every vertex of 𝐺 is in a (maximal) independent set of cardinality at least 𝑘. Therefore, in the complement 𝐺 of 𝐺, every vertex is in a 𝑘-clique. Thus, by Lemma 8.26, we have

𝑚 = 𝑛2 − 𝑚(𝐺) ≤ 𝑛2 − 12 𝑛(𝑘 − 1) − 12 𝑟 (𝑘 − 𝑟), as desired.

258

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8.5 Summary In this chapter, we presented relationships between the size of a graph and its domination, total domination, and independent domination numbers. This chapter concludes a sequence of four chapters on domination bounds, including general bounds for the three core domination numbers, bounds in terms of minimum degree and order, probabilistic bounds, and bounds in terms of size and order.

Chapter 9

Efficient Domination in Graphs 9.1

Introduction

In this chapter, we consider the concept of efficient domination, that is, the concept of dominating every vertex exactly once. An efficient dominating set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is a dominating set with the additional property that the closed neighborhood N[𝑣] of every vertex 𝑣 ∈ 𝑉 contains exactly one vertex in 𝑆. It should be noted at the outset that not every graph has an efficient dominating set, the cycles 𝐶4 and 𝐶5 being two small examples. Thus, the study of efficient domination in graphs focuses primarily on families of graphs each member of which has an efficient dominating set, and then algorithms for finding such sets, or on families of graphs for which it can be determined in polynomial time which members do and do not have efficient dominating sets. The main focus of this chapter is on two types of efficient domination in graphs, namely, efficient domination and efficient total domination. Also, perfect domination will be discussed.

9.1.1

Efficient Dominating Sets

We can formally define an efficient dominating set in a graph 𝐺 in several different ways. Here is a standard way. Definition 9.1 A set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is an efficient dominating set if for every vertex 𝑣 ∈ 𝑉, we have |N[𝑣] ∩ 𝑆| = 1, or equivalently, the closed neighborhood N[𝑣] contains exactly one vertex in 𝑆. For brevity, we will call an efficient dominating set 𝑆 an efficient set, and a graph 𝐺 having an efficient dominating set an efficient graph.  If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is an efficient set of 𝐺, then by definition N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] is a partition of 𝑉 into 𝑘 pairwise disjoint closed neighborhoods, we will call this an efficient partition. An efficient partition need not be unique, for example, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_9

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260

 N[𝑣 1 ], N[𝑣 4 ] and N[𝑣 2 ], N[𝑣 5 ] are two distinct efficient partitions of the path 𝑃5 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 . The observation that neither the cycle 𝐶4 nor the cycle 𝐶5 is an efficient graph immediately partitions all graphs into two classes: (i) efficient graphs, and (ii) those not having an efficient dominating set, called inefficient graphs. The fact that not all graphs are efficient immediately raises the question of the complexity of the following decision problem, to be discussed later in this chapter. 

EFFICIENT DOMSET (ED)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient (dominating) set? Observation 9.2 If 𝑆 is an efficient set of a graph 𝐺, then for any two vertices 𝑢, 𝑣 ∈ 𝑆, 𝑑 (𝑢, 𝑣) > 2. Thus, an efficient set 𝑆 is both an independent set and a packing. From this observation, it follows that every efficient set is an independent dominating set, sometimes called an independent perfect dominating set. In 1988 Bange et al. [56] published what many graph theorists, as distinct from coding theorists, consider to be the first paper on efficient domination in graphs. But in this paper the authors state that the concept appeared in a Sandia Laboratories technical report ten years earlier by the same authors [55]. As we shall see later in this chapter, the concept of efficient domination was introduced from the aspect of coding theory even earlier in 1973. As originally noted in [56], every efficient set of a graph 𝐺 is a 𝛾-set of 𝐺. Proposition 9.3 ([56]) For any graph 𝐺, if 𝑆 and 𝑆 ′ are two distinct efficient sets, then |𝑆| = |𝑆 ′ | = 𝛾(𝐺). ′ Proof Let 𝑆 and 𝑆 ′ be two distinct efficient sets, and assume that  |𝑆| < |𝑆 |. Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } and consider the efficient partition 𝜋 = N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] . By the Pigeonhole Principle, if |𝑆 ′ | > |𝑆|, there must be two vertices 𝑢 and 𝑣 in 𝑆 ′ that belong to the same closed neighborhood in 𝜋. But this means that 𝑑 (𝑢, 𝑣) ≤ 2, a contradiction. If |𝑆| is an efficient dominating set, then by definition 𝛾(𝐺) ≤ |𝑆|. But since every 𝛾-set of 𝐺 must contain at least one vertex in each closed neighborhood, then 𝛾(𝐺) ≥ |𝑆|. Thus, |𝑆| = 𝛾(𝐺).

9.1.2

Efficient Total Dominating Sets

Definition 9.4 A set 𝑆 ⊆ 𝑉 is an efficient total dominating set, also called an open efficient dominating set, of a graph 𝐺 = (𝑉, 𝐸) if for every vertex 𝑣 ∈ 𝑉, |N(𝑣) ∩ 𝑆| = 1, or equivalently, the open neighborhood N(𝑣) of 𝑣 contains exactly one vertex in 𝑆. For brevity, we will call an efficient total dominating set a total efficient set.  If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is a total efficient set, then by definition N(𝑣 1 ), N(𝑣 2 ), . . . , N(𝑣 𝑘 ) is a partition of 𝑉 into 𝑘 pairwise disjoint open neighborhoods, called a

Section 9.1. Introduction

261

total efficient partition. Notice that if 𝑆 is a total efficient set, then the subgraph 𝐺 [𝑆] induced by 𝑆 is a disjoint union of 𝐾2 subgraphs. For a simple illustration, consider the  path 𝑃6 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 with total efficient set 𝑆 = {𝑣 1 , 𝑣 2 , 𝑣 5 , 𝑣 6 }, where N(𝑣 1 ),  N(𝑣 2 ), N(𝑣 5 ), N(𝑣 6 ) = {𝑣 2 }, {𝑣 1 , 𝑣 3 }, {𝑣 4 , 𝑣 6 }, {𝑣 5 } is a partition of 𝑉 (𝑃6 ) into four pairwise disjoint open neighborhoods. Notice in this case that the induced subgraph 𝐺 [𝑆] is a disjoint union of two 𝐾2 subgraphs. As with efficient sets, not every graph has a total efficient set and graphs having one are called total efficient graphs. Similar to efficient sets, every total efficient set of a graph 𝐺 is a 𝛾t -set of 𝐺.

9.1.3

Perfect Dominating Sets

Definition 9.5 A set 𝑆 ⊆ 𝑉 is a perfect dominating set of a graph 𝐺 = (𝑉, 𝐸) if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, we have |N(𝑣) ∩ 𝑆| = 1, or equivalently, every vertex in 𝑉 \ 𝑆 has exactly one neighbor in 𝑆. As with efficient sets and total efficient sets, we will call a perfect dominating set a perfect set. Definition 9.6 A set 𝑆 ⊆ 𝑉 is a perfect total dominating set of a graph 𝐺 = (𝑉, 𝐸) if 𝑆 is a total dominating set and every vertex in 𝑉 \ 𝑆 has exactly one neighbor in 𝑆. A perfect total dominating set is called a total perfect set. Notice that by definition, every (total) efficient set is automatically a (total) perfect set, but not conversely. In other words, if 𝑆 is a perfect set, then 𝑆 is not necessarily an independent set or a packing.

9.1.4 Examples Consider the 4-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 1 . This cycle does not have an efficient set, but the set 𝑆 = {𝑣 1 , 𝑣 2 } is both a total efficient and a perfect set. By comparison, the 5-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 does not have an efficient set, nor does it have a total efficient set, but the set 𝑆 = {𝑣 1 , 𝑣 2 , 𝑣 3 } is both a perfect set and a total perfect set. The 6-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 𝑣 1 has an efficient set 𝑆 = {𝑣 1 , 𝑣 4 }, but does not have a total efficient set. The set 𝑆 = {𝑣 1 , 𝑣 4 } is also a perfect set, but so is the set 𝑆 ′ = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }. This shows that a graph 𝐺 can have perfect sets of different cardinalities, while every efficient set in a graph 𝐺 must have the same cardinality, namely 𝛾(𝐺). The seven graphs in Figure 9.1 illustrate the distinctions between efficient, total efficient, and perfect sets. In an efficient set 𝑆, no two vertices in 𝑆 can be within distance 2 of each other, while in total efficient sets 𝑆, the vertices in 𝑆 appear in adjacent pairs, but no vertex in one pair is within distance 2 of a vertex in another pair. As previously mentioned for perfect domination, there are no constraints on the vertices within a perfect set; indeed, in the example given in Figure 9.1, they induce a connected subgraph. In closing this section, we refer the reader to the 2015 survey by Klostermeyer [529] of types of efficient domination. We should point out that in the literature different terminology has been used for different types of efficient dominating sets. For example, Fellows and Hoover [290] used the term “perfect dominating” for “efficient dominating,” “weakly perfect dominating” for “efficient total dominating,” and “semiperfect dominating” for “perfect dominating.”

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262

efficient

total efficient

total efficient

efficient

total efficient

perfect

total efficient

Figure 9.1 Efficient, total efficient, and perfect sets

9.2

Efficient Domination

In 1973 Biggs [76] wrote what is generally considered to be the first paper on efficient domination in graphs, although it was written from a coding theory perspective. His opening paragraph is the following: The problem of the existence of 𝑒-error correcting perfect codes of block length 𝑚 over GF(𝑞) is set in the vector space 𝑉 (𝑚, 𝑞), endowed with the Hamming metric; we shall refer to this as the classical perfect code question. It is possible to replace the vector space by a graph Γ(𝑚, 𝑞), whose vertices are vectors and whose edges join vectors which differ in precisely one coordinate. In fact, there is an analogous graph for all natural numbers 𝑞, not just the prime powers, so that we have a slight gain in generality. Using graph theory terminology, Biggs [76] defined a perfect 𝑒-code as follows. Let 𝐺 = (𝑉, 𝐸) be a connected graph. For each non-negative integer 𝑒 and each  vertex 𝑢 ∈ 𝑉, define the distance-𝑒 neighborhood of 𝑢 to be the set N𝑒 [𝑢] = 𝑣 : 𝑑 (𝑢, 𝑣) ≤ 𝑒 . A perfect 𝑒-code is a subset 𝑆 ⊆ 𝑉 such that the sets N𝑒 [𝑣], for all 𝑣 ∈ 𝑆, form a partition of 𝑉. Any connected graph 𝐺 therefore always has a perfect 0-code (let 𝑆 = 𝑉) and a perfect 𝑑-code where 𝑑 = diam(𝐺). These are called the two trivial codes. Hence, an efficient dominating set is a perfect 1-code. A classic example of perfect 1-codes occurs in the 𝑛-cubes 𝑄 𝑛 , the 2𝑛 vertices of which correspond 1-to-1 with the set of 2𝑛 𝑛-tuples of zeroes and ones, where two 𝑛-tuples are adjacent if and only if they differ in precisely one position, called Hamming distance-1. For 𝑛-cubes, a perfect 1-code, that is, an efficient dominating set 𝑆, corresponds to a set of code words which are to be transmitted from one location to another. If in transmission a single error occurs, then a bit string like 101 might be received as either 001 or 111 or 100. Consider the 3-cube 𝑄 3 with vertices: 𝑣 0 = (0, 0, 0), 𝑣 1 = (0, 0, 1), 𝑣 2 = (0, 1, 0), . . . , 𝑣 7 = (1, 1, 1). The set 𝑆 = {𝑣 0 , 𝑣 7 } is an efficient set of 𝑄 3 , since N[𝑣 0 ] = {𝑣 0 , 𝑣 1 ,

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263

𝑣 2 , 𝑣 4 } and N[𝑣 7 ] = {𝑣 3 , 𝑣 5 , 𝑣 6 , 𝑣 7 }. Thus, if in transmitting the code word (0, 0, 0) a single error occurs, and either (0, 0, 1) or (0, 1, 0) or (1, 0, 0) is received, it can be corrected uniquely to (0, 0, 0). It is well known in coding theory that the 𝑛-cube 𝑄 𝑛 has an efficient set if and only if 𝑛 = 2 𝑘 − 1, for some positive integer 𝑘. Thus, the 𝑛-cube is efficient if and only if 𝑛 = 2 𝑘 − 1. By Proposition 9.3, an efficient dominating set in 𝐺 has cardinality 𝛾(𝐺). In 2012 Brandstädt et al. [103] generalized this as follows, where N = {1, 2, . . .} is the set of positive integers. For a graph 𝐺 = (𝑉, 𝐸), define the following weighted Í vertex function w : 𝑉 → N by w(𝑣) = deg(𝑣) + 1. For 𝑆 ⊆ 𝑉, define w(𝑆) = 𝑣 ∈𝑆 w(𝑣). For any graph 𝐺 = (𝑉, 𝐸), the graph 𝐺 2 = (𝑉, 𝐸 2 ) is the graph in which 𝑢𝑣 ∈ 𝐸 2 if and only if 𝑑 (𝑢, 𝑣) ≤ 2 in 𝐺. Proposition 9.7 ([103]) For any graph 𝐺 = (𝑉, 𝐸) and corresponding weight function w : 𝑉 → N, and any subset 𝑆 ⊆ 𝑉, the following hold: (a) If 𝑆 is an efficient set in 𝐺, then w(𝑆) ≥ |𝑉 |. (b) If 𝑆 is an independent set in 𝐺 2 , then w(𝑆) ≤ |𝑉 |. Proposition 9.8 ([103]) For any graph 𝐺 = (𝑉, 𝐸) and corresponding weight function w : 𝑉 → N, the following are equivalent for any subset 𝑆 ⊆ 𝑉: (a) 𝑆 is an efficient set. (b) 𝑆 is a minimum weight dominating set in 𝐺 and w(𝑆) = |𝑉 |. (c) 𝑆 is a maximum weight independent set in 𝐺 2 and w(𝑆) = |𝑉 |.  Proof (a) ⇒ (b): If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is an efficient set in 𝐺, then N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] is a partition of 𝑉 and hence w(𝑆) = |𝑉 |. Also, by Proposition 9.7, there is no efficient set 𝑆 ′ ⊆ 𝑉 with w(𝑆 ′ ) < w(𝑆) = |𝑉 |. (b) ⇒ (c): If 𝑆 is a minimum weight dominating set in 𝐺 with w(𝑆) = |𝑉 | and the closed neighborhoods of the vertices in 𝑆 partition 𝑉, then 𝑆 is a maximum independent set in 𝐺 2 with w(𝑆) = |𝑉 |, since, by Proposition 9.7, there is no independent set 𝑆 ′ ⊂ 𝑉 in 𝐺 2 with w(𝑆 ′ ) > w(𝑆) = |𝑉 |. (c) ⇒ (b): If 𝑆 is a maximum independent set in 𝐺 2 with w(𝑆) = |𝑉 |, then the closed neighborhoods of the vertices in 𝑆 partition 𝑉, and thus, 𝑆 is an efficient set in 𝐺.

9.2.1

Efficient Graphs

It is easy to see that every graph with domination number 1 is efficient. For other simple examples, we note that all paths are efficient, while a cycle 𝐶𝑛 , for 𝑛 ≥ 3, is efficient if and only if 𝑛 ≡ 0 (mod 3). In 1986 Harary and Livingston [385] characterized efficient trees in terms of forbidden subgraphs. In 2000 Cockayne et al. [184] gave a different characterization of efficient trees in terms of vertex subsets that must be contained in all minimum dominating sets and in all minimum independent dominating sets in trees. In 1990 Livingston and Stout [568] studied the problem of finding efficient sets in a variety of families of graphs considered in the study of parallel interconnection

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networks. For trees, directed acyclic graphs, and series-parallel graphs, the authors presented a linear algorithm for deciding if an efficient set exists, and if it does, the algorithm constructs one. For 2- and 3-dimensional grid graphs, toroidal graphs, hypercubes 𝑄 𝑛 , and cube-connected paths, they characterized which graphs have efficient sets and they characterized their structure. The authors also studied efficient sets in higher-dimensional grids, cube-connected cycles, and de Bruijn graphs. In 1993 Clark [178] studied the probabilities that a random graph 𝐺 ∈ G(𝑛, 𝑝) is efficient. He showed that for a wide range of probabilities 𝑝 that an edge exists between any two vertices, almost every graph in G(𝑛, 𝑝) is inefficient, that is, has no efficient set. In addition, he showed that almost every tree in the set T𝑛 of all trees of order 𝑛 is inefficient.

9.2.2 Efficient Grid Graphs and Efficient Toroidal Graphs  An 𝑚 × 𝑛 grid graph 𝐺 𝑚,𝑛 has a vertex set 𝑉 = (𝑖, 𝑗) : 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛] , where (𝑖, 𝑗) is adjacent to (𝑘, 𝑙) if 𝑖 = 𝑘 and | 𝑗 − 𝑙 | = 1 or 𝑗 = 𝑙 and |𝑖 − 𝑘 | = 1. We often refer to a grid graph as simply a grid. For a fixed value of 𝑖, the set of vertices of the form (𝑖, 𝑗), 1 ≤ 𝑗 ≤ 𝑛, is called the 𝑖 th row of 𝐺 𝑚,𝑛 , and for a fixed value of 𝑗, the set of vertices of the form (𝑖, 𝑗), 1 ≤ 𝑖 ≤ 𝑚, is called the 𝑗 th column of 𝐺 𝑚,𝑛 . Thus, the vertex (𝑖, 𝑗) is placed in the 𝑖 th row and 𝑗 th column of the grid. For example, in Figure 9.2(a), an efficient set consists of the highlighted vertices (1, 1), (2, 3), and (1, 5); while in Figure 9.2(b), an efficient set consists of the highlighted vertices (1, 2), (2, 4), (3, 1), and (4, 3), where we adopt the convention that vertex (1, 1) appears in the lower left corner of the grid. While this way of labeling and drawing is commonly used by those who do research in grid graphs, it is clear that a grid graph is the Cartesian product 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 , where sometimes a different convention is used (as we shall see in Chapter 18). A toroidal graph, or just a torus, is a Cartesian product of the form 𝐶𝑚 □ 𝐶𝑛 . We use similar notation for the vertices of a torus 𝐶𝑚 □ 𝐶𝑛 as we do for a grid. That is, vertex (𝑖, 𝑗), where 𝑖 ∈ [𝑚] and 𝑗 ∈ [𝑛], is in the 𝑖 th row and 𝑗 th column of the torus. We note that the vertices (𝑖, 1), (𝑖, 2), . . . , (𝑖, 𝑛) induce a cycle 𝐶𝑛 for all 𝑖 ∈ [𝑚], while the vertices (1, 𝑗), (2, 𝑗), . . . , (𝑚, 𝑗) induce a cycle 𝐶𝑚 for all 𝑗 ∈ [𝑛].

(a) 𝑃2 □ 𝑃5 efficient

(b) 𝑃4 □ 𝑃4 efficient

Figure 9.2 Efficient sets in grids

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As previously mentioned, paths are efficient, that is, 𝐺 1,𝑛 = 𝑃𝑛 is efficient. Figure 9.2 illustrates the only two types of efficient sets in grids 𝐺 𝑚,𝑛 , for 𝑚, 𝑛 ≥ 2, which exist only in 2 × 𝑛 grids for odd 𝑛 and in the 4 × 4 grid, as shown by Livingston and Stout [568] in the following theorem. Theorem 9.9 ([568]) The grid 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 , for 𝑚, 𝑛 ≥ 2, has an efficient set if and only if either 𝑚 = 𝑛 = 4 or 𝑚 = 2 and 𝑛 = 2𝑘 + 1, where 𝑘 ≥ 1. Furthermore, the efficient set  𝑆 = (1, 2), (2, 4), (3, 1), (4, 3) for 𝐺 4,4 is unique up to isomorphism and the efficient set 
𝑆 ′ = 1 + 𝑖 (mod 2), 1 + 2𝑖 : 0 ≤ 𝑖 ≤ 𝑘 for 𝐺 2,2𝑘+1 is unique up to isomorphism. Proof It is easy to verify that the sets 𝑆 and 𝑆 ′ are efficient sets of 𝐺 4,4 and 𝐺 2,2𝑘+1 , respectively. Let 𝑆2 be an efficient dominating set of 𝐺 2,𝑛 . If (1, 1) is not in 𝑆2 , then exactly one of (1, 2) and (2, 1) must be in 𝑆2 . If (1, 2) is in 𝑆2 , then since 𝑆2 is an efficient set, none of (2, 1), (2, 2), and (1, 1) is in 𝑆2 . But then (2, 1) cannot be efficiently dominated, a contradiction. If (2, 1) is in 𝑆2 , then, by symmetry, the set 𝑆2 will be isomorphic to an efficient set not containing (2, 1) but containing (1, 1). Therefore, we can assume, without loss of generality, that (1, 1) ∈ 𝑆2 . But if (1, 1) ∈ 𝑆2 , then all other vertices in 𝑆2 are uniquely determined. In particular, (2, 3) ∈ 𝑆2 , and it follows that 𝑛 ≡ 1 (mod 2) is odd. We proceed further with the following claim. Claim 9.9.1 No graph 𝐺 3,𝑛 has an efficient set for 𝑛 ≥ 3. Proof Suppose, to the contrary, that 𝑆3 is an efficient set of 𝐺 3,𝑛 , where 𝑛 ≥ 3. Since it is easy to see by inspection that 𝐺 3,3 does not have an efficient set, we can assume that 𝑛 ≥ 4. There are only three ways to dominate vertex (1, 1). Case 1. (1, 1) ∈ 𝑆3 . In this case, vertices (1, 2), (1, 3), (2, 1), (2, 2), and (3, 1) cannot be in 𝑆3 , else some vertex will be dominated twice. The only way to dominate vertex (3, 1) is to include vertex (3, 2) in 𝑆3 . But this implies that vertices (3, 3) and (2, 3) are not in 𝑆3 . The only way to dominate (2, 3) is with (2, 4) ∈ 𝑆3 . But now vertex (1, 3) can no longer be efficiently dominated. Case 2. (2, 1) ∈ 𝑆3 . This implies that (1, 1), (3, 1), (1, 2), (2, 2), (3, 2), and (2, 3) cannot be in 𝑆3 . Hence, (1, 3) and (3, 3) are in 𝑆3 in order to dominate (1, 2) and (3, 2), respectively, contradicting the efficiency of the set 𝑆3 since (2, 3) is dominated twice. Case 3. (1, 2) ∈ 𝑆3 . This implies that (1, 1), (2, 1), (2, 2), (3, 2), (2, 3), (1, 3), and (1, 4) cannot be in 𝑆3 , else some vertex will be dominated twice. This implies that (3, 1) ∈ 𝑆3 in order to dominate (2, 1). Hence, (3, 4) and (2, 4) are in 𝑆3 in order to dominate (3, 3) and (2, 3), respectively, contradicting the efficiency of the set 𝑆3 since (2, 4) and (3, 4) are adjacent. Since all three cases produce a contradiction and since exactly one of these three cases must occur, this completes the proof of Claim 9.9.1.

266

Chapter 9. Efficient Domination in Graphs

By Claim 9.9.1, no graph 𝐺 3,𝑛 has an efficient set for 𝑛 ≥ 3. Let 𝑆 𝑚 be an efficient set in 𝐺 𝑚,𝑛 for 𝑚, 𝑛 ≥ 4. To dominate vertex (1, 1), exactly one of (1, 1), (1, 2), and (2, 1) is in 𝑆 𝑚 . By the symmetry of vertices (1, 2) and (2, 1), there are only two cases to consider, namely, (1, 1) ∈ 𝑆 𝑚 or (1, 2) ∈ 𝑆 𝑚 . If (1, 1) ∈ 𝑆 𝑚 , then (1, 2), (1, 3), (2, 1), (2, 2), and (3, 1) cannot be in 𝑆 𝑚 , else some vertex will be dominated twice. Thus, only vertices (3, 2) or (2, 3) are available to dominate vertex (2, 2). By symmetry, we only need to consider one of these two cases, so assume that (2, 3) ∈ 𝑆 𝑚 . This implies that vertex (3, 2) ∉ 𝑆 𝑚 and (3, 3) ∉ 𝑆 𝑚 . But this means that both (4, 1) and (4, 2) are in 𝑆 𝑚 to dominate (3, 1) and (3, 2), respectively, contradicting the efficiency of the set 𝑆 𝑚 since (4, 1) and (4, 2) are adjacent. If (1, 2) ∈ 𝑆 𝑚 , then (1, 1), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), and (3, 2) cannot be in 𝑆3 , else some vertex will be dominated twice. This forces vertex (3, 1) to be in 𝑆 𝑚 in order to dominate vertex (2, 1). This implies that (3, 3), (4, 1), and (4, 2) are not in 𝑆 𝑚 . This in turn implies that (2, 4) ∈ 𝑆 𝑚 in order to dominate (2, 3). Vertex (4, 3) must then be in 𝑆 𝑚 in order to efficiently dominate (3, 3). Now if 𝑚 = 𝑛 = 4, then 𝑆 𝑚 is an efficient set in 𝐺 4,4 , as illustrated in Figure 9.2(b). If 𝑛 = 5, then vertex (𝑖, 5), for 𝑖 ∈ [4], is not in 𝑆 𝑚 , else some vertex will be dominated twice. Hence, vertices (1, 5) and (3, 5) cannot be efficiently dominated by 𝑆 𝑚 . If 𝑛 ≥ 6, then vertex (1, 6) must be in 𝑆 𝑚 in order to dominate (1, 5). And this then means that vertex (3, 5), which cannot be in 𝑆 𝑚 , can only be dominated by (3, 6), again contradicting the efficiency of the set 𝑆 𝑚 . It is easy to see that the two vertices (1, 1, 1) and (2, 2, 2) form an efficient set in the 3-dimensional grid graph, known as the cube 𝑄 3 . It is somewhat surprising that there are no other efficient 3-dimensional grid graphs. Theorem 9.10 ([568]) The 3-dimensional grid graph 𝐺 𝑚1 ,𝑚2 ,𝑚3 has an efficient set if and only if 𝑚 1 = 𝑚 2 = 𝑚 3 = 2. We conclude this subsection with a characterization of efficient toroidal graphs given by Klavžar and Seifter [526] in 1995. Theorem 9.11 ([526]) For 𝑚 ≥ 3 and 𝑛 ≥ 3, the torus 𝐶𝑚 □ 𝐶𝑛 has an efficient set if and only if 𝑚, 𝑛 ≡ 0 (mod 5). Figure 9.3 illustrates an efficient set in the torus 𝐶5 □ 𝐶5 .

9.2.3

Efficient Cube-connected Cycles

In 1993 Van Wieren et al. [729] studied efficient sets in cube-connected cycles, a family of cubic graphs having relatively small diameters and a regular structure making them useful models for parallel computer architectures. A cube-connected cycle 𝐶𝐶𝐶𝑛 is a graph formed from an 𝑛-cube 𝑄 𝑛 by replacing each of the 2𝑛 vertices of 𝑄 𝑛 with a cycle of order 𝑛, as illustrated in Figure 9.4. The existence of efficient sets in such graphs facilitates the design of efficient algorithms. Van Wieren et al. [729] gave a simple method of constructing efficient

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Figure 9.3 Efficient set (red vertices) in the torus 𝐶5 □ 𝐶5 (2, 011)

(1, 011)

(2, 111)

(3, 011)

(2, 010)

(1, 010)

(3, 111)

(2, 110)

(3, 010)

(1, 110)

(3, 110)

(2, 001)

(1, 001)

(2, 101)

(3, 001)

(2, 000)

(1, 000)

(3, 000)

(1, 111)

(1, 101)

(3, 101)

(2, 100)

(1, 100)

(3, 100)

Figure 9.4 Cube-connected cycle of dimension 3

sets in such graphs if they exist and proved that they do not exist otherwise. Efficient sets were shown to exist in cube-connected cycles of order 𝑘 for 𝑘 ≠ 5. For example, the set  (1, 000), (1, 111), (2, 011), (2, 100), (3, 010), (3, 101) is an efficient set of the cube-connected cycle 𝐶𝐶𝐶3 shown in Figure 9.4.

9.2.4

Efficient Vertex-transitive Graphs

A graph 𝐺 is vertex-transitive if for any two vertices 𝑢 and 𝑣 of 𝐺, there is an automorphism 𝑓 : 𝐺 → 𝐺 such that 𝑓 (𝑢) = 𝑣. In other words, a graph 𝐺 is

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vertex-transitive if its automorphism group acts transitively on 𝑉 (𝐺). For a simple example, cycles are vertex-transitive. In 2012 Knor and Potočnik [532] studied efficient domination in cubic vertextransitive graphs. The authors considered the general problem: characterize vertextransitive graphs that have efficient sets. Noting that 2-regular graphs are efficient if and only if they are isomorphic to a disjoint union of cycles of lengths divisible by 3, they considered 3-regular vertex-transitive graphs. They presented the number # of cubic vertex-transitive graphs of a given order |𝑉 |, followed by the number #ED of these having an efficient set as shown in Table 9.1. |𝑉 |

4

1 # #ED 1

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 2 1

4 2

4 3

7 11 6 10 12 12 7 32 10 16 38 26 12 37 11 2 6 2 9 3 6 1 23 2 8 4 25 2 19 1

Table 9.1 Number of efficient cubic vertex-transitive graphs The Möbius ladder 𝑀𝑛 is the cubic graph obtained from a cycle 𝐶2𝑛 by adding 𝑛 chords of a perfect matching which connect pairs of opposite vertices. One can note that the smallest Möbius ladders 𝑀2 and 𝑀3 are isomorphic to 𝐾4 and 𝐾3,3 , respectively. Theorem 9.12 ([532]) For 𝑚 ≥ 2, if 𝐺 is a connected cubic vertex-transitive graph of order 𝑛 = 2𝑚 , then 𝐺 does not have an efficient set if and only if 𝑚 ≥ 3 and 𝐺 = 𝑀2𝑚−1 . The authors also noted the following. Proposition 9.13 ([532]) For 𝑛 ≥ 3, a prism 𝐶𝑛 □ 𝐾2 is efficient if and only if 𝑛 ≡ 0 (mod 4). Proposition 9.14 ([532]) The Möbius ladder 𝑀𝑛 is efficient if and only if 𝑛 ≡ 2 (mod 4), or equivalently, if and only if |𝑉 (𝑀𝑛 )| ≡ 4 (mod 8).

9.2.5

Efficient Cayley Graphs

In 2001 Lee [556] studied efficient Cayley graphs. These graphs are defined as follows. Given a finite group Γ with an identity element 𝑒 (or 0) and an inverse-closed, symmetric subset 𝑆 of Γ not containing the identity element, that is, if 𝑎 ∈ 𝑆, then 𝑎 −1 ∈ 𝑆, the Cayley graph Cay(Γ, 𝑆) on Γ relative to the connection set 𝑆 is the graph with vertex set Γ such that vertices 𝑢 and 𝑣 are adjacent if and only if 𝑣𝑢 −1 ∈ 𝑆. This graph is regular of degree |𝑆|, and is connected if and only if 𝑆 is a generating set of Γ. In the special case when Γ = Z𝑛 , the additive group of integers modulo 𝑛, a Cayley graph Cay(Z𝑛 , 𝑆) on Z𝑛 is called a circulant. A graph 𝐺˜ is called a covering of 𝐺 with projection 𝑓 : 𝐺˜ → 𝐺 if there is a ˜ → 𝑉 (𝐺) such that, for any vertex 𝑣 ∈ 𝑉 (𝐺), 𝑓 | N( 𝑣) surjection 𝑓 : 𝑉 ( 𝐺) ˜ : N( 𝑣˜ ) →

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N(𝑣) is a bijection, where 𝑣˜ ∈ 𝑓 −1 𝑣. In addition, the projection 𝑓 : 𝐺˜ → 𝐺 is said to be an 𝑛-fold covering if 𝑓 is 𝑛-to-one. Theorem 9.15 ([556]) A Cayley graph of an abelian group is efficient if and only if it is a covering graph of a complete graph. As an application, Lee [556] proved the following. Theorem 9.16 ([556]) The following are equivalent: (a) The hypercube 𝑄 𝑛 is efficient. (b) 𝑄 𝑛 is a regular covering of the complete graph 𝐾𝑛+1 . (c) 𝑛 = 2𝑑 − 1 for some natural number 𝑑. In proving Theorems 9.15 and 9.16, Lee proved the following. Lemma 9.17 ([556]) If 𝑆1 , 𝑆2 , . . . , 𝑆 𝑛 are 𝑛 pairwise disjoint, efficient sets of a graph 𝐺, then the subgraph 𝐻 induced by 𝑆1 ∪ 𝑆2 ∪ · · · ∪ 𝑆 𝑛 is an 𝑚-fold covering graph of the complete graph 𝐾𝑛 , where 𝑚 = |𝑆𝑖 | for each 𝑖 ∈ [𝑛]. Proof It is easy to show that for any two disjoint efficient sets 𝑆𝑖 and 𝑆 𝑗 , the induced subgraph 𝐺 [𝑆𝑖 ∪ 𝑆 𝑗 ] is isomorphic to the Cartesian product 𝐾2 □ 𝐾 𝑚 , where 𝑚 = |𝑆𝑖 | = |𝑆 𝑗 |. This implies that for each vertex 𝑢 ∈ 𝑆𝑖 and each 𝑗 ≠ 𝑖, there exists a unique vertex 𝑣 ∈ 𝑆 𝑗 such that 𝑢 and 𝑣 are adjacent in the subgraph 𝐻 of 𝐺 induced by 𝑆1 ∪ 𝑆2 ∪ · · · ∪ 𝑆 𝑛 . Let 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 be the vertices of a complete graph 𝐾𝑛 . Then the function 𝑓 : 𝐻 → 𝐾𝑛 defined by 𝑓 (𝑠) = 𝑣 𝑖 for each 𝑠 ∈ 𝑆𝑖 is a covering projection. Lemma 9.18 ([556]) If 𝑓 : 𝐺˜ → 𝐺 is a covering and 𝑆 is a perfect set of 𝐺, ˜ Moreover, if 𝑆 is independent, then 𝑓 −1 (𝑆) is then 𝑓 −1 (𝑆) is a perfect set of 𝐺. independent. Proof It follows from the definition of a covering projection 𝑓 that 𝑓 −1 (𝑆) is ˜ \ 𝑓 −1 (𝑆). Then 𝑓 ( 𝑣˜ ) = 𝑣 ∈ 𝑉 (𝐺) \ 𝑆. independent if 𝑆 is independent. Let 𝑣˜ ∈ 𝑉 ( 𝐺) Since 𝑆 is a perfect set, there exists a unique vertex 𝑠 ∈ 𝑆 such that 𝑣 and 𝑠 are adjacent. Since 𝑓 is a covering projection, there exists a vertex 𝑠˜ ∈ 𝑓 −1 (𝑠) such that 𝑣˜ and 𝑠˜ are adjacent. One can show that such a vertex 𝑠˜ is unique. Let 𝑠˜′ be a vertex in 𝑓 −1 (𝑆) such that 𝑣˜ and 𝑠˜′ are adjacent. Then 𝑓 ( 𝑣˜ ) = 𝑣 and 𝑓 ( 𝑠˜′ ) are adjacent in 𝐺. Since 𝑆 is a perfect set, 𝑓 ( 𝑠˜′ ) = 𝑠 = 𝑓 ( 𝑠˜). But since 𝑓 is a covering projection, 𝑠˜ = 𝑠˜′ . Theorem 9.19 ([556]) A graph 𝐺 is a covering of the complete graph 𝐾𝑛 if and only if 𝐺 has a vertex partition {𝑆1 , 𝑆2 , . . . , 𝑆 𝑛 } such that 𝑆𝑖 is an efficient set for all 𝑖 ∈ [𝑛]. Further results on efficient domination in Cayley graphs can be found in the 2019 survey by Tamizh Chelvam and Sivagami [701].

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9.2.6 Efficient Circulant Graphs    A circulant graph 𝐶𝑛 ⟨𝐿⟩ with a given list 𝐿 ⊆ 1, 2, . . . , 21 𝑛 is a graph on 𝑛 ≥ 3 vertices in which the 𝑖 th vertex is adjacent to the (𝑖 + 𝑗) th and (𝑖 − 𝑗) th vertices for each 𝑗 in the list 𝐿 and where is taken modulo 𝑛. More precisely, if 𝐿 = {𝑠1 , 𝑠2 ,  addition  . . . , 𝑠𝑟 } ⊆ 1, 2, . . . , 12 𝑛 , then thecirculant graph 𝐶𝑛 ⟨𝐿⟩ is the graph with vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑛−1 } and edge set 𝑣 𝑖 𝑣 𝑖+ 𝑗 (mod 𝑛) : 𝑖 ∈ [𝑛 − 1] 0 and 𝑗 ∈ {𝑠1 , 𝑠2 , . . . , 𝑠𝑟 } . Equivalently, if 𝐿 is a subset of the finite cyclic group Z𝑛 not containing the identity element 0 and 𝐿 = −𝐿, then a circulant graph is a Cayley graph Cay(Z𝑛 , 𝐿) on Z𝑛 with respect to 𝐿. As such, circulant graphs are the only family of vertextransitive graphs for which the number of vertices can be prime (see Turner [719]). For example, the circulant graphs 𝐶8 ⟨1, 2⟩ and 𝐶10 ⟨1, 2, 3⟩ are shown in Figure 9.5(a) and (b), respectively.

(a) 𝐶8 ⟨1, 2⟩

(b) 𝐶10 ⟨1, 2, 3⟩

Figure 9.5 The circulant graphs 𝐶8 ⟨1, 2⟩ and 𝐶10 ⟨1, 2, 3⟩ Circulant graphs form an important class of topological structures in interconnection networks, having been widely used in telecommunication networks, VLSI design, and distributed computation, in part because of their symmetry, fault-tolerance, and efficient routing capabilities. In 2007 Obradović et al. [621] studied efficient sets in circulant graphs having only two chord lengths, called 2-chord circulants. Lemma 9.20 ([621]) If 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is an efficient connected 4-regular graph such that gcd(𝑠1 , 𝑛) = 1 and/or gcd(𝑠2 , 𝑛) = 1, then |𝑠1 ± 𝑠2 | . 0 (mod 5) and 𝑠1 , 𝑠2 . 0 (mod 5) Lemma 9.21 ([621]) If 𝐶5𝑘 ⟨𝑠1 , 𝑠2 ⟩ is an efficient connected 4-regular graph such that gcd(𝑠1 , 𝑛) ≠ 1 and gcd(𝑠2 , 𝑛) ≠ 1, then |𝑠1 ± 𝑠2 | . 0 (mod 5) and 𝑠1 , 𝑠2 . 0 (mod 5). Theorem 9.22 ([621]) If 𝐺 = 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is a connected 4-regular graph, then 𝐺 is efficient if and only if each of the following holds: (a) 𝑛 = 5𝑘 (b) |𝑠1 ± 𝑠2 | . 0 (mod 5) (c) 𝑠1 , 𝑠2 . 0 (mod 5). Proof If 𝐺 = 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is a connected 4-regular graph and 𝐺 is efficient, then conditions (a), (b), and (c) follow from Lemmas 9.20 and 9.21 and the definition of efficient domination, which implies that 𝑛 ≡ 0 (mod 5).

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271

 Conversely, let 𝐷 = 0, 5, 10, . . . , 5(𝑖 − 1) and let 𝑢 ∈ 𝑉 \ 𝐷. Since 𝑠1 , 𝑠2 . 0 (mod 5), the set 𝐷 is an independent set in 𝐺. Vertex 𝑢 can be written as 𝑢 = 5𝑞 + 𝑟, for 0 ≤ 𝑞 ≤ (𝑖 − 1) and 1 ≤ 𝑟 ≤ 4. Since |𝑠1 ± 𝑠2 | . 0 (mod 5), the chord lengths 𝑠1 , 𝑠2 , −𝑠1 , −𝑠2 must be in four different congruence classes modulo 5 and none of these classes are 0 (mod 5) since 𝑠1 , 𝑠2 . 0 (mod 5). Therefore, exactly one chord length is in the same congruence class modulo 5 as −𝑟, and 𝑢 has a single neighbor in 𝐷. Hence, 𝐷 is an efficient set of 𝐺. The four graphs in Figure 9.6 show the four possible efficient sets in the 2-chord circulants of order 𝑛 = 10.

(a) 𝐶10 ⟨1, 2⟩

(b) 𝐶10 ⟨1, 3⟩

(c) 𝐶10 ⟨2, 4⟩

(d) 𝐶10 ⟨3, 4⟩

Figure 9.6 Efficient sets in 2-chord circulants

In a remark at the end of their paper, Obradović et al. [621] gave an example of an 8-regular circulant graph 𝐺 = 𝐶27 ⟨2, 7, 9, 11⟩ for which the sets in the form {𝑎, 𝑏, 𝑐}, where 𝑎 ∈ {0, 9, 18}, 𝑏 ∈ {3, 12, 21}, and 𝑐 ∈ {6, 15, 24}, are efficient sets of 𝐺. Note that some of these efficient sets contain vertices that are not equally spaced in 𝐺. In 2013 Reji Kumar and MacGillivray [656] considered the construction of infinitely many efficient circulant graphs in which the elements of an efficient set need not be equally spaced. They showed that if a circulant graph of sufficiently large degree has an efficient set 𝐷, then either the elements of 𝐷 are equally spaced in 𝐺 or 𝐺 = 𝐻 ◦ 𝐾𝑚 , where 𝐻 is a smaller efficient circulant graph. They also gave the following result. Theorem 9.23 ([656]) If 𝐹 = 𝐶𝑚𝑛 ⟨𝑇⟩, where 𝑇 ∪ {0} is a union of cosets of 𝑛Z𝑛 , then 𝐹 is efficient if and only if 𝐶𝑛 ⟨𝑇⟩ is efficient. The lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 is the graph with 𝑉 (𝐺 ◦ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) (𝑢, 𝑤) ∈ 𝐸 (𝐺) or (ii) 𝑢 = 𝑤 and (𝑣, 𝑥) ∈ 𝐸 (𝐻). In 2009 Taylor [705] proved the following result, which was also proven independently by Reji Kumar and MacGillivray [656]. Theorem 9.24 ([705]) If 𝐺 is an isolate-free graph and 𝐻 is any graph, then the lexicographic product 𝐺 ◦ 𝐻 is efficient if and only if 𝐺 is efficient and 𝐻 is efficient with 𝛾(𝐻) = 1.

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9.2.7 Efficient Graphs with Efficient Complements In 1998 Harary et al. [384] characterized the coronas and the caterpillars having efficient sets. In addition, they characterized the family of graphs 𝐺 for which both 𝐺 and 𝐺 have efficient sets, as follows. Let F be the family of graphs containing the path 𝑃4 and every graph that can be obtained from the union of a path 𝑃4 : 𝑢 𝑥 𝑦 𝑣 and an arbitrary graph 𝐻 by adding edges such that each vertex of 𝐻 is adjacent to exactly one of 𝑢 and 𝑣 and to exactly one of 𝑥 and 𝑦. Equivalently, 𝐺 ∈ F if and only if 𝐺 has an induced path 𝑃4 : 𝑢 𝑥 𝑦 𝑣 such that each vertex in 𝑉 \ {𝑢, 𝑥, 𝑦, 𝑣} has exactly one neighbor in {𝑢, 𝑣} and exactly one neighbor in {𝑥, 𝑦}. We note that if 𝐺 ∈ F , then 𝐺 ∈ F . Theorem 9.25 ([384]) A graph 𝐺 and its complement 𝐺 are isolate-free efficient graphs if and only if 𝐺 ∈ F . Proof It is straightforward to see that if 𝐺 ∈ F , then 𝐺 is efficient. If 𝐺 ∈ F , then 𝐺 ∈ F , so both 𝐺 and 𝐺 are efficient. Let 𝐺 be a graph such that both 𝐺 and 𝐺 are isolate-free efficient graphs. Since 𝐺 has no isolated vertices, it follows that 𝛾(𝐺) ≥ 2. Similarly, 𝛾(𝐺) ≥ 2. Moreover, since the distance between any two vertices in an efficient set of 𝐺 (respectively 𝐺) is at least 3, diam(𝐺) ≥ 3 and diam(𝐺) ≥ 3. Note that any two vertices at distance 3 or more apart in 𝐺 form a dominating set of 𝐺, implying that 𝛾(𝐺) ≤ 2 and so 𝛾(𝐺) = 2. Similarly, 𝛾(𝐺) = 2.  Let {𝑢, 𝑣} be an efficient set of 𝐺. Then N𝐺 [𝑢], N𝐺 [𝑣] is a partition of the  vertices of 𝐺, implying that N𝐺 (𝑢), N𝐺 (𝑣) is a vertex partition in 𝐺. In other words, 𝑢𝑣 is a dominating edge of 𝐺, that is, an adjacent pair of vertices which forms a dominating set in a graph 𝐺. Similarly, for any efficient set {𝑥, 𝑦} of 𝐺, 𝑥𝑦 is a dominating edge of 𝐺. Since no vertex in N𝐺 [𝑢] dominates 𝑣 and no vertex in N𝐺 [𝑣] dominates 𝑢 in 𝐺, it follows that neither 𝑢 nor 𝑣 can be incident to a dominating edge in 𝐺. Hence, {𝑢, 𝑣} ∩ {𝑥, 𝑦} = ∅. Moreover, 𝑢 is adjacent to exactly one of 𝑥 and 𝑦, say 𝑥, in 𝐺. This implies that 𝑣 is not adjacent to 𝑥 but is adjacent to 𝑦 in 𝐺. Thus, 𝑢 𝑥 𝑦 𝑣 is an induced 𝑃4 in 𝐺.  Since {𝑥, 𝑦} is an efficient set of 𝐺, N𝐺 [𝑥], N𝐺 [𝑦] is a vertex partition of 𝐺 and so, N𝐺 (𝑥), N𝐺 (𝑦) is a vertex partition in 𝐺. Thus, every vertex in 𝑉 \ {𝑢, 𝑥, 𝑦, 𝑣} has exactly one neighbor in {𝑢, 𝑣} and exactly one neighbor in {𝑥, 𝑦} in 𝐺. Hence, 𝐺 ∈ F . The path 𝑃4 ∈ F is an example of an efficient self-complementary graph. For another example of a graph 𝐺 in F , see Figure 9.7, where the blue vertices form an efficient set in the graph 𝐺, while the red vertices form an efficient set in its complement 𝐺.

9.3

Efficient Total Domination

In 2002 Gavlas and Schultz [328] and in 2003 Gavlas et al. [329] studied efficient total domination in graphs and presented the following results.

Section 9.3. Efficient Total Domination

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Figure 9.7 A graph 𝐺 such that both 𝐺 and 𝐺 are efficient

Theorem 9.26 ([328]) A path 𝑃𝑛 has a total efficient set if and only if 𝑛 . 1 (mod 4). Theorem 9.27 ([328]) A cycle 𝐶𝑛 has a total efficient set if and only if 𝑛 ≡ 0 (mod 4).

9.3.1

Total Efficient Trees

Gavlas and Schultz [328] characterized the class of trees having a total efficient set as follows. Let T denote the class of trees with total efficient sets 𝑆 that can be constructed from the tree 𝑇1 = 𝐾2 and set 𝑆1 = 𝑉 (𝐾2 ) by a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 = 𝑇 of trees and total efficient sets 𝑆1 , 𝑆2 , . . . , 𝑆 𝑘 = 𝑆, where 𝑘 ≥ 1, and if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying one of the following two operations for all 𝑖 ∈ [𝑘 − 1]: Operation 1. Attach a leaf to a vertex of 𝑆𝑖 , and let 𝑆𝑖+1 = 𝑆𝑖 . Operation 2. Add a path 𝑢 𝑣 𝑤 𝑥, where 𝑢 ∈ 𝑉 (𝑇𝑖 ) \ 𝑆𝑖 and 𝑣, 𝑤, and 𝑥 are new vertices not in 𝑇𝑖 , and let 𝑆𝑖+1 = 𝑆𝑖 ∪ {𝑤, 𝑥}. Note that 𝑢 𝑣 𝑤 𝑥 induces a path 𝑃4 in 𝑇𝑖+1 and that vertex 𝑢 must be dominated by a vertex in 𝑆𝑖 . Theorem 9.28 ([328]) A nontrivial tree 𝑇 has a total efficient set 𝑆 if and only if 𝑇 ∈ T. Proof We proceed by induction on the order 𝑛 ≥ 2 of a tree 𝑇. It is easy to see that all trees of order 𝑛 with 2 ≤ 𝑛 ≤ 6 having a total efficient set are in T . Assume that all nontrivial total efficient trees of order at most 𝑛′ are in T and let 𝑇 be a tree of order 𝑛 = 𝑛′ + 1 having a total efficient set 𝑆. We show that 𝑇 ∈ T . If the tree 𝑇 has a leaf 𝑣 not in 𝑆, then the tree 𝑇 ′ = 𝑇 − 𝑣 has a total efficient set 𝑆. Hence, by the inductive hypothesis, 𝑇 ′ ∈ T , and by applying Operation 1 to 𝑇 ′ , attaching the leaf 𝑣, we can construct 𝑇. Therefore, 𝑇 ∈ T . Hence, we may assume that every leaf in 𝑇 belongs to 𝑆. Thus, no vertex can have two leaf neighbors, implying that 𝑇 is not a star. Hence, 𝑇 has diameter at least 3. Let 𝑃 be a longest path in 𝑇 and let 𝑥, 𝑤, 𝑣, 𝑢 be the first four vertices of 𝑃. By our earlier assumptions, all leaves of 𝑇 belong to 𝑆. In particular, the vertex 𝑥 belongs to 𝑆. Therefore, vertex 𝑤 must also be in 𝑆. Hence, the vertices 𝑣 and 𝑢 cannot be in 𝑆. If deg(𝑤) > 2, then the vertex 𝑤 has a neighbor 𝑦 not on 𝑃. Since 𝑃 is a longest path, the vertex 𝑦 is a leaf and therefore belongs to the set 𝑆, which means that

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the vertex 𝑤 is dominated twice, once by 𝑥 and once by 𝑦, a contradiction. Hence, deg(𝑤) = 2. Suppose that deg(𝑣) > 2 and so the vertex 𝑣 has a neighbor 𝑧 not on 𝑃. If the vertex 𝑧 belongs to 𝑆, then the vertex 𝑣 is dominated twice, a contradiction. Hence, 𝑧 cannot be in 𝑆. Thus, the vertex 𝑧 cannot be a leaf, and therefore must have a neighbor 𝑧 ′ different from 𝑣. Since 𝑃 is a longest path, the vertex 𝑧 ′ is a leaf and therefore belongs to the set 𝑆. However, since 𝑧 ∉ 𝑆, the vertex 𝑧 ′ has no neighbor in 𝑆, a contradiction. Hence, deg(𝑣) = 2. By our earlier observations, {𝑤, 𝑥} ⊆ 𝑆 and neither 𝑣 nor 𝑢 belong to 𝑆. Further, 𝑥 is a leaf and deg(𝑤) = deg(𝑣) = 2. Deleting the vertices 𝑣, 𝑤, 𝑥 from 𝑇 results in a tree 𝑇 ′ having a total efficient set 𝑆 ′ = 𝑆 \ {𝑤, 𝑥}. By induction 𝑇 ′ ∈ T and therefore 𝑇 ∈ T by virtue of applying Operation 2 to 𝑇 ′ . Conversely, it follows the construction that 𝑇 = 𝑇𝑘 has a total efficient set.

9.3.2

Total Efficient Grid Graphs

In 2006 Klostermeyer and Goldwasser [530] characterized the class of all grid graphs 𝑃𝑚 □ 𝑃𝑛 having a total perfect code, that is, a total efficient set. This extended the work of Gavlas and Schultz [328] for 𝑚 = 1 and the work of Cowen et al. [204], who solved this problem for odd-by-odd grids. Theorem 9.29 ([204]) For 𝑚, 𝑛 > 1, if 𝑚 and 𝑛 are both odd, then 𝑃𝑚 □ 𝑃𝑛 is not total efficient. Theorem 9.30 ([530]) The following hold: (a) The 1 × 𝑛 grid is total efficient if and only if 𝑛 . 1 (mod 4). (b) For 𝑚, 𝑛 > 1, the grid 𝑃𝑚 □ 𝑃𝑛 is total efficient if and only if 𝑚 is even and 𝑛 (mod (𝑚 + 1)) ∈ {1, 𝑚 − 2, 𝑚}. For example, the red vertices form a total efficient set in 𝑃6 □ 𝑃11 illustrated in Figure 9.8.

Figure 9.8 Total efficient set of red vertices in the grid 𝑃6 □ 𝑃11

Section 9.3. Efficient Total Domination

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In 2008 Dejter [219] studied properties of perfect sets and total perfect sets in rectangular grids. In Figure 9.9, given in [219], the set of red vertices form a perfect set, since every vertex has exactly one red neighbor. This set of red vertices, however, is not an efficient set nor a total efficient set, since there are vertices having more than one red neighbor. For example, vertex (8, 4) has two red neighbors and vertex (2, 2) has three red neighbors.

Figure 9.9 Perfect set of red vertices in the grid 𝑃11 □ 𝑃16

9.3.3

Total Efficient Cylindrical Graphs

A cylindrical graph, or just a cylinder, is a Cartesian product of the form 𝑃𝑚 □ 𝐶𝑛 , or 𝐶𝑚 □ 𝑃𝑛 . The following three results on total efficient sets in cylinders 𝑃𝑚 □ 𝐶𝑛 are due to Kuziak et al. [549] in 2014. Theorem 9.31 ([549]) The cylinder 𝑃2 □ 𝐶𝑛 is total efficient when 𝑛 ≡ 0 (mod 3). Theorem 9.32 ([549]) The cylinder 𝑃2𝑚+1 □ 𝐶4𝑛 is total efficient for all 𝑚, 𝑛 ≥ 1. Let 𝑃𝑚 : 𝑢 0 𝑢 1 . . . 𝑢 𝑚−1 and 𝐶𝑛 : 𝑣 0 𝑣 1 . . . 𝑣 𝑛−1 𝑣 0 . For example, the set {𝑢 0 , 𝑢 4 , . . . , 𝑢 2𝑚−2 } × {𝑣 0 , 𝑣 1 , 𝑣 4 , 𝑣 5 , . . . , 𝑣 4𝑚−4 , 𝑣 4𝑚−3 } ∪ {𝑢 2 , 𝑢 6 , . . . , 𝑢 2𝑚 } × {𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 , . . . , 𝑣 4𝑚−2 , 𝑣 4𝑚−1 } is a total efficient set of 𝑃2𝑚+1 □ 𝐶4𝑛 when 2𝑚 + 1 ≡ 3 (mod 4).

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Theorem 9.33 ([549]) For 3 ≤ 𝑛 ≤ 7, the following hold: (a) 𝑃𝑚 □ 𝐶3 is total efficient if and only if 𝑚 = 2. (b) 𝑃𝑚 □ 𝐶4 is total efficient if and only if 𝑚 is odd. (c) 𝑃𝑚 □ 𝐶5 is total efficient if and only if 𝑚 = 4. (d) 𝑃𝑚 □ 𝐶6 is total efficient if and only if 𝑚 = 2. (e) 𝑃𝑚 □ 𝐶7 is total efficient if and only if 𝑚 = 6.

9.3.4 Total Efficient Toroidal Graphs Some initial results on total efficient sets in toroidal graphs were given in 2008 by Dejter [219], who defined a total efficient set 𝑆 to be parallel if all edges in 𝐺 [𝑆] are parallel in orientation (either horizontal or vertical). Theorem 9.34 (Dejter’s Theorem [219]) A torus 𝐶𝑚 □ 𝐶𝑛 has a parallel total efficient set if and only if 𝑚, 𝑛 ≡ 0 (mod 4). An illustration of Theorem 9.34 is given in Figure 9.10.

Figure 9.10 A parallel total efficient set in the torus 𝐶4 □ 𝐶8 Dejter’s Theorem prompted Kuziak et al. [549] to make the following conjecture. Conjecture 9.35 ([549]) The torus 𝐶𝑚 □ 𝐶𝑛 is total efficient if and only if 𝑚, 𝑛 ≡ 0 (mod 4). The authors established partial results in support of their conjecture. The following result is based on the observation that if 𝐺 is an 𝑟-regular graph of order 𝑛 having a total efficient set, then 𝑛 ≡ 0 (mod 2𝑟). Theorem 9.36 ([549]) The torus 𝐶4 □ 𝐶𝑛 , for 𝑛 ≥ 4, is total efficient if and only if 𝑛 ≡ 0 (mod 4). Theorem 9.37 ([549]) The torus 𝐶𝑚 □ 𝐶𝑛 , for 𝑚 ≤ 𝑛 and 𝑚 ∈ {3, 5, 6, 7}, does not have a total efficient set. Proof We proceed with the following three claims. Claim 9.37.1 The torus 𝐶3 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 3.

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Proof Let 𝐺 = 𝐶3 □ 𝐶𝑛 where 𝑛 ≥ 3. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. We note that in this case when 𝑚 = 3 that the vertices (1, 𝑗), (2, 𝑗), (3, 𝑗) induce a triangle in 𝐺 for all 𝑗 ∈ [𝑛]. No vertical edge (i.e., two adjacent vertices in 𝑆 that belong to the same triangle) can appear in any total efficient set since then the third vertex in that triangle is totally dominated twice. Hence, all edges that belong to 𝐺 [𝑆] must be horizontal, that is, between two adjacent vertices in 𝑆 in the same row. If 𝑛 = 3, then every horizontal edge also belongs to a triangle and, therefore, 𝑆 would contain no horizontal edge either, a contradiction. Hence, 𝑛 ≥ 4. Without loss of generality, we can assume by symmetry that the set 𝑆 contains the (horizontal) edge joining (1, 1) and (1, 2). Since every vertex is totally dominated exactly once by the set 𝑆, this means that no neighbor of (1, 1) different from (1, 2), and no neighbor of (1, 2) different from (1, 1), belongs to the set 𝑆. Thus, none of the vertices (2, 1), (3, 1), (2, 2), (3, 2), (1, 3), and (1, 𝑛) belongs to 𝑆. Further, since (2, 2) and (3, 2) are uniquely totally dominated by (1, 2), the vertices (2, 3) and (3, 3) do not belong to 𝑆. In order to totally dominate the vertices (2, 3) and (3, 3), the vertices (2, 4) and (3, 4) must belong to 𝑆, that is, the vertical edge joining (2, 4) and (3, 4) belongs to 𝑆. This is illustrated in Figure 9.11, where the vertices (1, 1), (1, 2), (2, 4), and (3, 4) are colored red. This, however, contradicts our earlier observation that no vertical edge belongs to 𝑆.

Figure 9.11 No total efficient set in the torus 𝐶3 □ 𝐶𝑛

Claim 9.37.2 The torus 𝐶5 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 5. Proof Let 𝐺 = 𝐶5 □ 𝐶𝑛 where 𝑛 ≥ 5. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. Without loss of generality, we can assume that vertex (1, 1) belongs to the set 𝑆. Exactly one neighbor of every vertex must be in 𝑆. In particular, one neighbor of (1, 1) must be in 𝑆. We show that there must exist a vertical edge in 𝑆. Suppose, to the contrary, that every edge in 𝑆 is a horizontal edge. In particular, such a horizontal edge is incident with vertex (1, 1), that is, (1, 2) ∈ 𝑆 or (1, 𝑛) ∈ 𝑆, which by rotational symmetry are equivalent. So assume, without loss of generality, that vertex (1, 2) ∈ 𝑆. Since every vertex is totally dominated exactly once by the set 𝑆, this means that the vertices in the first two columns that belong to 𝑆 are the two vertices (1, 1) and (1, 2), that is, none of the vertices (2, 1), (3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2), and (5, 2) belongs to 𝑆. From this it follows that the vertices (3, 3) and (4, 3) must be in 𝑆 in

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Chapter 9. Efficient Domination in Graphs

order to totally dominate the vertices (3, 2) and (4, 2), respectively. However, the edge incident with (3, 3) and (4, 3) is a vertical edge in 𝑆, a contradiction. Therefore, there must exist at least one vertical edge in 𝑆. Without loss of generality, we can assume that such a vertical edge is incident with vertex (1, 1), that is, (2, 1) ∈ 𝑆 or, by symmetry, (5, 1) ∈ 𝑆. Without loss of generality, we can assume that (2, 1) ∈ 𝑆. Since every vertex is totally dominated exactly once by the set 𝑆, this means that no neighbor of (1, 1) different from (2, 1) belongs to the set 𝑆, that is, none of the vertices (5, 1), (1, 2), and (1, 𝑛) belongs to 𝑆. Moreover, since no neighbor of (2, 1) different from (1, 1) belongs to the set 𝑆, none of the vertices (3, 1), (2, 2), and (2, 𝑛) belongs to 𝑆. Further, since (5, 1) is uniquely totally dominated by (1, 1), the vertices (4, 1), (5, 2), and (5, 𝑛) do not belong to 𝑆. Since (3, 1) is uniquely totally dominated by (2, 1), the vertices (3, 2) and (3, 𝑛) do not belong to 𝑆. We now consider vertex (4, 1), which as observed earlier cannot be in 𝑆. Thus, it must be dominated by either (4, 2) or (4, 𝑛), which by rotational symmetry are equivalent. So assume, without loss of generality, that vertex (4, 2) ∈ 𝑆. From this it follows that vertex (4, 3) must be in 𝑆 in order to totally dominate (4, 2). Since (4, 2) is the unique neighbor of (4, 3) that belongs to 𝑆, the vertices (3, 3), (5, 3), and (4, 4) do not belong to 𝑆. Consider next the two vertices (1, 3) and (2, 3) (see the two green vertices in Figure 9.12), which cannot be in 𝑆 and can only be totally dominated by the two red vertices (1, 4) and (2, 4) in Figure 9.12. Thus, both (1, 4) and (2, 4) belong to 𝑆. At this point, all vertices in the first four columns have been totally dominated exactly once. In particular, the vertices in the first four columns that belong to 𝑆 are the six red vertices in Figure 9.12, namely (1, 1), (2, 1), (4, 2), (4, 3), (1, 4), and (2, 4). Further, we note that neither (1, 5) nor (2, 5) belongs to 𝑆. At this point, if 𝑛 = 5, then no vertex in column 5 can be in 𝑆 and vertices (3, 5), (4, 5), and (5, 5) cannot be totally dominated, a contradiction. Thus, we can assume that 𝑛 ≥ 6. But in this case, vertices (3, 5), (4, 5), and (5, 5) must be totally dominated by the three vertices (3, 6), (4, 6), and (5, 6). However, then vertex (4, 6) has two neighbors in 𝑆, contradicting the fact that 𝑆 is a total efficient set.

Figure 9.12 No total efficient set in the torus 𝐶5 □ 𝐶𝑛 Claim 9.37.3 The torus 𝐶6 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 6.

Section 9.3. Efficient Total Domination

279

Proof Let 𝐺 = 𝐶6 □ 𝐶𝑛 , where 𝑛 ≥ 6. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. Without loss of generality, we can assume that vertex (1, 1) belongs to the set 𝑆. We note that exactly one neighbor of (1, 1) must be in 𝑆. We show that there is no horizontal edge in 𝑆. Suppose, to the contrary, that such an edge exists. Without loss of generality, we can assume (1, 2) ∈ 𝑆, that is, the edge joining (1, 1) and (1, 2) is a horizontal edge. At this point, the only vertex in the first column that can be in 𝑆 different from (1, 1) is vertex (4, 1) and the only vertex in the second column that can be in 𝑆 different from (1, 2) is vertex (4, 2). Assume that (4, 1) ∈ 𝑆. It can only be totally dominated by (4, 𝑛) or (4, 2). Assume that it is totally dominated by vertex (4, 𝑛) in the rightmost column of the torus. Thus, (4, 2) is not in 𝑆, implying that vertex (3, 3) belongs to 𝑆 in order to totally dominate vertex (3, 2). Further, vertex (5, 3) belongs to 𝑆 in order to totally dominate vertex (5, 2). However, then vertex (4, 3) (see the green vertex (4, 3) in Figure 9.13) is totally dominated twice, a contradiction. Hence, (4, 𝑛) ∉ 𝑆 and so (4, 2) ∈ 𝑆 to totally dominate (4, 1). But then no vertex of column 3 is in 𝑆, that is, (𝑖, 3) ∉ 𝑆 for all 𝑖 ∈ [6]. This implies that (2, 4), (3, 4), (5, 4), and (6, 4) are in 𝑆 to totally dominate (2, 3), (3, 3), (5, 3), and (6, 3), respectively. But then each of (1, 4) and (4, 4) has two neighbors in 𝑆, contradicting the fact that 𝑆 is a total efficient set. Hence, (4, 1) ∉ 𝑆.

Figure 9.13 No total efficient set in the torus 𝐶6 □ 𝐶𝑛

Since (4, 1) ∉ 𝑆, the only vertex in the first column that belongs to 𝑆 is vertex (1, 1). Further, since neither (3, 2) nor (5, 2) is in 𝑆, the only vertex in 𝑆 that can totally dominate vertex (5, 1) is vertex (5, 𝑛) and the only vertex in 𝑆 that can totally dominate vertex (3, 1) is vertex (3, 𝑛). But then vertex (4, 𝑛) is totally dominated twice, a contradiction. Therefore, there is no horizontal edge in 𝑆. Recall that by our earlier assumption, (1, 1) ∈ 𝑆. Since every edge in 𝑆 is a vertical edge, we note that (2, 1) ∈ 𝑆 or (6, 1) ∈ 𝑆. By symmetry, we may assume (2, 1) ∈ 𝑆. Since exactly one neighbor of every vertex belongs to 𝑆, the vertices (1, 1) and (2, 1) are the only two vertices in the first column in 𝑆.

Chapter 9. Efficient Domination in Graphs

280

In order to totally dominate vertex (4, 1), either (4, 2) or (4, 𝑛) belongs to 𝑆. In order to totally dominate vertex (5, 1), either (5, 2) or (5, 𝑛) belongs to 𝑆. Since every edge in 𝐺 [𝑆] is a vertical edge, there are two possible cases to consider, namely (4, 2) and (5, 2) are in 𝑆, or (4, 𝑛) and (5, 𝑛) are in 𝑆. If (4, 2) and (5, 2) belong to 𝑆, then vertex (3, 4) belongs to 𝑆 in order to totally dominate the vertex (3, 3). But then (1, 3) cannot be totally dominated by 𝑆 without creating a vertex which is totally dominated a second time. If (4, 𝑛) and (5, 𝑛) belong to 𝑆, then vertices (1, 𝑛 − 2) and (2, 𝑛 − 2) belong to 𝑆 in order to totally dominate the vertices (1, 𝑛 − 1) and (2, 𝑛 − 1), respectively. But then (3, 𝑛 − 1) cannot be totally dominated by 𝑆 without creating a vertex which is totally dominated a second time. Since both cases produce a contradiction. The proof of the case when 𝑚 = 7 is similar to the proofs we give for the cases when 𝑚 ∈ {3, 5, 6} but involves a slightly deeper case analysis. We therefore omit the details of the proof when 𝑚 = 7. The proof of Theorem 9.37 now follows from Claims 9.37.1, 9.37.2, and 9.37.3.

9.3.5

Total Efficient Product Graphs

In 2014 Kuziak et al. [549] studied efficient total domination in the lexicographic, strong, disjunctive, and Cartesian products of graphs. We mentioned in Sections 9.3.3 and 9.3.4 some of their results on Cartesian products, namely, cylinders and toroidal graphs. In this section, we present results for the other graph products. Recall that the lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 is the graph with 𝑉 (𝐺 ◦ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) (𝑢, 𝑤) ∈ 𝐸 (𝐺) or (ii) 𝑢 = 𝑤 and (𝑣, 𝑥) ∈ 𝐸 (𝐻). Notice that in a lexicographic product  𝐺 ◦ 𝐻, for any fixed vertex 𝑔 ∈ 𝑉 (𝐺), the set of vertices (𝑔, ℎ) : ℎ ∈ 𝑉 (𝐻) induces a subgraph of 𝐺 ◦ 𝐻 that is isomorphic to 𝐻; we call this an 𝐻-layer of 𝐺 ◦ 𝐻. Theorem 9.38 ([549]) The lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if either (a) 𝐺 is an empty graph and 𝐻 has a total efficient set, or (b) 𝐺 has a total efficient set and 𝐻 contains an isolated vertex. Proof If 𝐺 is an empty graph of order 𝑛, then 𝐺 ◦ 𝐻 is isomorphic to 𝑛 disjoint copies of 𝐻. If 𝐻 has a total efficient set, then 𝑛 copies of 𝐻 is a graph having a total efficient set. Thus, if (a) holds, then 𝐺 ◦ 𝐻 has a total efficient set. Suppose that (b) holds, that is, 𝐺 has a total efficient set 𝐷 𝐺 and there is an isolated vertex 𝑣 0 in 𝐻. We note that in 𝐺 ◦ 𝐻, we have N𝐺◦𝐻 ((𝑔, 𝑣 0 )) = N𝐺 (𝑔) × 𝑉 (𝐻) and Ø N((𝑔, 𝑣 0 )) = 𝑉 (𝐺 × 𝐻). 𝑔∈𝐷𝐺

If 𝑔, 𝑔 ′ ∈ 𝐷 𝐺 and 𝑔 ≠ 𝑔 ′ , then N𝐺◦𝐻 ((𝑔, 𝑣 0 )) ∩ N𝐺◦𝐻 ((𝑔 ′ , 𝑣 0 )) ≠ ∅ implies that N𝐺 (𝑔) ∩ N𝐺 (𝑔 ′ ) ≠ ∅, which is a contradiction. Therefore, 𝐷 𝐺 × {𝑣 0 } is a total efficient set of 𝐺 ◦ 𝐻. Thus, if (b) holds, then 𝐺 ◦ 𝐻 has a total efficient set.

Section 9.3. Efficient Total Domination

281

Conversely, let 𝐺 ◦ 𝐻 have a total efficient set 𝐷 and let (𝑔, ℎ) and (𝑔 ′ , ℎ′ ) be adjacent vertices in 𝐷. Suppose first that there exists an edge with 𝑔 ≠ 𝑔 ′ . If ℎ′′ ∈ N 𝐻 (ℎ), then (𝑔, ℎ′′ ) ∈ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )), which is a contradiction. Hence, ℎ (and by symmetry also ℎ′ ) is an isolated vertex of 𝐻. Since 𝐻 contains an isolated vertex, it follows that 𝐺 has no isolated vertices, otherwise 𝐺 ◦ 𝐻 would contain isolated vertices, which is impossible for a graph having a total efficient set. Thus, if (𝑔, ℎ) and (𝑔 ′ , ℎ′ ) are adjacent vertices in 𝐷, then ℎ and ℎ′ are isolated vertices of 𝐻 (notice that it can happen that ℎ = ℎ′ ). If (𝑔, ℎ) ∈ 𝑉 (𝐺 ◦ 𝐻), let p𝐺 ((𝑔, ℎ)) = {𝑔}, the projection of (𝑔, ℎ) onto the vertex 𝑔 in the graph 𝐺. Similarly, for a total efficient set 𝐷 of 𝐺 ◦ 𝐻, let p𝐺 (𝐷) denote the set of vertices that are projections of vertices in 𝐷 onto the vertices of the graph 𝐺. If 𝑔 ′′ ∈ N𝐺 (𝑔) ∩N𝐺 (𝑔 ′ ) for some 𝑔, 𝑔 ′ ∈ p𝐺 (𝐷), then 𝑔 ′′ × 𝐻 ⊆ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )) for (𝑔, ℎ), (𝑔 ′ , ℎ′ ) ∈ 𝐷, which is a contradiction. In addition, Ø N𝐺 (𝑔 ′′ ) = 𝑉 (𝐺), 𝑔′′ ∈p𝐺 (𝐷)

since Ø

N𝐺◦𝐻 ((𝑔1 , ℎ1 )) (𝑔1 ,ℎ1 ) ∈𝐷

= 𝑉 (𝐺 ◦ 𝐻),

and 𝐷 is a total efficient set of 𝐺 ◦ 𝐻. Thus, 𝐺 has a total efficient set p𝐺 (𝐷). Now we can assume that adjacent vertices in 𝐷 have the same first coordinate, that is, any edge between two vertices in 𝐷 has the form (𝑔, ℎ) (𝑔, ℎ′ ). Thus, 𝑔 is an isolated vertex of 𝐺, otherwise {𝑔 ′ } × 𝐻 ⊆ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔, ℎ′ )) for any neighbor 𝑔 ′ of 𝑔 in 𝐺, which is not possible. Since N𝐺◦𝐻 ((𝑔, ℎ)) : (𝑔, ℎ) ∈ 𝐷 forms a partition of 𝑉 (𝐺 ◦ 𝐻), every vertex (𝑔 ′′ , ℎ′′ ) is in some N𝐺◦𝐻 ((𝑔, ℎ)). Again, (𝑔 ′′ , ℎ′′ ) is in some N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )) and we have 𝑔 = 𝑔 ′ = 𝑔 ′′ . Hence, every vertex of 𝐺 is an isolated vertex. Assume that 𝐺 has order 𝑛. Every 𝐻-layer is isomorphic to 𝐻 and 𝐺 ◦ 𝐻 is isomorphic to 𝑛 disjoint copies of 𝐻. Since 𝐺 ◦ 𝐻 has a total efficient set, every component of 𝐺 ◦ 𝐻 has a total efficient set. Therefore, 𝐻 has a total efficient set. The strong product 𝐺 ⊠ 𝐻 of two graphs 𝐺 and 𝐻 is a graph with 𝑉 (𝐺 ⊠ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣 = 𝑥, or (ii) 𝑢 = 𝑤 and 𝑣𝑥 ∈ 𝐸 (𝐻), or (iii) 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣𝑥 ∈ 𝐸 (𝐻). Note that in any strong product 𝐺 ⊠ 𝐻, we have |N(𝑢, 𝑤) ∩ N(𝑣, 𝑥)| ≥ 2 for any two adjacent vertices (𝑢, 𝑣) and (𝑤, 𝑥), where both vertices are not isolated vertices of 𝐺 and 𝐻, respectively. Thus, the following becomes clear. Theorem 9.39 ([549]) The strong product 𝐺 ⊠ 𝐻 of two graphs has a total efficient set if and only if one factor is the empty graph and the other graph has a total efficient set. The disjunctive product 𝐺 ⊕ 𝐻 is a graph with 𝑉 (𝐺 ⊕ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever 𝑢𝑤 ∈ 𝐸 (𝐺) or 𝑣𝑥 ∈ 𝐸 (𝐻).

282

Chapter 9. Efficient Domination in Graphs

Theorem 9.40 ([549]) The disjunctive product 𝐺 ⊕ 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if one graph has a total efficient set and the other graph contains an isolated vertex. In 2008 Abay-Asmerom et al. [1] added the following result about total efficient sets in direct products (also called tensor products). The direct product 𝐺 × 𝐻 is a graph with 𝑉 (𝐺 × 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣𝑥 ∈ 𝐸 (𝐻). Notice that in the direct product 𝐺 × 𝐻, N((𝑢, 𝑤)) = N𝐺 (𝑢) × N 𝐻 (𝑤). Theorem 9.41 ([1]) The direct product 𝐺 × 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if both 𝐺 and 𝐻 have a total efficient set. Corollary 9.42 ([1]) If graphs 𝐺 and 𝐻 both have a total efficient set, then 𝛾t (𝐺 × 𝐻) = 𝛾t (𝐺)𝛾t (𝐻).

9.3.6

Total Efficient Circulant Graphs

In 2014 Castle et al. [139] characterized the graphs in two subclasses of Cayley graphs which have total efficient sets, thereby extending the results in 2002 by Gavlas and Schultz [328], who characterized which cycles have total efficient sets. In 2017 Feng et al. [291] characterized the total efficient 𝑝-regular Cayley graphs, where 𝑝 is an odd prime, as follows. Theorem 9.43 ([291]) For 𝑛 ≥ 1 and 𝑝 an odd prime, a connected 𝑝-regular circulant graph Cay(Z𝑛 , 𝑆) has a total efficient set if and only if 𝑝 divides 𝑛 and 𝑠 . 𝑠′ (mod 𝑝), for distinct 𝑠, 𝑠′ ∈ 𝑆 ∪ {0}. In 2020 Kwon et al. [551] gave necessary and sufficient conditions for the existence of total efficient sets in 4-regular circulant graphs. Note that if a 4-regular circulant graph has a total efficient set, then its order must be a multiple of 8. Theorem 9.44 ([551]) A connected circulant graph 𝐶8𝑚 ⟨𝑎, 𝑏⟩ has a total efficient set if and only if (a) 𝑏 ≡ 3 (mod 8) or

(b) 𝑏 ≡ 1 (mod 8) and gcd 8𝑚, |𝑎 − 𝑏| = gcd 4𝑚, |𝑎 − 𝑏| . One can observe from this theorem that any connected 4-regular circulant graph of the form 𝐶8𝑚 ⟨𝑎, 𝑏⟩ is isomorphic to a circulant graph of the form 𝐶8𝑚 ⟨𝑐, 𝑑⟩, where 𝑐 ≡ 1 (mod 8) and 𝑑 (mod 8) ∈ [4] 0 . We note in closing this section that also in 2020 Kwon and Sohn [550] characterized 5-regular circulant graphs 𝐶10𝑚 ⟨𝑎, 𝑏, 5𝑚⟩ having total efficient sets.

9.4

Algorithms and Complexity of Efficient Domination

The most common decision problems corresponding to efficient domination are the following.

Section 9.4. Algorithms and Complexity of Efficient Domination

283

EFFICIENT DOMSET (ED)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient dominating set? EFFICIENT TOTAL DOMSET (ETD)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient total dominating set? PERFECT DOMSET (PD)

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a perfect dominating set of cardinality at most 𝑘? PERFECT TOTAL DOMSET (PTD)

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a perfect total dominating set of cardinality at most 𝑘? There are also weighted versions of these decision problems, where weights (positive integers) are assigned to the vertices. If a graph 𝐺 has an efficient dominating set, efficient total dominating set, or perfect dominating set, then one seeks to minimize the sum of the weights of the vertices in such a set. The weighted versions of these efficient domination problems are denoted WED, WETD, and WPD, respectively. In this section, we review only a few of the many NP-completeness results and algorithms for special classes of graphs. Bange et al. [56] showed that ED is NP-complete for arbitrary graphs, using a straightforward transformation from 3SAT. They constructively characterized trees that have an efficient dominating set and showed how to determine in linear time the maximum number of vertices in a tree 𝑇 that can be efficiently dominated. They also presented a recursive characterization of trees having two disjoint efficient dominating sets. In 1991 Fellows and Hoover [290] studied efficient domination, efficient total domination, and perfect domination and showed that all three corresponding decision problems are NP-complete for planar graphs, even for maximum degree at most 3, while the efficient domination problem ED can be solved in linear time for trees. In particular, for efficient domination they proved the following. Theorem 9.45 ([290]) EFFICIENT DOMSET (ED) is NP-complete for planar graphs having maximum degree 3. Proof The ED problem is clearly in the class NP since it is easy to guess a possible solution 𝑆 and verify if 𝑆 is an efficient dominating set in polynomial time.

284

Chapter 9. Efficient Domination in Graphs

To show that ED is NP-complete, one can construct a transformation from the following NP-complete problem to ED. THREE-DIMENSIONAL MATCHING (3DM)

Instance: Three disjoint sets 𝑅, 𝐵, and 𝑌 of equal cardinality 𝑞 and a set 𝑇 of triples, 𝑇 ⊆ 𝑅 × 𝐵 × 𝑌 , where each element of 𝑅 ∪ 𝐵 ∪ 𝑌 belongs to at least one triple in 𝑇. Question: Does there exist a set of 𝑞 triples in 𝑇 that contains all elements of 𝑅 ∪ 𝐵 ∪ 𝑌? Associated with each instance 𝐼 = (𝑇, 𝑅, 𝐵, 𝑌 ) of 3DM, one can construct a bipartite graph 𝐺 𝐼 = (𝑇, 𝑅 ∪ 𝐵 ∪ 𝑌 , 𝐸) with partite sets 𝑇 and 𝑅 ∪ 𝐵 ∪ 𝑌 , where a vertex 𝑥 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 is adjacent to a vertex 𝑡 ∈ 𝑇 if and only if the triple 𝑡 contains 𝑥. To the graph 𝐺 𝐼 , add a copy 𝑇 ′ of the set 𝑇 and the edges of a perfect matching between the vertices of 𝑇 and the vertices of 𝑇 ′ , whereby a vertex 𝑡 ∈ 𝑇 is joined to its copy 𝑡 ′ ∈ 𝑇 ′ . Thus, every vertex in 𝑇 ′ becomes a leaf. Denote the resulting bipartite graph by 𝐺 ′𝐼 . If 𝑃 ⊆ 𝑇 is a solution to the instance 𝐼 = (𝑇, 𝑅, 𝐵, 𝑌 ) of 3DM, then 𝑃 ∪ {𝑡 ′ : 𝑡 ∈ 𝑇 − 𝑃} is an efficient dominating set of 𝐺 ′𝐼 . Conversely, if 𝐺 ′𝐼 has an efficient dominating set 𝑆, then no vertex in 𝑅 ∪ 𝐵 ∪ 𝑌 can be in 𝑆, for if a vertex 𝑥 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 is in 𝑆, then there is at least one triple 𝑡 ∈ 𝑇 to which 𝑥 belongs. This means that 𝑡 ∉ 𝑆. But now vertex 𝑡 ′ must be in 𝑆 or else it is not dominated; but if 𝑡 ′ ∈ 𝑆 then vertex 𝑡 is dominated twice and 𝑆 is not an efficient dominating set. Thus, any efficient dominating set 𝑆 in 𝐺 ′𝐼 satisfies 𝑆 ⊆ 𝑇 ∪ 𝑇 ′ . This must constitute a solution set for 𝐼 since each element of 𝑅 ∪ 𝐵 ∪ 𝑌 must have exactly one neighbor in 𝑆, that is, one triple to which it belongs. In order to complete the proof of the theorem, Fellows and Hoover [290] note that Dyer and Frieze showed in [253] that 3DM is NP-complete even when restricted to instances for which the associated bipartite graph 𝐺 𝐼 is planar. Thus, given an instance of 3DM in which the bipartite graph 𝐺 ′𝐼 is planar, one can, in polynomial time, transform 𝐺 ′𝐼 to a planar bipartite graph 𝐺 ′′𝐼 having maximum degree at most 3 such that 𝐺 ′𝐼 has an efficient dominating set if and only if the revised graph 𝐺 ′′𝐼 has an efficient dominating set. This transformation can be described as follows. Let 𝑆1,1,3 be a tree obtained from a star 𝐾1,3 by subdividing one edge twice; thus, it becomes a spider having a central vertex 𝑤 𝑖 of degree 3 to which are attached two leaves, labeled 𝑥𝑖 and 𝑦 𝑖 and a path of length 3, ending in a leaf labeled 𝑧𝑖 . Suppose, for example, that the graph 𝐺 ′𝐼 has a vertex 𝑣 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 of degree 8. Label its neighbors 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 , 𝑔, ℎ. Create a path 𝑃 : 𝑟 0 𝑟 1 𝑟 2 . . . . Let 𝑟 0 = 𝑎. Attach a spider 𝑆1,1,3 to vertex 𝑟 1 , so 𝑟 1 = 𝑧1 , and identify 𝑥1 = 𝑏 and 𝑦 1 = 𝑐. Attach a second spider to vertex 𝑟 4 = 𝑧 2 , and identify 𝑥 2 = 𝑑 and 𝑦 2 = 𝑒. Attach a third spider to vertex 𝑟 7 = 𝑧3 and identify 𝑥3 = 𝑓 and 𝑦 3 = 𝑔. Finally, let 𝑟 11 = ℎ. In this way each attached spider has two leaves which correspond to two of the neighbors of

Section 9.4. Algorithms and Complexity of Efficient Domination

285

the original vertex 𝑣. The first neighbor of vertex 𝑣 is the initial vertex of this path 𝑟 0 = 𝑎, and if the degree of 𝑣 is even, the path 𝑃 will have an ending path, attached to the last spider vertex 𝑧 𝑘 , of length 4, whose leaf is the final neighbor of the vertex 𝑣.
Fellows and
Hoover further showed that ED in trees can be solved in O ln(|𝑉 |) time with O |𝑉 | processors in the CREW PRAM (Concurrent Read Exclusive Write,
Parallel Random Access Model) model of parallel computation, and also in O |𝑉 | time sequentially. In her 1994 PhD thesis, McRae [588] obtained the following eight NP-completeness results, none of which have been published. We present two of her proofs here. Theorem 9.46 ([588]) ED is NP-complete, even when restricted to the class of chordal graphs, the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.47 ([588]) ETD is NP-complete, even when restricted to the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.48 ([588]) PD is NP-complete, even when restricted to the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.49 ([588]) Unless P = NP, there is no polynomial algorithm that can take as input a bipartite graph 𝐺 and find a perfect dominating set for 𝐺 with cardinality that is within a factor 𝑓 of a minimum perfect dominating set. McRae’s NP-completeness results are based on simple transformations from the following well-known NP-complete problem. EXACT COVER BY 3-SETS (X3C)

Instance: Set 𝑋 = {𝑥 1 , 𝑥2 , . . . , 𝑥 3𝑞 } of elements, with |𝑋 | = 3𝑞, a collection 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3-element subsets of 𝑋. Question: Does 𝐶 contain an exact cover for 𝑋, that is, a subcollection 𝐶 ′ ⊆ 𝐶 such that every element of 𝑋 occurs in exactly one subset in 𝐶 ′ ? Theorem 9.50 ([588]) ED is NP-complete for bipartite graphs and for chordal graphs. Proof It is easy to see that ED is in NP since one can verify in polynomial time whether an arbitrary set 𝑆 is an efficient dominating set. We construct a polynomial time transformation from X3C to ED as follows. Given an instance (𝑋, 𝐶) of X3C, we construct the following instance 𝐺 of ED. For each element 𝑥𝑖 ∈ 𝑋, construct a vertex 𝑥 𝑖 ∈ 𝑉 (𝐺). For each subset 𝐶𝑖 = {𝑥 𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 } ∈ 𝐶, construct a copy of 𝑃2 with two adjacent vertices 𝑐 𝑖1 and 𝑐 𝑖2 . From each vertex 𝑐 𝑖1 , add three edges to the three vertices in the subset 𝐶𝑖 . Note that this graph 𝐺 is bipartite. One can show that 𝐺 has an efficient dominating set if and only the instance of X3C has a solution. Notice that if 𝐺 has an efficient dominating set 𝑆, then 𝑆

286

Chapter 9. Efficient Domination in Graphs

cannot contain any of the 𝑥 𝑗 vertices, else some of the 𝑐 𝑖2 vertices cannot be efficiently dominated. Notice, as well, that if a 𝑐 𝑖1 vertex is not in 𝑆, then 𝑐 𝑖2 must be in 𝑆. If the graph 𝐺 is modified so that all possible edges between 𝑥 𝑗 vertices are added to form a complete subgraph of order 𝑛, then the resulting graph 𝐺 ′ is chordal. The same argument can then be used to show that 𝐺 ′ has an efficient dominating set if and only if the instance of X3C has a solution. Thus, ED is NP-complete for bipartite graphs and for chordal graphs. Theorem 9.51 ([588]) ETD is NP-complete for bipartite graphs and for chordal graphs. Proof It is easy to see that ETD is in NP since one can verify in polynomial time whether an arbitrary set 𝑆 is an efficient total dominating set. We construct a polynomial time transformation from X3C to ETD as follows. Given an instance (𝑋, 𝐶) of X3C, we construct the following instance 𝐺 of ETD. For each element 𝑥 𝑖 ∈ 𝑋, construct a vertex 𝑥𝑖 ∈ 𝑉 (𝐺). For each subset 𝐶𝑖 = {𝑥𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 } ∈ 𝐶, construct a copy of 𝑃3 with vertices 𝑐 𝑖1 , 𝑐 𝑖2 , and 𝑐 𝑖3 . From each vertex 𝑐 𝑖1 , add three edges to the three vertices in the subset 𝐶𝑖 . Note that this graph 𝐺 is bipartite. One can show that 𝐺 has an efficient total dominating set if and only if the instance of X3C has a solution. Notice that if 𝐺 has an efficient total dominating set 𝑆, then 𝑆 cannot contain any of the 𝑥 𝑗 vertices, else some of the 𝑐 𝑖2 and 𝑐 𝑖3 vertices cannot be efficiently total dominated. Notice, as well, that if a 𝑐 𝑖1 vertex is in 𝑆 in order to efficiently dominate three 𝑥 𝑗 vertices, then 𝑐 𝑖2 must also be in 𝑆. Similarly, if 𝑐 𝑖1 is not in 𝑆, then 𝑐 𝑖2 and 𝑐 𝑖3 must be in 𝑆. If the graph 𝐺 is modified so that all possible edges between 𝑥 𝑗 vertices are added to form a complete subgraph of order 𝑛, then the resulting graph 𝐺 ′ is chordal. The same argument can then be used to show that 𝐺 ′ has an efficient total dominating set if and only if the instance of X3C has a solution. Thus, ETD is NP-complete for bipartite graphs and for chordal graphs. In 2002 Lu and Tang [574] showed that ED is NP-complete for planar bipartite graphs and chordal bipartite graphs. In 1996 Yen and Lee [763] were among the first to study perfect domination in graphs. The authors studied the following three variants of perfect dominating sets 𝑆: • 𝑆 is an independent perfect dominating set if 𝑆 is an independent set. Note that an independent perfect dominating set is equivalent to an efficient dominating set. • 𝑆 is a total perfect dominating set if the subgraph 𝐺 [𝑆] induced by 𝑆 is isolate-free. • 𝑆 is a connected perfect dominating set if the subgraph 𝐺 [𝑆] induced by 𝑆 is connected. Yen and Lee [763] showed that these three variants of perfect domination are NPcomplete for bipartite graphs, and independent perfect domination and total perfect domination are NP-complete for chordal graphs. But for the family of block graphs, the authors constructed linear algorithms for weighted perfect domination and each of these three variants of perfect domination. Tables 9.2, 9.3, and 9.4 give a brief summary of algorithm and complexity results for efficient domination and the several variations of efficient domination discussed

Section 9.4. Algorithms and Complexity of Efficient Domination

287

in this chapter. We also refer the reader to the 2018 survey by Brandstädt [94]. Let the graphs in the table have order 𝑛, size 𝑚, minimum degree 𝛿, and maximum degree Δ.

Problem

Family

Author(s)

Year

Ref

ED

general cubic planar 3-regular planar bipartite, chordal general

Bange et al. Kratochvíl and Křivánek Livingston and Stout Yen and Lee Bakker and Van Leeuwen Fellows and Hoover McRae

1988 1988 1990 1990 1991

[56] [541] [568] [761] [48]

1991 1994

[290] [588]

McRae

1994

[588]

Kratochvíl et al. Smart and Slater

1995 1995

[542] [681]

Yen and Lee Yen and Lee Bange et al. Lu and Tang

1996 1996 1996 1998

[763] [763] [54] [573]

Lu and Tang

2002

[574]

Schaudt Eschen and Wang Brandstädt et al. Abrishami and Rahbarnia Brandstädt and Mosca

2012 2014 2018 2018

[668] [262] [98] [6]

2020

[108]

PD, ED ED PD ETD ED ED, ETD

PD

ETD ED ED TPD ED ED ED ETD ED WED ED ED

planar, Δ ≤ 3 bipartite, chordal line graphs, line graphs of bipartite line graph, line graphs of bipartite, NP-approximation chordal chordal, 2𝑃3 -free chordal chordal bipartite, chordal general claw-free, line graphs of bipartite chordal bipartite, planar bipartite planar bipartite, Δ ≤ 3 chordal 2𝑃3 -free diam = 3𝑘 or 3𝑘 + 2 𝐾1,5 -free chordal, 𝐾3 + 3𝐾1 -free chordal

Table 9.2 NP-completeness results for efficient domination

Chapter 9. Efficient Domination in Graphs

288 Problem ED WPD WED WED WED WPD WED ETD WPD WED WED

WED WED WED WED ED WED

ED WED WETD

ETD WED WED

Family

Author(s)

Bange et al. trees trees Yen and Lee Chang and Liu split graphs circular-arc Chang and Liu interval Chang and Liu cocomparability Chang et al. interval Chang et al. interval Kratochvíl et al. series-parallel Yen and Lee block graphs Yen and Lee trapezoid Liang et al. graphs permutation cocomparability Chang trapezoid Lin circular-arc Chang Courcelle et al. bounded clique-width distance-hereditary Hsieh bipartite Lu and Tang permutation distancehereditary simplicial graphs Barbosa and Slater convex bipartite Brandstädt et al. interval bigraphs chordal bipartite, Schaudt balanced, odd-sun-free chordal, strongly chordal 𝑇3 -free chordal, Schaudt claw-free (𝑆1,2,2 , Net)-free Milanič 𝑃5 -free Brandstädt et al.

Year Ref

Time

1988 [56] linear 1990 [761] linear 1993 [148] linear 1994 [149] O (𝑛2 + 𝑚) 1994 [149] O (𝑛 + 𝑚) 1995 [145] O (𝑛𝑚) 1995 [145] O (𝑛 + 𝑚) 1995 [542] O (𝑛3 ) 1995 [762] linear 1996 [763] O (𝑛 + 𝑚)
1997 [562] O★ 𝑛 ln ln(𝑛)
+𝑚 O (𝑛 + 𝑚) 1997 [146] O (𝑛2 ) 1998 [565] O★ (𝑛 ln(𝑛)) 1998 [147] linear 2000 [203] polynomial 2002 [501] O (𝑛 + 𝑚) 2002 [574] O★ (𝑛) O★ (𝑛) 2012 [59] 2012 [103]

polynomial polynomial

2012 [668]

polynomial

2012 [668]

O (𝑛3 )

2013 [591] 2013 [104]

polynomial polynomial

Table 9.3 Polynomial results for efficient domination

Section 9.4. Algorithms and Complexity of Efficient Domination Problem

Family

Author(s)

Brandstädt and Le (𝑆1,2,2 , xNet)-free (𝑃5 ∪ 𝑘 𝑃2 )-free Brandstädt and Giakoumakis WED 𝑃5 -free Brandstädt AT-free Brandstädt et al. WED dually chordal ED (𝑃6 , bull)-free Brandstädt and Karthick (𝑃6 , 𝑆1,1,3 )-free WED 𝑃6 -free Lokshtanov et al. WED 𝑃5 -free, Brandstädt and Mosca 𝑃6 -free WED (𝑃6 , banner)-free Karthick WED net-free chordal, Brandstädt and Mosca 𝑆1,2,3 -free chordal WED (𝑃6 , house, hole, Brandstädt et al. domino)-free 𝑃6 -free chordal (𝑃6 , net)-free WED diam-3 bipartite Abrishami and Rahbarnia diam-3 planar WED 2𝑃2 -free Brandstädt et al. (𝑃4 ∪ 𝑃2 )-free (2𝑃3 , 𝑘 𝑃2 )-free connected (𝐾3 , 𝑆1,2,3 )-free (2𝑃3 , 𝐾3 )-free WED 𝑃6 -free Lokshtanov et al. Brandstädt and Mosca WED 𝐻-free bipartite for 𝐻 = 𝑃7 or 𝐻 = 𝑃9 , Δ ≤ 3 or 𝐻 = 𝑆2,2,4 WED 𝑆1,2,3 -free chordal Brandstädt and Mosca net-free chordal extended-gem-free chordal WED 𝐻-free chordal, Brandstädt and Mosca for 𝐻 chordal with |𝑉 (𝐻)| ≤ 5 with four exceptions WED WED

289

Year Ref 2014 2014 2015 2015 2016 2016 2016 2016 2017

Time

[101] polynomial [97] polynomial [93] linear [96] polynomial O (𝑚 + 𝑛) [99] polynomial polynomial [570] polynomial [105] linear O (𝑛5 𝑚) [517] O (𝑛3 ) [106] polynomial

2017 [95] polynomial polynomial O (𝑛2 𝑚) 2018 [6] O (𝑛 + 𝑚) O (𝑛5 ) 2018 [98] linear O (𝛿𝑚) polynomial polynomial O (𝑛3 𝑚) 2018 [571] polynomial 2019 [107] polynomial polynomial O (𝑛6 ) 2020 [108] polynomial polynomial polynomial 2020 [108] polynomial

Table 9.4 Polynomial results for efficient domination

Chapter 10

Domination and Forbidden Subgraphs 10.1 Introduction In this chapter, we study the three core domination parameters in graphs with specific structural restrictions imposed, such as forbidding certain cycles or claws. First we review some terminology. A graph 𝐺 is 𝐹-free if it does not contain a given graph 𝐹 as an induced subgraph. Thus, if 𝐺 is 𝐶 𝑘 -free for an integer 𝑘 ≥ 3, then 𝐺 contains no induced cycle 𝐶 𝑘 and we say that 𝐺 has no induced 𝑘-cycle. In the special case when 𝐹 = 𝐾1,3 , an 𝐹-free graph is called claw-free, while if 𝐹 = 𝐾4 − 𝑒, then an 𝐹-free graph is called diamond-free. Similarly, a 𝐾3 -free graph is called triangle-free. A graph 𝐺 is (𝐹1 , 𝐹2 , . . . , 𝐹𝑘 )-free, for a collection of graphs 𝐹1 , 𝐹2 , . . . , 𝐹𝑘 , if 𝐺 contains no induced 𝐹𝑖 for any 𝑖 ∈ [𝑘]. For example, a (𝐶4 , 𝐶5 )-free graph has no induced 4-cycle and no induced 5-cycle. In the more general case where the restriction forbids a subgraph 𝐻 that is not necessarily induced, we simply say that 𝐺 does not contain 𝐻 as a subgraph, or that 𝐺 has no 𝐻 subgraph.

10.2

Domination and Forbidden Cycles

In this section, we show that if certain cycles are forbidden, then the upper bounds on the three core domination parameters established in Chapter 6 can often be improved. Hence, the lack of certain cycles in a sense decreases these domination numbers.

10.2.1

Domination Number

We present improved bounds on the domination number of a graph when certain cycles are forbidden. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_10

291

292

Chapter 10. Domination and Forbidden Subgraphs

Forbidden 4- and 5-Cycles and Minimum Degree Two Recall that in Chapter 6, we defined the family Gdom to consist of all graphs that can be obtained from a graph in the family Fdom by adding edges, including the possibility of none, joining link vertices. For example, a graph 𝐺 in the family Gdom constructed from a graph in the family Fdom with two cycle units and two key units is shown in Figure 10.1, where the highlighted vertices are link vertices.

Figure 10.1 A graph in the family Gdom As shown in Theorem 6.19, if 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2, then the extremal graphs 𝐺 achieving the upper bound of 𝛾(𝐺) = 25 𝑛 for the domination number, are precisely the graphs in the family Gdom . We note that every graph in the family Gdom contains a 4-cycle or a 5-cycle. Hence, it is a natural question to ask if the 25 -bound on the domination number can be improved if we forbid 4- and 5-cycles, or if this bound is still asymptotically best possible. In 2011 Henning et al. [477] showed that the 25 -bound can indeed be significantly improved by forbidding 4- and 5-cycles. In fact, they proved a much stronger result, which requires some additional terminology. Definition 10.1 A vertex 𝑣 is called a bad-4-cut-vertex of a graph 𝐺 if 𝐺 −𝑣 contains a component 𝐶𝑣 , which is an induced 4-cycle, and 𝑣 is adjacent to at least one but at most three vertices on 𝐶𝑣 . Let bc4 (𝐺) denote the number of bad-4-cut-vertices in 𝐺. Definition 10.2 A cycle 𝐶 is called a special-cycle if 𝐶 is a 5-cycle in 𝐺 such that if 𝑢 and 𝑣 are consecutive vertices on 𝐶, then at least one of 𝑢 and 𝑣 has degree 2 in 𝐺. Let sc(𝐺) be the maximum number of vertex-disjoint special-cycles in 𝐺 that contain no bad-4-cut-vertices. As observed in [477], if 𝐺 is a graph of order 𝑛, then sc(𝐺) + bc4 (𝐺) ≤ 15 𝑛. By contracting two vertices 𝑥 and 𝑦 in a graph 𝐺, we mean replacing the vertices 𝑥 and 𝑦 by a new vertex 𝑣 𝑥 𝑦 and joining 𝑣 𝑥 𝑦 to all vertices in 𝑉 \ {𝑥, 𝑦} that were adjacent to 𝑥 or 𝑦 in 𝐺. The authors in [477] defined two types of reducible graphs. Using these reductions, they defined a family F≤13 of graphs of order at most 13. Definition 10.3 If a graph 𝐺 has a path 𝑣 1 𝑢 1 𝑢 2 𝑣 2 such that deg𝐺 (𝑢 1 ) = deg𝐺 (𝑢 2 ) = 2, then we call the graph 𝐺 (𝑣 1 , 𝑣 2 ) obtained from 𝐺 by contracting 𝑣 1 and 𝑣 2 and deleting {𝑢 1 , 𝑢 2 } a Type-1 𝐺-reducible graph. We note that if 𝐺 has order 𝑛, then the resulting graph 𝐺 (𝑣 1 , 𝑣 2 ) has order 𝑛 − 3.

Section 10.2. Domination and Forbidden Cycles

293

Definition 10.4 If a graph 𝐺 has a path 𝑥 1 𝑤 1 𝑤 2 𝑤 3 𝑥2 such that deg𝐺 (𝑤 2 ) = 2 and N𝐺 (𝑤 1 ) = N𝐺 (𝑤 3 ) = {𝑥1 , 𝑥2 , 𝑤 2 }, then we call the graph 𝐺 (𝑥1 , 𝑥2 ) obtained from 𝐺 by deleting {𝑤 1 , 𝑤 2 , 𝑤 3 } and adding the edge 𝑥 1 𝑥2 if the edge is not already present in 𝐺 a Type-2 𝐺-reducible graph. If 𝐺 has order 𝑛, then the resulting graph 𝐺 (𝑥 1 , 𝑥2 ) has order 𝑛 − 3. Definition 10.5 Let F4 = {𝐶4 }. For every 𝑖 ≡ 1 (mod 3) with 𝑖 > 4, we define the family F𝑖 as follows. A graph 𝐺 belongs to F𝑖 if and only if 𝛿(𝐺) ≥ 2 and there is a Type-1 or Type-2 𝐺-reducible graph in F𝑖−3 . By construction, for every 𝑖 ≥ 4 with 𝑖 ≡ 1 (mod 3), if 𝐺 ∈ F𝑖 , then 𝐺 has order 𝑖. Recall that in Chapter 6, we defined the family Bdom = {𝐵1 , 𝐵2 , . . . , 𝐵7 } of seven exceptional connected graphs with the property that if 𝐺 ∈ Bdom has order 𝑛, then 𝛿(𝐺) ≥ 2 and 𝛾(𝐺) > 52 𝑛. The family Bdom is featured in the statement of the classic McCuaig-Shepherd result, namely Theorem 6.18. We repeat here a drawing of the graphs in the family Bdom in Figure 10.2. The 4-cycle, labeled 𝐵1 in Figure 10.2, is a Type-1 𝐵𝑖 -reducible graph for all 𝑖 ∈ {2, 3, . . . , 7} and so 𝐵𝑖 ∈ F7 for all 𝑖 ∈ {2, 3, . . . , 7}. Note that the 4-cycle 𝐵1 is also a Type-2 𝐵𝑖 -reducible graph for 𝑖 ∈ {6, 7}. Henning et al. [477] observed that the family F7 consists of precisely the six graphs {𝐵2 , 𝐵3 , . . . , 𝐵7 } in the family Bdom of order 7. Hence, F4 = {𝐶4 }

and

F7 = Bdom \ {𝐶4 },

and |F4 | = 1 and |F7 | = 6. Equivalently, Bdom = F4 ∪ F7 .

𝐵1

𝐵4

𝐵2

𝐵5

𝐵3

𝐵6

𝐵7

Figure 10.2 The family Bdom

Definition 10.6 Let F≤13 = F4 ∪ F7 ∪ F10 ∪ F13 . As shown in [477], if 𝐺 ∈ F≤13 has order 𝑛, then 𝑛 ∈ {4, 7, 10, 13} and 𝛾(𝐺) = 13 (𝑛 + 2). A computer search shows that there are 28,076 non-isomorphic graphs in the family F≤13 . Furthermore, 41 of these graphs have bad-4-cut-vertices.

294 Therefore, let

Chapter 10. Domination and Forbidden Subgraphs

 Fforbid = 𝐺 ∈ F≤13 : bc4 (𝐺) = 0}

consist of the 28,035 graphs in the family F≤13 that do not have a bad-4-cut-vertex. We are now in a position to state the main result in [477]. Theorem 10.7 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then either 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 83 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). As a consequence of Theorem 10.7, we have the following result. Corollary 10.8 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, no special cycle, and no bad-4-cut-vertex, then 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 38 𝑛. If 𝐺 is a graph with 𝛿(𝐺) ≥ 3, then 𝐺 contains no special cycle and no bad4-cut-vertex and 𝐺 ∉ Fforbid . Hence, Theorem 6.20 in Chapter 6 due to Reed is an immediate consequence of Corollary 10.8. As observed earlier, every graph in the family Fforbid has order at most 13, implying the following corollary of Theorem 10.7. Corollary 10.9 ([477]) If 𝐺 is a connected graph of order 𝑛 ≥ 14 with 𝛿(𝐺) ≥ 2, no special cycle, and no bad-4-cut-vertex, then 𝛾(𝐺) ≤ 38 𝑛. The McCuaig-Shepherd result given in Theorem 6.18 in Chapter 6 is an immediate corollary of Theorem 10.7. Corollary 10.10 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 is one of the seven exceptional graphs in the family Bdom . Proof By Theorem 10.7, 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 38 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). Suppose that 𝐺 ∈ Fforbid . Recall that F≤13 = F4 ∪ F7 ∪ F10 ∪ F13 , where F4 = {𝐶4 } and Bdom = F4 ∪ F7 . If 𝐺 ∈ F10 , then 𝑛 = 10 and 𝛾(𝐺) = 13 (𝑛 + 2) = 4 = 25 𝑛, while if 𝐺 ∈ F13 , then 𝑛 = 13 and 𝛾(𝐺) = 13 (𝑛 + 2) = 5 < 25 𝑛. Hence, if 𝐺 ∈ Fforbid , then either 𝐺 ∈ Bdom or 𝛾(𝐺) ≤ 25 𝑛. Suppose next that 𝛾(𝐺) ≤ 38 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). Since sc(𝐺) + bc4 (𝐺) ≤ 15 𝑛, this simplifies to 𝛾(𝐺) ≤ 38 𝑛 + 18 · 15 𝑛 = 25 𝑛. If 𝐺 is (𝐶4 , 𝐶5 )-free, then bc4 (𝐺) = sc(𝐺) = 0. Thus, we have the following consequence of Theorem 10.7. Corollary 10.11 ([477]) If 𝐺 is a (𝐶4 , 𝐶5 )-free connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then the following hold: (a) If 𝐺 ∉ Fforbid , then 𝛾(𝐺) ≤ 38 𝑛. (b) If 𝑛 ≥ 14, then 𝛾(𝐺) ≤ 38 𝑛. An (infinite) family of (𝐶4 , 𝐶5 )-free connected graphs achieving equality in the bound of Corollary 10.11(b) can be constructed as follows. Define an 8-key to be a graph of order 8 obtained from a 7-cycle by adding a pendant edge to one of its vertices. Define a unit to be a graph that is isomorphic to an 8-cycle or to an 8-key.

Section 10.2. Domination and Forbidden Cycles

295

We call a unit a cycle unit or a key unit if it is an 8-cycle or an 8-key, respectively. In each cycle unit, we select an arbitrary vertex 𝑣 and the two vertices at distance 3 from 𝑣 in the unit and we call these three vertices the link vertices of the cycle-unit, while in a key-unit we call the vertex of degree 1 the link vertex of the unit. Let Hdom be the family of all graphs 𝐺 such that either 𝐺 = 𝐶8 or 𝐺 can be obtained from the disjoint union of at least two units, each of which is a cycle unit or a key unit, by adding edges between link vertices so that the resulting graph is a (𝐶4 , 𝐶5 )-free connected graph where each added edge is a bridge of 𝐺. Every dominating set in 𝐺 contains at least three vertices from each unit of 𝐺. Thus, if 𝐺 has 𝑘 ≥ 2 units, then 𝑛 = 8𝑘 ≥ 16 and 𝛾(𝐺) ≥ 3𝑘 = 83 𝑛. By Corollary 10.11(b), 𝛾(𝐺) ≤ 38 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛. A graph 𝐺 in the family Hdom with two cycle units and two key units is shown in Figure 10.3, where the highlighted vertices are the link vertices.

Figure 10.3 A graph 𝐺 in the family Hdom We note that if 𝐺 is 2-connected graph of order 𝑛 ≥ 5, then bc4 (𝐺) = 0 since 𝐺 can have no cut-vertices. Also, if deg𝐺 (𝑢) + deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 in a graph 𝐺, then sc(𝐺) = 0. Hence, we have the following corollary of Theorem 10.7. Corollary 10.12 ([477]) If 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and deg𝐺 (𝑢)+ deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺, then 𝛾(𝐺) ≤ 38 𝑛. Corollary 10.12 can be restated as follows. Corollary 10.13 ([477]) If 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and the set of vertices of degree 2 in 𝐺 is an independent set, then 𝛾(𝐺) ≤ 38 𝑛. That the bound of Corollary 10.12 is tight may be seen as follows. Let 𝑘 ≥ 2 be an integer and let G2conn be the family of all graphs that can be obtained from a 2-connected graph 𝐹 of order 2𝑘 that contains a perfect matching 𝑀 using the following procedure. Replace each edge 𝑢𝑣 in the matching 𝑀 by an 8-cycle 𝐶𝑢𝑣 : 𝑣 1 𝑣 2 . . . 𝑣 8 𝑣 1 with two chords 𝑣 1 𝑣 4 and 𝑣 2 𝑣 5 , where 𝑢 = 𝑣 8 and 𝑣 = 𝑣 6 . Let 𝐺 𝑢𝑣 denote the resulting subgraph and let 𝐺 denote the resulting 2-connected graph of order 𝑛 = 8𝑘. Every dominating set in 𝐺 contains at least three vertices from 𝐺 𝑢𝑣 for each edge 𝑢𝑣 ∈ 𝑀, and so 𝛾(𝐺) ≥ 3𝑘 = 38 𝑛. Since 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and deg𝐺 (𝑢) + deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺,

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296

by Corollary 10.12, we have 𝛾(𝐺) ≤ 83 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛. A graph in the family G2conn with 𝑘 = 4 that is obtained from an 8-cycle 𝐹 is shown in Figure 10.4.

𝑣7 𝑣

𝑢

𝑣5

𝑣1 𝑣2

𝑣4 𝑣3

Figure 10.4 A graph in the family G2conn

Domination and Large Girth In this section, we study bounds on the domination number when there are no small cycles. The girth of a graph that contains a cycle is the length of a shortest cycle in the graph. In 1990 Brigham and Dutton [119] observed that the deletion of a shortest cycle 𝐶 of length 𝑔 from a graph 𝐺 of order 𝑛 with minimum degree at least 2 and girth at least 5 produces an isolate-free graph 𝐺 ′ of order 𝑛− 𝑔,implying  𝑔  by  𝑛 Ore’s 𝑔 bound given in Theorem 6.2 that 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 𝛾(𝐶) ≤ 𝑛−𝑔 + ≤ 2 3 2 − 6 . Theorem 10.14 ([119]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   𝑛 𝑔 𝛾(𝐺) ≤ − . 2 6 In 2005 Volkmann [735, 736] strengthened the bound of Theorem 10.14 slightly as follows. Theorem 10.15 ([735, 736]) For 𝑖 ∈ [2], if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   𝑛 𝑔 𝑖+1 𝛾(𝐺) ≤ − − , 2 6 2 unless 𝐺 is a cycle or 𝐺 belongs to a set G𝑖 of graphs, where |G1 | = 2 and |G2 | = 40. Motivated by these results, in 2008 Rautenbach [650] proved the following result. Theorem 10.16 ([650]) For every 𝑘 ∈ N, there exists a finite set G𝑘 of graphs such that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then 𝛾(𝐺) ≤

𝑛 𝑔 − − 𝑘, 2 6

unless 𝐺 is a cycle or 𝐺 belongs to the set G𝑘 . As discussed in Chapter 6, in 1996 Reed  [655] conjectured that if 𝐺 is a connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛 . Although the conjecture is false, all known

Section 10.2. Domination and Forbidden Cycles

297

counterexamples, including the constructions given by Kostochka and Stodolsky [538] and Kelmans [521], contain small cycles. Much discussion therefore centered on when Reed’s conjecture becomes true with the additional condition that the girth 𝑔 of the graph is sufficiently large. The first such result was presented in 2006 by Kawarabayashi et al. [518]. Recall that a 2-factor of a graph 𝐺 is a spanning 2-regular subgraph of 𝐺, that is, a collection of vertex-disjoint cycles that contain all the vertices of 𝐺. Theorem 10.17 ([518]) If 𝐺 is a connected cubic graph of order 𝑛 with girth 𝑔 ≥ 3 and a 2-factor, then   1 1 𝛾(𝐺) ≤ 𝑛. + 3 9⌊𝑔/3⌋ + 3 In 2009 Kostochka and Stodolsky [539] improved the bound of Theorem 10.17 for graphs without short cycles. Their proof exploits the ideas and techniques of vertex-disjoint path covers used in Reed’s seminal paper [655] and in addition uses intricate discharging arguments. Theorem 10.18 ([539]) If 𝐺 is a connected cubic graph of order 𝑛 with girth 𝑔 ≥ 3, then   1 8 𝛾(𝐺) ≤ + 𝑛. 3 3𝑔 2 In 2008 Rautenbach and Reed [651] and in 2012 Král et al. [540] established further upper bounds on the domination number of a cubic graph in terms of its order and girth. In 2008 Löwenstein and Rautenbach [572] proved a best possible upper bound on the domination number of graphs of minimum degree at least 2 and girth at least 5. Theorem 10.19 ([572]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then ! 1 2 𝛾(𝐺) ≤ +  
𝑛. 3 3 3 𝑔+1 + 1 3

The bound of Theorem 10.19 is best possible for the union of cycles 𝐶3⌊ (𝑔+1)/3⌋+1 .   Since 3 𝑔+1 + 1 ≥ 𝑔 for 𝑔 ≥ 3, as an immediate consequence of Theorem 10.19, 3 we have the following result. The bound of Corollary 10.20 is best possible for the union of cycles 𝐶𝑔 , where 𝑔 ≡ 1 (mod 3). Corollary 10.20 ([572]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   1 2 𝛾(𝐺) ≤ + 𝑛. 3 3𝑔 As a corollary of Theorem 10.19, Löwenstein and Rautenbach [572] proved the following upper bound on the domination number of a cubic graph of girth at least 5.

Chapter 10. Domination and Forbidden Subgraphs

298

Theorem 10.21 ([572]) If 𝐺 is a cubic graph of order 𝑛 with girth 𝑔 ≥ 5, then   44 82 𝛾(𝐺) ≤ + 𝑛. 135 135𝑔 44 Since 135 + Theorem 10.21.

82 135𝑔


13 𝑛, as shown in Figure 10.5, where the highlighted vertices form a 𝛾-set of 𝐺.

Figure 10.5 The generalized Petersen graph 𝑃(7, 2) Another closely related 13 -conjecture for domination in cubic graphs can be attributed to Kostochka [536], who announced the following question in the open problem session at the Third International Conference on Combinatorics, Graph Theory and Applications, held at Elgersburg, Germany, March 2009: Is it true that the domination number of a bipartite cubic graph is at most one-third its order? Kostochka and Stodolsky commented in their paper in [539] that it would be interesting to answer this question. This intriguing question of Kostochka was posed seven years later as a formal conjecture by Henning in [459]. Conjecture 10.24 ([459]) If 𝐺 is a bipartite cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 1 3 𝑛.

Section 10.2. Domination and Forbidden Cycles

299

Both Conjectures 10.23 and 10.24 seem to be very challenging. The following conjecture, posed independently in 2006 by Kelmans [521] and in 2005 by Kostochka and Stodolsky [538], claims that Reed’s conjecture is true for 3-connected cubic graphs. Conjecture   10.25 ([521, 538]) If 𝐺 is a 3-connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 𝑛3 . At the same Elgersburg’s conference, Kostochka also posed the following question: Is it true that the vertex set of every bipartite cubic graph can be partitioned into three dominating sets? If this is true, then it would be a stronger result than the truth of Conjecture 10.24. This question of Kostochka was subsequently posed seven years later as a formal conjecture by Henning [459] in 2016. Conjecture 10.26 ([459]) The vertex set of every bipartite cubic graph can be partitioned into three dominating sets.

10.2.2

Total Domination Number

In this section, we present improved bounds on the total domination number of a graph when certain cycles are forbidden. Forbidden Induced 6-cycles and Minimum Degree Two Recall that Gtdom consists of all graphs 𝐺 that can be constructed from a connected graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 6-cycle 𝐶𝑣 and join 𝑣 either to one vertex of 𝐶𝑣 or to two vertices at distance 2 apart on the cycle 𝐶𝑣 . The graph 𝐹 is called the underlying graph of 𝐺. A graph 𝐺 ∈ Gtdom with the underlying graph 𝐹 = 𝐶5 is illustrated in Figure 10.6. 𝐹 = 𝐶5

Figure 10.6 A graph in the family Gtdom As shown in Theorem 6.47, if 𝐺 is a connected graph of order 𝑛 > 14 and 𝛿(𝐺) ≥ 2, then the extremal graphs 𝐺 achieving the upper bound of 47 𝑛 for the total domination number are precisely the graphs in the family Gtdom . We note that every graph in the family Gdom contains an induced 6-cycle. Hence, it is a natural question to ask if the 47 -bound on the total domination can be improved if we forbid induced 6-cycles, or if this bound is still asymptotically best possible. In 2009 Henning and Yeo [485] showed that the 47 -bound can indeed be improved if

300

Chapter 10. Domination and Forbidden Subgraphs

we forbid induced 6-cycles. They proved a much stronger result. Some additional terminology is needed to present it. Definition 10.27 For vertex-disjoint subsets 𝑋 and 𝑌 of a graph 𝐺, an (𝑋, 𝑌 )-total dominating set, abbreviated (𝑋, 𝑌 )-TD-set, of 𝐺 is defined in [485] as a set 𝑆 of vertices of 𝐺 such that 𝑋 ∪ 𝑌 ⊆ 𝑆 and 𝑉 \ 𝑌 ⊆ N(𝑆). Thus, an (𝑋, 𝑌 )-TD-set is a set 𝑆 ⊆ 𝑉 such that 𝑆 contains 𝑋 ∪ 𝑌 and 𝑆 totally dominates the set 𝑉 \ 𝑌 . In particular, every vertex in 𝑋 is required to belong to 𝑆 and to have a neighbor in the (𝑋, 𝑌 )-TD-set 𝑆, while the vertices in 𝑌 are only required to belong to 𝑆 but not necessarily have a neighbor in 𝑆. The (𝑋, 𝑌 )-total domination number of 𝐺, denoted by 𝛾t (𝐺; 𝑋, 𝑌 ), is the minimum cardinality of an (𝑋, 𝑌 )-TD-set in 𝐺. Note that (∅, ∅)-TD-sets in 𝐺 are precisely the TD-sets in 𝐺. Hence, 𝛾t (𝐺) = 𝛾t (𝐺; ∅, ∅). As remarked in [485], the concept of an (𝑋, 𝑌 )-TD-set is related to the concept of restricted domination in graphs when certain vertices are specified to belong to the dominating set, introduced in 1997 by Sanchis in [664] and studied further, for example, in [351, 455]. Restricted total domination in graphs was first introduced and studied by Henning in [457]. Definition 10.28 Let 𝐺 be a graph and let 𝑋 and 𝑌 be disjoint vertex sets in 𝐺. A vertex 𝑥 in 𝐺 is called an (𝑋, 𝑌 )-cut-vertex in [485] if the following hold: (a) The graph 𝐺 − 𝑥 contains a component which is an induced 6-cycle 𝐶 𝑥 and which does not contain any vertices from 𝑋 or 𝑌 . (b) The vertex 𝑥 is adjacent to exactly one vertex on 𝐶 𝑥 or to exactly two vertices at distance 2 apart on 𝐶 𝑥 . Next we define a bad-6-cut-vertex in a graph. Definition 10.29 When 𝑋 = 𝑌 = ∅, we call an (𝑋, 𝑌 )-cut-vertex of 𝐺 a bad-6-cutvertex of 𝐺 and we denote the number of bad-6-cut-vertices in 𝐺 by bc(𝐺; 𝑋, 𝑌 ) (standing for “bad cut-vertex”) or simply by bc6 (𝐺). Thus, bc6 (𝐺) is the number of bad-6-cut-vertices in 𝐺. In the graph 𝐺 shown in Figure 10.6, for example, when the underlying graph 𝐹 of 𝐺 is a 5-cycle, let 𝑋 and 𝑌 be disjoint vertex sets in 𝐺 such that 𝑋 ∪ 𝑌 ⊆ 𝑉 (𝐹). Then, each vertex in 𝑉 (𝐹) is an (𝑋, 𝑌 )-cut-vertex of 𝐺 and bc(𝐺; 𝑋, 𝑌 ) = 5. In particular, when 𝑋 = 𝑌 = ∅, we have bc6 (𝐺) = 5. Next we define what is meant by a 𝐺-reducible graph, and thereafter define a family of Ftdom,𝑖 of forbidden graphs. Definition 10.30 If there is a path 𝑣 1 𝑢 1 𝑢 2 𝑢 3 𝑣 3 in a graph 𝐺 such that deg(𝑢 1 ) = deg(𝑢 2 ) = deg(𝑢 3 ) = 2 in 𝐺, then we call the graph obtained from 𝐺 by contracting 𝑣 1 and 𝑣 3 and deleting {𝑢 1 , 𝑢 2 , 𝑢 3 } a 𝑃5 -reduction of 𝐺. Note that it is possible that 𝑣 1 𝑣 3 is an edge of 𝐺. Definition 10.31 For 𝑘 ∈ {3, 5, 6}, let Ftdom,𝑘 = {𝐶 𝑘 }. For notational convenience, let Ftdom,4 = ∅. For every 𝑖 > 6, we define Ftdom,𝑖 as follows. A graph 𝐺 belongs to Ftdom,𝑖 if and only if 𝛿(𝐺) ≥ 2 and 𝐺 has a 𝑃5 -reduction to a graph that belongs to Ftdom,𝑖−4 .

Section 10.2. Domination and Forbidden Cycles

(a)

(b)

(c)

(d)

301

(e)

(f)

Figure 10.7 The family Ftdom,9

To illustrate Definition 10.31, the six graphs in the family Ftdom,9 are shown in Figure 10.7. ′ and 𝐶 ′′ , shown in Figure 6.19 in Chapter 6, belong to the The graphs 𝐶10 10 family Ftdom,10 . Every graph in the family Ftdom,𝑖 is a connected graph of order 𝑖. Further, the total domination number of a graph in the family Ftdom,𝑖 cannot be too large, as the following lemma shows.   Lemma 10.32 ([485]) For every integer 𝑖 ≥ 3, if 𝐺 ∈ Ftdom,𝑖 , then 𝛾t (𝐺) ≤ 𝑖+2 2 . Let 𝐺 7 be the graph shown in Figure 10.8 (and also shown in Figure 6.17(a)).

Figure 10.8 The graph 𝐺 7 We now define a family Ftdom of (forbidden) graphs. Let I = {3, 5, 6, 7, 9, 10, 14, 18} and let ! Ø Ftdom = {𝐾2 , 𝐺 7 } ∪ Ftdom,𝑖 . 𝑖∈I

 ′ , 𝐶 ′′ . The Recall that in Section 6.3.2, we defined Btdom = 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 10 following result is proven in [485]. Lemma 10.33 ([485]) If 𝐺 ∈ Ftdom has order 𝑛, then the following hold:   (a) 𝛾t (𝐺) ≤ 12 𝑛 + 1. (b) If 𝐺 ≠ 𝐾2 and 𝐺 ∉ Btdom , then 𝛾t (𝐺) ≤ 47 𝑛. As before, let 𝑋 and 𝑌 be vertex-disjoint sets in a graph 𝐺. Let 𝛿1 (𝐺; 𝑋, 𝑌 ) denote the number of vertices of degree 1 in 𝐺 that do not belong to the set 𝑌 , and let 𝛿2,1 (𝐺; 𝑋, 𝑌 ) denote the number of vertices of degree 2 in 𝐺 that do not belong to 𝑋 ∪ 𝑌 and are adjacent to a degree-1 vertex in 𝐺 that does not belong to 𝑋 ∪ 𝑌 . We are now in a position to state the main result in [485]. We remark that the proof of this result given by Henning and Yeo [485] relies heavily on an interplay between total domination in graphs and transversals in hypergraphs.

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Theorem 10.34 ([485]) If 𝐺 is a connected graph of order 𝑛 ≥ 2 and 𝑋 and 𝑌 are two vertex-disjoint sets of 𝐺, then either 𝑋 = 𝑌 = ∅ and 𝐺 ∈ Ftdom or 11𝛾t (𝐺; 𝑋, 𝑌 ) ≤ 6𝑛 + 8|𝑋 | + 5|𝑌 | + 2bc(𝐺; 𝑋, 𝑌 ) + 2𝛿1 (𝐺; 𝑋, 𝑌 ) + 2𝛿2,1 (𝐺; 𝑋, 𝑌 ). If 𝐺 is a graph with 𝛿(𝐺) ≥ 2 and 𝑋 and 𝑌 are vertex-disjoint sets in 𝐺, then 𝛿1 (𝐺; 𝑋, 𝑌 ) = 𝛿2,1 (𝐺; 𝑋, 𝑌 ) = 0. Hence, setting 𝑋 = 𝑌 = ∅ in Theorem 10.34, we have the following immediate consequence of Theorem 10.34. Theorem 10.35 ([485]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then either 𝐺 ∈ Ftdom or 6 2 𝛾t (𝐺) ≤ 11 𝑛 + 11 bc6 (𝐺). As a consequence of Lemma 10.33 and Theorem 10.35, we have the following result, which is a restatement of Theorem 6.46. Corollary 10.36 ([485]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. We note that the number of bad-6-cut-vertices in a graph 𝐺 of order 𝑛 is bc6 (𝐺) ≤ 17 𝑛. By Theorem 10.35, either 𝐺 ∈ Ftdom or 𝛾t (𝐺) ≤

6 11 𝑛

+

2 11 bc6 (𝐺)

≤

6 11 𝑛

+

2 11

· 17 𝑛 = 47 𝑛.

If 𝐺 ∈ Ftdom and 𝐺 ∉ Btdom , then by Lemma 10.33(b), we have 𝛾t (𝐺) ≤ Hence, 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 ∈ Btdom .

4 7 𝑛.

If 𝐺 contains no induced 6-cycle, then bc6 (𝐺) = 0. Since every graph in the family Ftdom has order at most 18, as a consequence of Theorem 10.35, we have the following result. Theorem 10.37 ([485]) If 𝐺 is a connected 𝐶6 -free graph of order 𝑛 ≥ 19 with 6 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 11 𝑛. Thus, by Theorem 10.37, if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum degree at least 2 can be 6 improved from the 47 -bound in Theorem 6.46 to a 11 -bound. To illustrate the tightness of Theorem 10.37, let Gtdom,6 be the family of graphs 𝐺 that can be constructed from a connected 𝐶6 -free graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 10-cycle 𝐶𝑣 and join 𝑣 to one vertex of 𝐶𝑣 . The graph 𝐹 is called the underlying graph of 𝐺. Each graph 𝐺 ∈ Gtdom,6 is a connected 6 𝐶6 -free graph of order 𝑛 with 𝛾t (𝐺) = 11 𝑛. A graph 𝐺 ∈ Gtdom,6 when the underlying graph 𝐹 = 𝐶4 is illustrated in Figure 10.9. Forbidden 4- and 6-cycles and Minimum Degree Three In the previous section, we showed that if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum degree 2 can be

Section 10.2. Domination and Forbidden Cycles

303

𝐹 = 𝐶4

Figure 10.9 A graph in the family Gtdom,6

6 improved from a 47 -bound (see Theorem 6.46) to a 11 -bound (see Theorem 10.37). In this section, we study the effect of forbidden 4- and 6-cycles on the total domination number of a graph with minimum degree at least 3. As shown in Theorems 6.52 and 6.59, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 ∈ Gcubic ∪ Hcubic or 𝐺 is the generalized Petersen graph 𝑃(8, 3), where the families Gcubic and Hcubic are defined in Section 6.3.3. An example of a graph 𝐺 4 ∈ Gcubic and a graph 𝐻4 ∈ Hcubic is illustrated in Figure 10.10(a) and (b), respectively, while the generalized Petersen graph 𝑃(8, 3) is shown in Figure 10.10(c). We note that every vertex that belongs to a graph in the family Gcubic ∪ Hcubic belongs to an induced 6-cycle, as does every vertex of the generalized Petersen graph 𝑃(8, 3). We also note that every vertex that belongs to a graph in the family Gcubic ∪ Hcubic belongs to a 4-cycle. Hence, it is a natural question to ask if the 12 -bound on the domination number can be significantly improved if we forbid induced 4-cycles and/or forbid induced 6-cycles, or if this 12 -bound is still asymptotically best possible.

(a) 𝐺 4

(b) 𝐻4

(c) 𝑃(8, 3)

Figure 10.10 The graphs 𝐺 4 ∈ Gcubic , 𝐻4 ∈ Hcubic , and 𝑃(8, 3)

In 2021 Henning and Yeo [494] made partial progress on this problem by relaxing the condition, simply forbidding all 4-cycles and all 6-cycles (not necessarily induced), to show that that the 12 -bound can indeed be improved. To present their result, we shall need some additional hypergraph terminology. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. Two edges in a hypergraph 𝐻

304

Chapter 10. Domination and Forbidden Subgraphs

overlap if they intersect in at least two vertices. A linear hypergraph therefore has no overlapping edges. A finite affine plane AG(2, 𝑞) of dimension 2 and order 𝑞 ≥ 2 is a collection of 𝑞 2 points and 𝑞 2 + 𝑞 lines such that each line contains 𝑞 points and each point is contained in 𝑞 + 1 lines. The affine plane AG(2, 3) of dimension 2 and order 3 is illustrated in Figure 10.11. This is equivalent to a linear 3-uniform 4-regular hypergraph 𝐻 on nine vertices, where the lines of AG(2, 3) correspond to the 3-edges of 𝐻.

Figure 10.11 The affine plane AG(2, 3) The authors in [494] showed that if 𝐻 is a linear hypergraph that does not contain a subhypergraph isomorphic to the affine plane AG(2, 3) of order 3 with two vertices deleted, then 17𝜏(𝐻) ≤ 5𝑛(𝐻) + 3𝑚(𝐻). As a consequence of this hypergraph result, they proved the following graph theory result using the interplay between total domination in graphs and transversals in hypergraphs, where 𝐺 12 is the graph shown in Figure 10.12.

Figure 10.12 The graph 𝐺 12

Theorem 10.38 ([494]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 8 no 𝐺 12 subgraph, then 𝛾t (𝐺) ≤ 17 𝑛. We remark that in the statement of Theorem 10.38, all 4-cycles are forbidden, even if they are not induced. Further, both conditions, namely the conditions that the graph contains no 4-cycle and no 𝐺 12 as a subgraph, are essential. As observed earlier, there are infinitely many connected cubic graphs 𝐺 of order 𝑛 that contain 4-cycles and satisfy 𝛾t (𝐺) = 12 𝑛. Moreover, the generalized Petersen graph 𝐺 = 𝑃(8, 3) contains no 4-cycle but does contain 𝐺 12 as a subgraph and satisfies 𝛾t (𝐺) = 12 𝑛. We note that if 𝐺 is a graph that contains no 4-cycles and no 6-cycles, then 𝐺 12 is not a subgraph

Section 10.2. Domination and Forbidden Cycles

305

of 𝐺. Hence, as an immediate consequence of Theorem 10.38, we have the following result. Theorem 10.39 ([494]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 8 no 6-cycle, then 𝛾t (𝐺) ≤ 17 𝑛. We close this section on forbidden 4- and 6-cycles with the following two conjectures. Conjecture 10.40 ([496]) If 𝐺 is not the generalized Petersen graph 𝑃(8, 3) and 𝐺 8 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3 and no 4-cycle, then 𝛾t (𝐺) ≤ 17 𝑛. If Conjecture 10.40 is true, then the upper bound is tight as may be seen by considering the following family Gtdom,4 of graphs with minimum degree at least 3 and with no 4-cycles, that can be constructed as follows: let 𝐹 be a connected graph with 𝛿(𝐹) ≥ 2 that contains no 4-cycles and for each vertex 𝑣 of 𝐹, add a copy 𝐺 𝑣 of the generalized Petersen graph 𝑃(8, 3) and join 𝑣 to one vertex of 𝐺 𝑣 . A graph 𝐺 in the family Gtdom,4 is illustrated in Figure 10.13.

𝐹

Figure 10.13 A graph 𝐺 ∈ Gtdom,4

Proposition 10.41 ([490]) If 𝐺 ∈ Gtdom,4 has order 𝑛, then 𝐺 is a connected graph 8 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 𝛾t (𝐺) = 17 𝑛. The following conjecture implies that if we forbid induced 6-cycles, then the upper bound on the total domination number can be improved from a 12 -bound (see Theorem 6.52) to a 49 -bound. Conjecture 10.42 If 𝐺 is a connected 𝐶6 -free graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 49 𝑛. If Conjecture 10.42 is true, then the bound is best possible. For example, the graph 𝐺 18 of order 𝑛 = 18, shown in Figure 10.14, has no induced 6-cycle and satisfies 𝛾t (𝐺) = 8 = 49 𝑛, where the red vertices are an example of a 𝛾t -set of 𝐺 18 .

Chapter 10. Domination and Forbidden Subgraphs

306

Figure 10.14 The graph 𝐺 18

Forbidden 4-cycles and Minimum Degree Four In the previous section, we showed that if we forbid 4- and 6-cycles, then the upper bound on the total domination number of a graph with minimum degree 3 can be 8 improved from a 12 -bound (see Theorem 6.52) to an 17 -bound (see Theorem 10.39). In this section, we study the effect of forbidden 4-cycles on the total domination number of a graph with minimum degree at least 4. As shown in Theorem 6.65, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛, with equality if and only if 𝐺 is the bipartite complement of the Heawood graph, which is shown in Figure 10.15(b) along with the Heawood graph in Figure 10.15(a). We note that the Heawood graph and its bipartite complement are also shown in Chapter 6.

(a) The Heawood graph

(b) The bipartite complement

Figure 10.15 The Heawood graph and its bipartite complement

Note that every vertex in the bipartite complement of the Heawood graph belongs to a 4-cycle. It is therefore a natural question to ask whether the upper bound of 37 𝑛 on the total domination number of a graph with minimum degree at least 4 can be improved if we restrict the graph to contain no 4-cycles, or if the 37 -bound is asymptotically best possible even with 4-cycles forbidden. In 2020 Henning and Yeo [492] proved the following upper bound on the transversal number of a linear 4-uniform hypergraph. Theorem 10.43 ([492]) If 𝐻 is a 4-uniform linear hypergraph of order 𝑛 and size 𝑚, then 𝜏(𝐻) ≤ 15 (𝑛 + 𝑚). As a consequence of Theorem 10.43, we have the following result which improves the 37 -bound on the total domination number of a graph of order 𝑛 with minimum degree at least 4 that contains no 4-cycle to a 25 -bound.

Section 10.2. Domination and Forbidden Cycles

307

Theorem 10.44 ([492]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 and no 4-cycle, then 𝛾t (𝐺) ≤ 52 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 that contains no 4-cycles. Consider the open neighborhood hypergraph ONH(𝐺) of 𝐺. Note that each edge of ONH(𝐺) has size at least 4. Since 𝐺 contains no 4-cycle, the hypergraph ONH(𝐺) contains no overlapping edges and is therefore a linear hypergraph. Consider the hypergraph 𝐻 obtained from ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 4 to produce a 4-uniform linear hypergraph 𝐻 of order 𝑛(𝐻) = 𝑛 and size 𝑚(𝐻) = 𝑛. Since every transversal of 𝐻 is a transversal of ONH(𝐺), by Observation 6.56 Theorem 10.43, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤
and 1 1 + 𝑚(𝐻) = 𝑛(𝐻) (𝑛 + 𝑛) = 25 𝑛. 5 5 That the bound in Theorem 10.44 is best possible may be seen by the 4-regular bipartite graph 𝐺 30 of order 𝑛 = 30 illustrated in Figure 10.16, which has no 4-cycle and satisfies 𝛾t (𝐺 30 ) = 12 = 25 𝑛. We note that the graph 𝐺 30 is the incidence bipartite graph of the linear 4-uniform hypergraph obtained by removing an arbitrary vertex from the affine plane AG(2, 4) of order 4.

Figure 10.16 The graph 𝐺 30

Total Domination and Large Girth In this section, we study bounds on the total domination number when there are no small cycles. In 2009 Haynes and Henning [423] provided the following upper bound on the total domination number of a graph in terms of its girth 𝑔 and order 𝑛. Theorem 10.45 ([423]) If 𝐺 ≠ 𝐶𝑛 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 7, then 𝛾t (𝐺) ≤ 23 𝑛 − 16 𝑔 unless 𝐺 = 𝐻14 , where 𝐻14 is the graph shown in Figure 10.17, in which case 𝛾t (𝐺) = 8 = 23 𝑛 + 13 − 16 𝑔. If we allow small girth, then we have the following more general result in 2008 by Henning and Yeo [484]. The proof of Theorem 10.46 relies heavily on the interplay between total domination in graphs and transversals in hypergraphs.

Chapter 10. Domination and Forbidden Subgraphs

308

Figure 10.17 The graph 𝐻14

Theorem 10.46 ([484]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3,
then 𝛾t (𝐺) ≤ 12 + 𝑔1 𝑛.       For 𝑛 ≥ 3, we observe that 𝛾t (𝐶𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 . Thus, if 𝑛 ≡ 2 (mod 4)
1 1 and 𝐺 = 𝐶𝑛 , then 𝐺 has order 𝑛, girth 𝑔 = 𝑛, and 𝛾t (𝐺) = 𝑛+2 2 = 2 + 𝑔 𝑛. Hence, the bound in Theorem 10.46 is tight for cycles of length congruent to 2 modulo 4. If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3, then by Theorem 10.46,

𝛾t (𝐺) ≤ 12 + 𝑔1 𝑛 = 23 𝑛 − 16 𝑛 − 6𝑛 𝑔 . Therefore, Theorem 10.46 improves on the bound of Theorem 10.45 when 𝑛 − 6𝑛/𝑔 > 𝑔, that is, when 𝑛 ≥ 𝑔 2 /(𝑔 − 6). The upper bounds in both Theorem 10.45 and Theorem 10.46 on the total domination number of a graph with minimum degree at least 2 in terms of its order and girth were subsequently improved in 2012 by Henning and Yeo [487]. To state their result, recall that in Section 10.2.2 we defined the family Ftdom,𝑖 for 𝑖 ≥ 3. For 𝑖 ≥ 1 an integer, the family Htdom,𝑖 of graphs is defined as follows: ! ! ! 𝑖 𝑖+1 2𝑖+2 Ø Ø Ø Htdom,𝑖 = Ftdom,4𝑘+3 ∪ Ftdom,4𝑘+1 ∪ Ftdom,4𝑘+2 ∪ {𝐾2 }, 𝑘=0

𝑘=1

and let Htdom =

𝑘=1 ∞ Ø

Htdom,𝑖 .

𝑖=1

Every graph in the family Htdom,𝑖 has order at most 8𝑖 + 10. Note that the graph 𝐾2 has order 𝑛 = 2 and 𝛾t (𝐺) = 2 = 𝑛+2 2 . As a consequence of Lemma 10.32, we have the following result.   Lemma 10.47 ([485]) If 𝐺 ∈ Htdom has order 𝑛, then 𝛾t (𝐺) ≤ 𝑛+2 2 . We are now in a position to state the following result from [487]. Theorem 10.48 ([487]) For 𝑖 ≥ 1 an integer, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 4𝑖 + 3, then either 𝐺 ∈ Htdom,𝑖 or   2𝑖 + 4 𝛾t (𝐺) ≤ 𝑛. 4𝑖 + 7 As a consequence of Theorem 10.48, the upper bounds given in Theorem 10.45 and Theorem 10.46 can be improved as follows.

Section 10.2. Domination and Forbidden Cycles

309

Theorem 10.49 ([487]) If 𝐺 is a connected graph order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3, then   𝑛 𝑛 𝛾t (𝐺) ≤ + max 1, . 2 2(𝑔 + 1) Proof Let 𝐺 be a connected graph order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3. When 𝑔 ≤ 6, we have     1 1 1 1 4 + 𝑛≥ + 𝑛 = 𝑛. 2 2(𝑔 + 1) 2 14 7 By Theorem 6.46, we know that 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . However, each of the six graphs 𝐺 in the family Btdom satisfies 𝛾t (𝐺) ≤ 12 𝑛 + 1. Hence, if 𝑔 ≤ 6, then the result of the theorem is immediate. Therefore, we may assume that 𝑔 ≥ 7. Since 𝑔 ≥ 7, we can write 𝑔 = 4𝑖 + 𝑗 for some integers 𝑖 ≥ 1 and 𝑗 ∈ {3, 4, 5, 6}. In particular, 𝑔 ≥ 4𝑖 + 3. By Theorem 10.48, either 𝐺 ∈ Htdom , in which case 𝛾t (𝐺) ≤ 12 𝑛 + 1 by Lemma 10.47, or  𝛾t (𝐺) ≤

   2𝑖 + 4 1 1 𝑛≤ + 𝑛. 4𝑖 + 7 2 2(𝑔 + 1)

To illustrate the tightness of Theorems 10.48 and 10.49, let Hgirth be the family of all graphs that can be obtained from a connected graph 𝐻 of order at least 2 and girth at least 4𝑖 + 3 for 𝑖 ≥ 1 as follows: for each vertex 𝑣 of 𝐻, add a (4𝑖 + 6)-cycle and join 𝑣 to exactly one vertex of this cycle. Each graph 𝐺 ∈ Hgirth is a connected graph of order 𝑛 = (4𝑖 + 7)|𝑉 (𝐻)| and girth at least 4𝑖 + 3 with   2𝑖 + 4 𝛾t (𝐺) = (2𝑖 + 4)|𝑉 (𝐻)| = 𝑛. 4𝑖 + 7 A graph 𝐺 in the family Hgirth is illustrated in Figure 10.18. If 𝐻 has girth at least 𝑛 4𝑖 + 6, then the resulting graph 𝐺 has girth 𝑔 = 4𝑖 + 6 and satisfies 𝛾t (𝐺) = 𝑛2 + 2(𝑔+1) , thus achieving equality in the bound of Theorem 10.49.

𝐻

𝐶4𝑖+6

𝐶4𝑖+6

𝐶4𝑖+6

𝐶4𝑖+6

Figure 10.18 A graph in the family Hgirth
If 𝐺 is a graph of order 𝑛 and girth 𝑔, then 𝑛 ≥ 𝑔, implying that 12 + 𝑔1 𝑛 ≥ 12 𝑛 + 1. Therefore, Theorem 10.49 is a stronger result than Theorem 10.46.

Chapter 10. Domination and Forbidden Subgraphs

310

10.2.3

Independent Domination Number

In this section, we present improved bounds on the independent domination number of a cubic graph when certain cycles are forbidden. Forbidden 4-cycles The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 are shown in Figure 10.19(a) and (b), respectively.

(a) 𝐾3,3

(b) 𝐶5 □ 𝐾2

Figure 10.19 The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 In this section, we consider Conjecture 6.95 which we presented in Section 6.4.3. We repeat here the statement of the conjecture. Conjecture 10.50 ([236]) If 𝐺 ∉ {𝐾3,3 , 𝐶5 □ 𝐾2 } is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. We show that Conjecture 10.50 is true if we forbid 4-cycles. In order to prove this result, Dorbec et al. [236] gave a stronger result. Recall that a subcubic graph is a graph with maximum degree at most 3. For a subcubic graph 𝐺, let 𝑛𝑖 (𝐺) denote the number of vertices of degree 𝑖 in 𝐺 for 𝑖 ∈ [3] 0 . Theorem 10.51 ([236]) If 𝐺 is a 𝐾2,3 -free subcubic graph with no (𝐶5 □ 𝐾2 )component, then 8𝑖(𝐺) ≤ 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). The proof of Theorem 10.51 given in [236] uses a method of weighting vertices along with detailed counting arguments, which we do not discuss here. The condition in Theorem 10.51 that 𝐺 does not contain an induced 𝐾2,3 cannot be dropped. For example, the graph 𝐺 obtained by attaching a pendant edge to a vertex of degree 2 in 𝐾2,3 (or equivalently by deleting two adjacent edges in 𝐾3,3 ) has 𝑖(𝐺) = 3, 𝑛3 (𝐺) = 3, 𝑛2 (𝐺) = 2, and 𝑛1 (𝐺) = 1, implying in this case that 8𝑖(𝐺) = 24 > 22 = 8 · 0 + 5 · 1 + 4 · 2 + 3 · 3 = 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). In the special case of Theorem 10.51 when 𝐺 is a connected cubic graph, we have the following result showing that Conjecture 10.50 is true if there is no subgraph isomorphic to 𝐾2,3 . Theorem 10.52 ([236]) If 𝐺 ≠ 𝐶5 □ 𝐾2 is a connected 𝐾2,3 -free cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛.

Section 10.2. Domination and Forbidden Cycles

311

1 defined in Section 6.4.3 are 𝐾2,3 -free, Graphs that belong to the family Fcubic and by Proposition 6.96, achieve this bound exactly. An example of a graph in the 1 family Fcubic is illustrated in Figure 10.20. Hence, the bound in Theorem 10.52 is achieved for infinitely many graphs.

1 Figure 10.20 A graph in the family Fcubic

As an immediate consequence of Theorem 10.52, we observe that Conjecture 10.50 is true if we forbid induced 4-cycles. Corollary 10.53 ([236]) If 𝐺 is a connected 𝐶4 -free cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. As a consequence of Theorem 10.51, Dorbec et al. [236] proved the following upper bound on the domination number of a subcubic graph. Theorem 10.54 ([236]) If 𝐺 is a subcubic graph, then 8𝛾(𝐺) ≤ 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). A special case of Theorem 10.54 when 𝐺 is a cubic graph of order 𝑛 is Reed’s result [655] that 𝛾(𝐺) ≤ 38 𝑛 (see Corollary 10.55 below). This provides a different proof to that given by Reed’s vertex-disjoint path proof technique, where he chooses a vdp-cover of a graph to prove the 38 -bound on the domination number of a cubic graph. However, the new proof given in [236] for cubic graphs does not extend to general graphs with minimum degree at least 3 (where vertices of degree greater than 3 are thrown into the mix). Corollary 10.55 ([655]) If 𝐺 is a cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 38 𝑛. If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺) = 6𝑛 − 2𝑚 + 2𝑖. Hence, as an immediate consequence of Theorem 10.54 we have the following 1999 result due to Fisher et al. [303] and Rautenbach [650]. Corollary 10.56 ([303, 650]) If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 4𝛾(𝐺) ≤ 3𝑛 − 𝑚 + 𝑖.

312

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Bipartite Graphs In this section, we consider upper bounds on the independent domination number of a bipartite cubic graph. As a special case of Theorem 6.90, we have the following upper bound on the independent domination number. Theorem 10.57 If 𝐺 is a connected bipartite cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 1 2 𝑛, with equality if and only if 𝐺 = 𝐾3,3 . A natural question is whether we can improve the 12 -bound in Theorem 10.57 if we exclude the exceptional graph 𝐾3,3 . In 2013 Goddard and Henning [352] posed the following conjecture. Conjecture 10.58 ([352]) If 𝐺 ≠ 𝐾3,3 is a connected bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛. Southey [683], as part of his PhD thesis, confirmed by computer search that Conjecture 10.58 is true when 𝑛 ≤ 26. The upper bound in Conjecture 10.58 is achieved, 4 for example, by the bipartite cubic graph 𝐺 22 of order 𝑛 = 22 with 𝑖(𝐺 22 ) = 8 = 11 𝑛 shown in Figure 10.21, where the highlighted vertices form an 𝑖-set of 𝐺 22 .

Figure 10.21 The bipartite cubic graph 𝐺 22 In 2014 Henning et al. [468] showed that Conjecture 10.58 is true if the girth is at least 6. Theorem 10.59 ([468]) If 𝐺 is a bipartite cubic graph of order 𝑛 with girth at 4 least 6, then 𝑖(𝐺) ≤ 11 𝑛. In 2018 Abrishami and Henning [3] showed that the bipartite condition imposed on the cubic graph in the statement of Theorem 10.59 can be relaxed. Theorem 10.60 ([3]) If 𝐺 is a cubic graph of order 𝑛 with girth at least 6, then 4 𝑖(𝐺) ≤ 11 𝑛. Subsequently, in 2019 Abrishami and Henning [4] extended the result of Theo5 4 rem 10.60 and improved the 11 -bound to a 14 -bound. Theorem 10.61 ([4]) If 𝐺 is a cubic graph of order 𝑛 with girth at least 6, then 5 𝑖(𝐺) ≤ 14 𝑛.

Section 10.2. Domination and Forbidden Cycles

313

The condition in Theorem 10.61 that 𝐺 has girth at least 6 is essential. Indeed, if 𝐺 is the cubic bipartite graph 𝐺 22 of order 𝑛 = 22 shown in Figure 10.21, then 𝐺 4 5 has girth 4 and 𝑖(𝐺) = 8 = 11 𝑛 > 14 𝑛. The result of Theorem 10.59 was subsequently improved in 2019 by Brause and Henning [109]. In order to state their result, they defined three graphs 𝐵4 , 𝐵6 , and 𝐵12 called “bad graphs” shown in Figure 10.22. Further, they defined a bad component of a graph 𝐺 to be a component of 𝐺 isomorphic to 𝐵4 , 𝐵6 , or 𝐵12 .

(a) 𝐵4

(b) 𝐵6

(c) 𝐵12

Figure 10.22 The “bad graphs” 𝐵4 , 𝐵6 , and 𝐵12 Given a graph 𝐺, let 𝑏 1 (𝐺) be the number of components of 𝐺 isomorphic to 𝐵6 or 𝐵12 , and let 𝑏 2 (𝐺) be the number of components of 𝐺 isomorphic to 𝐵4 . Brause and Henning [109] established the following upper bound on the independent domination number of a bipartite subcubic graph that does not have an induced subgraph isomorphic to 𝐾2,3 , where as before we let 𝑛𝑖 (𝐺) denote the number of vertices of degree 𝑖 in 𝐺 for 𝑖 ∈ [3] 0 . Theorem 10.62 ([109]) If 𝐺 is a 𝐾2,3 -free bipartite subcubic graph, then 11𝑖(𝐺) ≤ 11𝑛0 (𝐺) + 7𝑛1 (𝐺) + 5𝑛2 (𝐺) + 4𝑛3 (𝐺) + 𝑏 1 (𝐺) + 2𝑏 2 (𝐺). The proof of Theorem 10.62 given in [109] uses vertex weighting arguments and assigns weights 4, 5, 7, and 11 to the vertices of degrees 3, 2, 1, and 0, respectively, in the subcubic graph 𝐺. They defined the 𝑡-weight of 𝐺 as the sum of the weights of the vertices of 𝐺 and the 𝑤-weight of 𝐺 as the sum of the 𝑡-weight and the terms 𝑏 1 (𝐺) + 2𝑏 2 (𝐺). Using the notion of a 𝑡-weight and 𝑤-weight, they proved the upper bound given in the statement of Theorem 10.62. In the special case of Theorem 10.62 when 𝐺 is a connected cubic graph, we have the following result, showing that Conjecture 10.58 is true if there is no subgraph isomorphic to 𝐾2,3 . Theorem 10.63 ([109]) If 𝐺 is a connected 𝐾2,3 -free bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛. As an immediate consequence of Theorem 10.63, we observe that Conjecture 10.58 is true if we forbid 4-cycles. Corollary 10.64 ([236]) If 𝐺 is a connected 𝐶4 -free bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛.

Chapter 10. Domination and Forbidden Subgraphs

314 Triangle-free Graphs

Independent domination in triangle-free graphs was investigated in 2008 by Haviland [401] and later in 2012 by Goddard and Lyle [360], who proved the following bounds on the independent domination number (where part (c) was also established in 2010 by Shiu et al. [674]). Theorem 10.65 ([360]) If 𝐺 is a triangle-free graph of order 𝑛, then the following hold: (a) There exist graphs 𝐺 with 𝑖(𝐺) = 𝑛 − O (𝑛). 3 (b) If 𝛿(𝐺) ≥ 20 𝑛, then 𝑖(𝐺) ≤ 12 𝑛 and this bound is tight for 𝛿(𝐺) ≤ 14 𝑛.  (c) If 𝛿(𝐺) ≥ 14 𝑛, then 𝑖 ≤ max 𝑛 − 2𝛿(𝐺), 𝛿(𝐺) and this bound is tight. Equality is obtained for graphs such as the following: take a path 𝑃4 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 and replace each 𝑣 𝑖 with an independent set 𝐴𝑖 with the same neighborhood, where | 𝐴1 | = | 𝐴4 | = 12 𝑛 − 𝛿 and | 𝐴2 | = | 𝐴3 | = 𝛿. Goddard and Lyle [360] constructed triangle-free graphs 𝐺 with 𝑖(𝐺) > 12 𝑛 for 1 all 0 < 𝛿(𝐺) < 10 𝑛 as follows: for a positive integer 𝛿, let 𝐺 𝛿 be obtained from the corona 𝐶5 ◦ 𝐾1 of a 5-cycle by replacing each endvertex by an independent set of size 15 𝑛 − 𝛿 and replacing each vertex of the 5-cycle by an independent set of size 𝛿. 1 Then, 𝑖(𝐺 𝛿 ) > 12 𝑛 for all 𝛿 < 10 𝑛. They posed the following question. Question 10.66 ([360]) Is it true that if 𝐺 is a triangle-free graph of order 𝑛 with 1 𝛿(𝐺) ≥ 10 𝑛, then 𝑖(𝐺) ≤ 12 𝑛?

10.3

Domination in Claw-free Graphs

Claw-free graphs are very well-studied in graph theory. The 1997 survey of claw-free graphs by Faudree et al. [271] remains a standard reference work on the topic. The class of claw-free graphs has important structural properties, and a full description is given by Chudnovsky and Seymour in a series of seven papers that appeared in the Journal of Combinatorial Theory (see for example [173]). For an overview, we refer the reader to their 2005 survey on the structure of claw-free graphs in [172]. In this section, we present selected results on the three core domination parameters in claw-free graphs.

10.3.1

Domination and Independent Domination Numbers

In 1978 Allan and Laskar [15] wrote the first paper on domination in claw-free graphs, in which they proved the well-known result that the domination number is equal to the independent domination number for all claw-free graphs. Theorem 10.67 ([15]) If 𝐺 is a claw-free graph, then 𝛾(𝐺) = 𝑖(𝐺). Proof Let 𝐺 be a claw-free graph. Among all 𝛾-sets of 𝐺, let 𝑆 be chosen so that the subgraph 𝐺 [𝑆] induced by 𝑆 contains the fewest edges. We show that 𝑆 is an independent set. Suppose, to the contrary, that there exist vertices 𝑢 and 𝑣

Section 10.3. Domination in Claw-free Graphs

315

in 𝑆 that are adjacent. Since 𝑆 is a minimal dominating set, it follows by Lemma 2.72 that ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. Since 𝑢 and 𝑣 are adjacent vertices, we note that ipn[𝑣, 𝑆] = ∅, implying that the 𝑆-external private neighborhood of 𝑣 is nonempty, that is, epn[𝑣, 𝑆] ≠ ∅. If there are two (distinct) vertices 𝑣 1 and 𝑣 2 in epn[𝑣, 𝑆] that are not adjacent, then the subgraph 𝐺 [{𝑢, 𝑣, 𝑣 1 , 𝑣 2 }] is a claw in 𝐺, a contradiction. Hence, the set epn[𝑣, 𝑆] forms a clique, implying that if we replace an arbitrary vertex 𝑣 ′ from the set epn[𝑣, 𝑆], the resulting set the vertex 𝑣 in
𝑆 with ′ ′ 𝑆 = 𝑆 \ {𝑣} ∪ {𝑣 } is a 𝛾-set of 𝐺. However, 𝐺 [𝑆 ′ ] contains fewer edges than 𝐺 [𝑆], a contradiction. Therefore, 𝑆 is an independent set and hence is an ID-set of 𝐺, and so 𝑖(𝐺) ≤ |𝑆| = 𝛾(𝐺). By definition, 𝛾(𝐺) ≤ 𝑖(𝐺). Consequently, 𝛾(𝐺) = 𝑖(𝐺). Surprisingly, it was another 20 years before domination in claw-free graphs was studied in more depth. In 1998 Dutton and Brigham [252] observed the following simple structure of minimal dominating sets in claw-free graphs. Lemma 10.68 ([252]) The subgraph induced by any minimal dominating set in a claw-free graph is the disjoint union of complete graphs. Proof Let 𝑆 be an arbitrary minimal dominating set in a claw-free graph 𝐺. Suppose, to the contrary, that the subgraph 𝐺 [𝑆] contains an induced path 𝑣 1 𝑣 2 𝑣 3 on three vertices. Since ipn[𝑣 2 , 𝑆] = ∅, Lemma 2.72 implies that epn[𝑣 2 , 𝑆] ≠ ∅. If 𝑣 4 is an arbitrary vertex in epn[𝑣 2 , 𝑆], then the subgraph 𝐺 [{𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }] is a claw in 𝐺, a contradiction. Hence, there is no induced path on three vertices in the subgraph 𝐺 [𝑆]. Equivalently, 𝐺 [𝑆] is the disjoint union of complete graphs. We note that if 𝐺 is the corona 𝐾 𝑘 ◦ 𝐾1 of a complete graph 𝐾 𝑘 of order 𝑘 ≥ 1, then 𝐺 is a connected claw-free graph of order 𝑛 = 2𝑘 ≥ 2 satisfying 𝛾(𝐺) = 12 𝑛. Hence, if 𝐺 is a connected claw-free graph of order 𝑛 ≥ 2, then the upper bound in Theorem 6.2 of 𝛾(𝐺) ≤ 12 𝑛 cannot be improved. If 𝐺 is a claw-free graph of order 𝑛 with minimum degree at least 2, then the upper bound in Theorem 6.18 of 𝛾(𝐺) ≤ 25 𝑛 cannot be improved. For example, for 𝑘 ≥ 1 an integer, let 𝐺 be obtained from the disjoint union of 𝑘 5-cycles as follows. In each 5-cycle, select one vertex and designate it as a joining vertex, and add an
edge between its two neighbors in the cycle. Thereafter, add all 𝑘2 edges between the 𝑘 joining vertices so that they form a clique. The resulting connected claw-free graph 𝐺 has order 𝑛 = 5𝑘 and satisfies 𝛾(𝐺) = 25 𝑛. For example, when 𝑘 = 4 the resulting graph 𝐺 is illustrated in Figure 10.23, where the highlighted vertices form a 𝛾-set of 𝐺. We observe that of the seven graphs that belong to the family Bdom (see Figure 10.2), only the graphs 𝐵1 = 𝐶4 and 𝐵4 = 𝐶7 are claw-free. Hence, as a consequence of Theorem 6.18 and the above observations, we have the following result. Theorem 10.69 If 𝐺 is a connected claw-free graph of order 𝑛, then the following hold: (a) If 𝛿(𝐺) ≥ 1, then 𝛾(𝐺) ≤ 12 𝑛, and this bound is tight. (b) If 𝛿(𝐺) ≥ 2 and 𝐺 ∉ {𝐶4 , 𝐶7 }, then 𝛾(𝐺) ≤ 25 𝑛, and this bound is tight.

316

Chapter 10. Domination and Forbidden Subgraphs

Figure 10.23 A connected claw-free graph 𝐺 of order 𝑛 with 𝛾(𝐺) = 25 𝑛

Recall that Reed’s result in Theorem 6.20 states that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 38 𝑛. Further, this bound is tight, as illustrated in Figure 6.10. In contrast to the result of Theorem 10.69, the tight 38 -bound in Theorem 6.20 on the domination number of a graph with minimum degree at least 3 can be improved to a 1 3 -bound in the class of claw-free graphs. In 2021 Babikir and Henning [45] proved the following result. Theorem 10.70 ([45]) If every vertex of a graph 𝐺 of order 𝑛 belongs to a triangle, then 𝛾(𝐺) ≤ 13 𝑛. If 𝐺 is a claw-free graph with minimum degree at least 3, then every vertex of 𝐺 belongs to a triangle. Hence, as an immediate consequence of Theorem 10.70, we have the following result. Theorem 10.71 ([45]) If 𝐺 is a claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 13 𝑛. The upper bound in Theorem 10.71 is tight, even if the maximum degree is arbitrarily large. To construct such a family, we first recall that by a 3-prism we mean the graph 𝐶3 □ 𝐾2 shown in Figure 10.24.

Figure 10.24 The 3-prism 𝐶3 □ 𝐾2 Let (𝐶3 □ 𝐾2 ) − be the graph obtained from the 3-prism 𝐶3 □ 𝐾2 by deleting one edge that does not belong to a triangle. We call the two vertices of degree 2 in (𝐶3 □ 𝐾2 ) − the gluing vertices of (𝐶3 □ 𝐾2 ) − . For 𝑘 ≥ 2, let 𝐿 𝑘 be obtained from 𝑘 disjoint copies of (𝐶3 □
𝐾2 ) − by selecting one gluing vertex from each copy of (𝐶3 □ 𝐾2 ) − and adding all 𝑘2 edges joining these 𝑘 gluing vertices, and thereafter
adding all 𝑘2 edges joining the remaining 𝑘 gluing vertices. We note that 𝐿 𝑘 is a connected claw-free graph on 𝑛 = 6𝑘 vertices with 𝛿(𝐺) = 3 and Δ(𝐺) = 𝑘 + 1.

Section 10.3. Domination in Claw-free Graphs

317

When 𝑘 = 4, the graph 𝐿 4 is illustrated in Figure 10.25, where the gluing vertices are highlighted. Let Lclaw-free be the family of all such graphs 𝐿 𝑘 , where 𝑘 ≥ 2. We note that there exist graphs in Lclaw-free with arbitrarily large maximum degree.

Figure 10.25 A graph 𝐿 4 in the family Lclaw-free

Let 𝐺 ∈ Lclaw-free have order 𝑛 = 6𝑘, and so 𝐺 = 𝐿 𝑘 for some 𝑘 ≥ 2. Every dominating set of 𝐺 contains at least two vertices from each of the 𝑘 copies of (𝐶3 □ 𝐾2 ) − used to construct the graph 𝐺, implying that 𝛾(𝐺) ≥ 2𝑘. Conversely, selecting two vertices from each copy of (𝐶3 □ 𝐾2 ) − , one vertex from each triangle, produces a dominating set of 𝐺, and so 𝛾(𝐺) ≤ 2𝑘. Consequently, 𝛾(𝐺) = 2𝑘 = 13 𝑛. Hence, we have the following observation, showing that the upper bound in Theorem 10.71 is tight for claw-free graphs with minimum degree 3 and arbitrarily large maximum degree. Proposition 10.72 ([43]) If 𝐺 ∈ Lclaw-free has order 𝑛, then 𝛾(𝐺) = 𝑖(𝐺) = 𝛾t (𝐺) = 13 𝑛. As a special case of Theorem 10.71, if 𝐺 is a claw-free cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛, thereby improving the upper bound of 𝛾(𝐺) ≤ 38 𝑛 given in Theorem 6.23 for general cubic graphs. The connected claw-free cubic graphs achieving equality in this bound were characterized by Babikir and Henning [43]. For 𝑘 ≥ 1 an integer, the connected claw-free cubic graph 𝑁Δ,𝑘 is constructed in [43] as follows. Take 2𝑘 disjoint copies 𝑇1 , 𝑇2 , . . . , 𝑇2𝑘 of a triangle 𝐾3 , where 𝑉 (𝑇𝑖 ) = {𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 }, and add the edges 𝑏 2𝑖−1 𝑏 2𝑖 and 𝑐 2𝑖−1 𝑐 2𝑖 for all 𝑖 ∈ [𝑘]. Further, add the edges 𝑎 2𝑖 𝑎 2𝑖+1 for all 𝑖 ∈ [𝑘], where addition is taken modulo 2𝑘, and so 𝑎 2𝑘 𝑎 2𝑘+1 is the edge 𝑎 2𝑘 𝑎 1 . Following the terminology in [43], we call the resulting graph 𝑁Δ,𝑘 a triangle-necklace with 2𝑘 triangles. We note that the triangle-necklace 𝑁Δ,1 , shown in Figure 10.26(a), is the 3-prism 𝐶3 □ 𝐾2 shown in Figure 10.24. The triangle-necklace 𝑁Δ,3 is shown in Figure 10.26(b). Let Ntriangle = {𝑁Δ,𝑘 : 𝑘 ≥ 1}. The graphs that achieve equality in the upper bound 𝛾(𝐺) ≤ 13 𝑛 for connected claw-free cubic graphs 𝐺 are precisely the graphs in the family Ntriangle . Theorem 10.73 ([43]) If 𝐺 is a connected claw-free cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛, with equality if and only if 𝐺 ∈ Ntriangle .

Chapter 10. Domination and Forbidden Subgraphs

318

(a) 𝑁Δ,1

(b) 𝑁Δ,3

Figure 10.26 The triangle-necklaces 𝑁Δ,1 and 𝑁Δ,3

10.3.2

Total Domination Number

In 1998 Dutton and Brigham [252] observed the following simple structure of minimal total dominating sets in claw-free graphs. Lemma 10.74 ([252]) Every component in the subgraph induced by a minimal TD-set in a claw-free graph is constructed from a clique of order at least 2 by adding at most one pendant edge to each vertex of the clique. Proof Let 𝑆 be an arbitrary minimal TD-set in a claw-free graph 𝐺. Suppose that the subgraph 𝐺 [𝑆] contains an induced path 𝑣 1 𝑣 2 𝑣 3 on three vertices. By Lemma 4.25, we have |epn(𝑣 2 , 𝑆)| ≥ 1 or |ipn(𝑣 2 , 𝑆)| ≥ 1. If epn(𝑣 2 , 𝑆) ≠ ∅ and 𝑣 4 is an arbitrary vertex in epn(𝑣 2 , 𝑆), then the subgraph 𝐺 [{𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }] is a claw in 𝐺, a contradiction. Hence, epn(𝑣 2 , 𝑆) = ∅, implying that |ipn(𝑣 2 , 𝑆)| ≥ 1. Hence, the vertex 𝑣 2 has a neighbor in 𝑆 of degree 1 that is uniquely totally dominated by 𝑣 2 but by no other vertex of 𝑆. Renaming vertices if necessary, we may assume that 𝑣 1 has degree 1 in 𝐺 [𝑆]. Let 𝐶 be the set of neighbors of 𝑣 2 in 𝐺 [𝑆] different from 𝑣 1 , together with the vertex 𝑣 2 , and so {𝑣 2 , 𝑣 3 } ⊆ 𝐶 = N𝐺 [𝑆 ] (𝑣 2 ) \ {𝑣 1 }. If there are two (distinct) vertices 𝑢 and 𝑤 in 𝐶 that are not adjacent, then the subgraph 𝐺 [{𝑢, 𝑤, 𝑣 1 , 𝑣 2 }] is a claw in 𝐺, a contradiction. Hence, the set 𝐶 forms a clique. Since 𝐺 is claw-free, the vertex 𝑣 2 has at most two neighbors of degree 1 in 𝐺 [𝑆]. Further, if 𝑣 2 has two neighbors of degree 1 in 𝐺 [𝑆], then 𝐶 = {𝑣 2 , 𝑣 3 } and the component of 𝐺 [𝑆] containing 𝑣 2 is a path 𝑃3 . Hence, we may assume that 𝑣 1 is the only neighbor of 𝑣 2 of degree 1 in 𝐺 [𝑆], for otherwise we have the desired structure of the component. Suppose that a vertex 𝑣 ∈ 𝐶 different from 𝑣 2 has a neighbor not in 𝐶. Similar arguments for the vertex 𝑣 2 ∈ 𝐶 show that such a neighbor of 𝑣 is unique and is a vertex of degree 1, and that every other neighbor of 𝑣 belongs to the clique 𝐶. This completes the proof of Lemma 10.74. We note that if 𝐺 is the 2-corona 𝐾 𝑘 ◦ 𝑃2 of a complete graph 𝐾 𝑘 of order 𝑘 ≥ 1, then 𝐺 is a connected claw-free graph of order 𝑛 = 3𝑘 ≥ 3 satisfying 𝛾t (𝐺) = 23 𝑛. Hence, if 𝐺 is a connected claw-free graph of order 𝑛 ≥ 2, then the upper bound in Theorem 4.27 of 𝛾t (𝐺) ≤ 23 𝑛 cannot be improved. Theorem 10.75 ([182]) If 𝐺 is a connected claw-free graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛, and this bound is tight. In contrast to the result of Theorem 10.75, the tight 47 -bound in Theorem 6.46 on the total domination number of a graph with minimum degree at least 2 can be

Section 10.3. Domination in Claw-free Graphs

319

improved to a 21 -bound in the class of claw-free graphs, if we exclude a family of graphs R ★ that we describe below. claw-free An elementary 4-subdivision of a nonempty graph 𝐺 is a graph obtained from 𝐺 by subdividing some edge four times. A 4-subdivision of 𝐺 is a graph obtained from 𝐺 by a sequence of zero or more elementary 4-subdivisions of edges of 𝐺. Adopting the notation in [281], we define a good edge of 𝐺 as an edge 𝑢𝑣 such that both N[𝑢] and N[𝑣] induce a clique in 𝐺 − 𝑢𝑣. A good 4-subdivision of 𝐺 is a 4-subdivision of 𝐺 obtained by a sequence of elementary 4-subdivisions of good edges (at each stage in the resulting graph). Let 𝑅1 , 𝑅2 , . . . , 𝑅7 be the seven graphs shown in Figure 10.27.

𝑅1

𝑅2

𝑅5

𝑅3

𝑅6

𝑅4

𝑅7

Figure 10.27 The seven graphs 𝑅1 , 𝑅2 , . . . , 𝑅7  Let R ★ 𝑖 = 𝐺 : 𝐺 is a good 4-subdivision of the graph 𝑅𝑖 for 𝑖 ∈ [7] , and let R★ be the family defined by claw-free R★ claw-free =

7 Ø

R★ 𝑖 .

𝑖=1

In 2008 Favaron and Henning [281] presented the following 12 -bound result on the total domination number of a connected claw-free graph with minimum degree at least 2 that improves the 47 -bound in Theorem 6.46 for general connected graphs with minimum degree at least 2. Theorem 10.76 ([281]) If 𝐺 is a connected claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then one of the following holds: (a) 𝛾t (𝐺) ≤ 12 𝑛. , in which case 𝛾t (𝐺) = 12 (𝑛 + 1). (b) The graph 𝐺 is an odd cycle or 𝐺 ∈ R ★ claw-free (c) The graph 𝐺 = 𝐶𝑛 where 𝑛 ≡ 2 (mod 4), in which case 𝛾t (𝐺) = 12 (𝑛 + 2). As an immediate consequence of Theorem 10.76, we have the following result. Corollary 10.77 ([281]) If 𝐺 is a connected claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 12 (𝑛 + 2), with equality if and only if 𝐺 is a cycle of length congruent to 2 modulo 4.

320

Chapter 10. Domination and Forbidden Subgraphs

We consider next claw-free cubic graphs. Let 𝑁2 be the claw-free cubic graph, called a diamond-necklace with two diamonds, shown in Figure 10.28. If 𝐺 = 𝐾4 , then 𝐺 has order 𝑛 = 4 and 𝛾t (𝐺) = 2 = 21 𝑛. If 𝐺 = 𝑁2 , then 𝐺 has order 𝑛 = 8 and 𝛾t (𝐺) = 4 = 12 𝑛.

Figure 10.28 A diamond-necklace 𝑁2 with two diamonds Recall that Archdeacon et al. [35] showed in Theorem 6.52 that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Further, recall that the infinite family of connected graphs that achieve equality in this bound are all cubic graphs and are characterized in Theorem 6.59. In 2004 Favaron and Henning [279] proved that the 1 2 -bound also holds for claw-free cubic graphs, but showed that in this case, equality only holds for two small graphs, namely 𝐾4 and 𝑁2 . Theorem 10.78 ([279]) If 𝐺 is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 ∈ {𝐾4 , 𝑁2 }. In 2010 Southey and Henning [684] showed that if we exclude the two graphs 𝐾4 and 𝑁2 , then the result of Theorem 10.78 can be strengthened. Theorem 10.79 ([684]) If 𝐺 ∉ {𝑁1 , 𝑁2 } is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 49 𝑛. In 2011 Lichiardopol [564] showed that equality in the bound of Theorem 10.79 is achieved if and only if 𝐺 ∈ {𝐺 18.1 , 𝐺 18.2 }, where 𝐺 18.1 and 𝐺 18.2 are the two claw-free cubic graphs of order 𝑛 = 18 shown in Figure 10.29(a) and (b), respectively. In particular, if 𝐺 ∈ {𝐺 18.1 , 𝐺 18.2 }, then 𝛾t (𝐺) = 8 = 49 𝑛, where the highlighted vertices in these two graphs are examples of 𝛾t -sets. Let F = {𝐾4 , 𝑁2 , 𝐺 18.1 , 𝐺 18.2 }. In 2021 Babikir and Henning [44] showed that if we exclude the four graphs in the family F , then the result of Theorem 10.79 can be improved. Theorem 10.80 ([44]) If 𝐺 ∉ F is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 37 𝑛. The bound in Theorem 10.80 is best possible, as may be seen by considering the graphs 𝐺 28.1 and 𝐺 28.2 , shown in Figure 10.30(a) and (b), respectively, constructed in [44]. If 𝐺 ∈ {𝐺 28.1 , 𝐺 28.2 }, then 𝐺 has order 𝑛 = 28 and 𝛾t (𝐺) = 12 = 37 𝑛, where the highlighted vertices of 𝐺 in the figure form a 𝛾t -set of cardinality 12. It remains, however, an open problem to characterize the graphs achieving equality in the upper bound of Theorem 10.80. Let 𝐺 30 and 𝐺 48 be the two claw-free cubic graphs shown in Figure 10.31(a) and (b), respectively. If 𝐺 = 𝐺 30 , then 𝐺 has order 𝑛 = 30 and 𝛾t (𝐺) = 12 = 25 𝑛,

Section 10.3. Domination in Claw-free Graphs

321

(a) 𝐺 18.1

(b) 𝐺 18.2

Figure 10.29 The graphs 𝐺 18.1 and 𝐺 18.2

(a) 𝐺 28.1

(b) 𝐺 28.2

Figure 10.30 The graphs 𝐺 28.1 and 𝐺 28.2

where the highlighted vertices in Figure 10.31(a) form a 𝛾t -set of 𝐺 of cardinality 12. If 𝐺 = 𝐺 48 , then 𝐺 has order 𝑛 = 48 and 𝛾t (𝐺) = 18 = 38 𝑛, where the highlighted vertices in Figure 10.31(b) form a 𝛾t -set of 𝐺 of cardinality 18.

(a) 𝐺 30

(b) 𝐺 48

Figure 10.31 The graphs 𝐺 30 and 𝐺 48

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Using matching results in cubic graphs, Favaron and Henning [280] in 2008 obtained the following upper bounds on the total domination number of a connected cubic graph that is claw-free and diamond-free. Theorem 10.81 ([280]) If 𝐺 is a connected (𝐾1,3 , 𝐾4 − 𝑒)-free cubic graph of order 𝑛 ≥ 6, then 𝛾t (𝐺) ≤ 52 𝑛, with equality if and only if 𝐺 = 𝐺 30 . Further, the authors in [280] obtained the following upper bound on the total domination of a connected cubic graph that is claw-free, diamond-free, and 𝐶4 -free. Theorem 10.82 ([280]) If 𝐺 is a connected (𝐾1,3 , 𝐾4 − 𝑒, 𝐶4 )-free cubic graph of order 𝑛 ≥ 6, then 𝛾t (𝐺) ≤ 38 𝑛, with equality if and only if 𝐺 = 𝐺 48 . It remains an open problem to determine a tight upper bound on the total domination number of a connected claw-free graph of sufficiently large order 𝑛 with 𝛿(𝐺) ≥ 3. Let 𝐹claw-free be the claw-free graph of order 7 shown in Figure 10.32.

Figure 10.32 The graph 𝐹claw-free For 𝑘 ≥ 2 an integer, let 𝐺 be obtained from the disjoint union of 𝑘 copies of
the graph 𝐹claw-free by adding all 𝑘2 edges between the 𝑘 vertices of degree 2 in each copy of 𝐹claw-free so that they form a clique. The resulting connected claw-free graph 𝐹𝑘 has order 𝑛 = 7𝑘 and satisfies 𝛾t (𝐹𝑘 ) = 3𝑘 = 37 𝑛. For example, when 𝑘 = 4 the resulting graph 𝐹4 is illustrated in Figure 10.33. Let Fclaw-free be the family of all such graphs 𝐹𝑘 , for 𝑘 ≥ 2.

Figure 10.33 A graph 𝐹4 in the family Fclaw-free Let 𝐺 ∈ Fclaw-free have order 𝑛, and so 𝐺 = 𝐹𝑘 for some 𝑘 ≥ 2. Every TD-set of 𝐺 contains at least three vertices from each of the 𝑘 copies of the graph 𝐹claw-free used to construct the graph 𝐺, implying that 𝛾t (𝐺) ≥ 3𝑘. Conversely, 𝛾t (𝐹claw-free ) = 3, and choosing a 𝛾t -set from each of the 𝑘 copies of the graph 𝐹claw-free in the graph 𝐺

Section 10.4. Summary

323

constructs a TD-set of 𝐺 of cardinality 3𝑘, and so 𝛾t (𝐺) ≤ 3𝑘. Consequently, 𝛾t (𝐺) = 3𝑘 = 73 𝑛. Hence, we have the following. Proposition 10.83 If 𝐺 ∈ Fclaw-free has order 𝑛, then 𝛾t (𝐺) = 37 𝑛. 𝑛 denote the family of all connected claw-free graphs of order 𝑛 with Let F≥3 𝛿(𝐺) ≥ 3. We note that the family Fclaw-free contains connected claw-free graphs with 𝛿(𝐺) = 3 having arbitrarily large order and maximum degree. Hence, by Proposition 10.83, ! 𝛾t (𝐺) 3 lim sup ≥ . 𝑛→∞ 𝐺 ∈ F 𝑛 𝑛 7 ≥3

It remains, however, an open problem to determine the limit of this supremum.

10.4

Summary

In this chapter, we presented improved bounds on the three core domination numbers of a graph with given structural restrictions, such as forbidding certain cycles or claws. We showed in Corollary 10.11 that if we forbid 4- and 5-cycles, then the upper bound on the domination number of a connected graph of order 𝑛 ≥ 14 with 𝛿(𝐺) ≥ 2 can be improved from the 25 -bound due to McCuaig-Shepherd to a 38 -bound, and this bound is tight. We showed in Theorem 10.37 that if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum 6 degree at least 2 can be improved from the 47 -bound in Theorem 6.46 to a 11 -bound. We established bounds on the domination and total domination numbers of a graph in terms of their order and girth. We discussed several outstanding conjectures, foremost of which are the Verstraëte 13 -conjecture, namely Conjecture 10.23, and the Kostochka’s inspired 13 -conjecture, namely Conjecture 10.24, for domination in cubic graphs. In Theorem 10.71 we showed that Reed’s tight 38 -bound in Theorem 6.20 on the domination number of a graph with minimum degree at least 3 can be improved to a 1 3 -bound in the class of claw-free graphs. In the case of claw-free cubic graphs, the graphs that achieve equality in this 13 -bound are characterized in Theorem 10.73. In Theorem 10.76 we showed that the 47 -bound in Theorem 6.46 on the total domination number of a general connected graphs with minimum degree at least 2 can be improved to a 12 -bound. In Theorem 10.80 we showed that if 𝐺 is a connected claw-free cubic graph of order 𝑛, and 𝐺 is not one of four exceptional graphs, then 𝛾t (𝐺) ≤ 37 𝑛, and this bound is best possible.

Chapter 11

Domination in Planar Graphs 11.1

Introduction

In this chapter, we present results on the core domination parameters in planar graphs. We shall adopt the following terminology for planar graphs. A graph is said to be planar if it can be drawn in the plane in such a way that no two edges intersect, except at a vertex to which they are both incident. A planar graph so drawn in the plane is said to be embedded in the plane and is called a plane graph. An embedding of a plane graph divides the plane into regions called faces. More formally, the faces of a plane graph 𝐺 are the connected pieces of the plane that remain after the points in the plane that correspond to the vertices and edges of 𝐺 are removed. Every plane graph contains exactly one unbounded face, called the outer face; the other faces are the inner faces. The boundary of a face 𝐹 (including the outer face) is the subgraph induced by the vertices and edges incident with the face 𝐹. Vertices and edges incident to the outer face of a plane graph 𝐺 are called external vertices and external edges, respectively, of 𝐺. Vertices of 𝐺 that are not external vertices are called internal vertices, and edges of 𝐺 that are not external edges are called internal edges of 𝐺. Two faces are adjacent if they have at least one edge in common. A facial triangle of a plane graph 𝐺 is a triangle whose interior is a face of 𝐺. A face bounded by a triangle is called a triangular face. For example, the plane graph 𝐺 in Figure 11.1 has four faces 𝑓𝑖 for 𝑖 ∈ [4], where 𝑓4 is its outer face and 𝑓3 is a triangular face. 𝑓4 𝑓1

𝑓2

𝑓3

Figure 11.1 A plane graph with four faces A triangulated disc (also called a near-triangulation in the literature) is a (simple) 2-connected plane graph all of whose faces are triangles, except possibly the outer © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_11

325

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face. A triangulated disc in which all faces, including the outer face, are triangles and any two face boundaries intersect in a single edge, a single vertex, or not at all is called a planar triangulation. A weak triangulated disc (also called a weak near-triangulation in the literature) is a plane graph all of whose faces are triangles, except possibly the outer face, and every vertex of 𝐺 is contained in a triangle. We remark that a weak triangulated disc may be disconnected. Moreover, every vertex of a weak triangulated disc is contained in a facial triangle. An outerplanar graph is a planar graph that has an embedding in the plane such that all vertices are external vertices. A planar (respectively, outerplanar) graph is maximal if 𝐺 + 𝑢𝑣 is not planar (respectively, outerplanar) for every two nonadjacent vertices 𝑢 and 𝑣 of 𝐺. Thus, a maximal outerplanar graph 𝐺, abbreviated mop in the literature, is a triangulated disc where every vertex of 𝐺 is an external vertex. We note that a planar triangulation is a maximal planar graph.

11.2 Domination in Planar Graphs In this section, we present results on domination in planar graphs. Grid graphs are a well-known class of planar graphs, but since they are covered in Chapter 17, we do not cover them here. Given a graph 𝐺 and an integer 𝑘, the DOMINATING SET problem is to decide if 𝐺 has a dominating set of cardinality at most 𝑘. As discussed in Chapter 3, the DOMINATING SET problem is a core NP-complete problem in combinatorial optimization and graph theory [325]. If the problem is restricted to planar graphs, it is known as the PLANAR DOMINATING SET problem, which remains NP-hard even when restricted to planar graphs of maximum degree three [325]. Hence, it is of interest to determine upper bounds on the domination number of a planar graph.

11.2.1

Domination in Planar Triangulations

An outward numbering of a triangulated disc 𝐺 of order 𝑛 is a numbering of the vertices from 1 to 𝑛 such that for every 𝑖 ∈ [𝑛], the set of vertices numbered 1 through 𝑖 induces a triangulated disc 𝐺 𝑖 for which each vertex of 𝑉 (𝐺) \ 𝑉 (𝐺 𝑖 ) is an external vertex of 𝐺 𝑖 , that is, lies on the outer face of 𝐺 𝑖 . In 1996 Matheson and Tarjan [585] studied dominating sets in triangulated discs and proved a tight upper bound on their domination numbers. Their key result is the following lemma, which shows that there is an outward numbering of a triangulated disc starting from any given triangle. Lemma 11.1 ([585]) Given an arbitrary triangle in a triangulated disc 𝐺, there is an outward numbering of 𝐺 that numbers the vertices of the given triangle 1, 2, and 3, in any desired order. Proof We proceed by induction on the order 𝑛 of a triangulated disc 𝐺 with a given triangle. We number the vertices of the given triangle arbitrarily 1, 2, and 3. For the

Section 11.2. Domination in Planar Graphs

327

inductive hypothesis, suppose that there is a numbering of the vertices 1, 2, . . . , 𝑖 for some 𝑖, where 3 ≤ 𝑖 < 𝑛, such that the vertex set [𝑖] induces a triangulated disk 𝐺 𝑖 and 1, 2, . . . , 𝑖 is an outward numbering of 𝐺 𝑖 . We describe how to extend this to an outward numbering of the vertices 1, 2, . . . , 𝑖 + 1. The triangulated disc 𝐺 properly contains the triangulated disc 𝐺 𝑖 and by definition, all vertices in 𝑉 (𝐺) \𝑉 (𝐺 𝑖 ) are external vertices of 𝐺 𝑖 . Since a triangulated disc is 2-connected, there exists a triangle 𝑇 of 𝐺 containing exactly two vertices of 𝐺 𝑖 . Thus, one vertex of 𝑇 is currently unnumbered (with no number assigned to it from [𝑖]), and two vertices of 𝑇 are numbered from [𝑖]. Let 𝑆 be the set of all such unnumbered vertices of 𝐺 that belong to a triangle with exactly one unnumbered vertex. For a vertex 𝑣 ∈ 𝑆, let {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be the set of vertices of 𝐺 𝑖 adjacent to 𝑣, indexed clockwise around the boundary of the outer face of 𝐺 𝑖 , with 𝑣 1 chosen so that none of the external edges of 𝐺 𝑖 from 𝑣 1 clockwise to 𝑣 𝑘 is an external edge of 𝐺. We note that this naming of the neighbors of 𝑣 is possible since the external edges of 𝐺 𝑖 are partitioned by 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 into 𝑘 parts, at most one of which can intersect the boundary of the outer face of 𝐺. We define 𝑇 (𝑣) as the set consisting of all triangles inside the cycle 𝐶𝑣 : 𝑣 𝑣 1 𝑣 2 . . . 𝑣 𝑘 𝑣, and so all vertices of such triangles belong to the cycle 𝐶𝑣 or to the interior region of the cycle 𝐶𝑣 . We define the score of the vertex 𝑣 to be the number of triangles in the set 𝑇 (𝑣). Among all vertices in the set 𝑆, we choose a vertex 𝑣 of minimum score. If there is some vertex of 𝑆 inside the cycle 𝐶𝑣 , then such a vertex would have a smaller score than 𝑣, contradicting our choice of the vertex 𝑣. Hence, the set 𝑇 (𝑣) must consist  exactly of the set {𝑣, 𝑣 1 , 𝑣 2 }, {𝑣, 𝑣 2 , 𝑣 3 }, . . . , {𝑣, 𝑣 𝑘−1 , 𝑣 𝑘 } , implying that the vertex 𝑣 can be numbered 𝑖 + 1. Recall that the domatic number dom(𝐺) is the maximum order of a partition of 𝑉 (𝐺) into dominating sets. We are now in a position to present the labeling argument given by Matheson and Tarjan [585] that the vertex set of a triangulated disc can be partitioned into three dominating sets. Theorem 11.2 ([585]) If 𝐺 is a triangulated disc, then dom(𝐺) ≥ 3. Proof By Lemma 11.1, there is an outward numbering of the vertices in the triangulated disc 𝐺 that starts from any given triangle. We describe next a labeling of the vertices with labels 𝑎, 𝑏, and 𝑐 such that the set of vertices with the same label is a dominating set of 𝐺. We label the vertices (using the labels 𝑎, 𝑏, and 𝑐) sequentially, taking care to preserve the following two properties: P1. The labeling constructed thus far yields three dominating sets of the subgraph induced by the current labeled vertices. P2. The boundary of the outer face of the triangulated disc defined by the vertices labeled so far has no two adjacent vertices with the same label. Initially, we label the vertices 1, 2, and 3 in the outward numbering of the vertices of 𝐺, with the labels 𝑎, 𝑏, and 𝑐, respectively. Since these three vertices form a triangle, properties P1 and P2 are true. In general, suppose that the vertices numbered 1, 2, . . . , 𝑖 are labeled, each with one of the labels 𝑎, 𝑏, and 𝑐, so that properties P1

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and P2 hold for some 𝑖, where 3 ≤ 𝑖 < 𝑛. Let 𝐺 𝑖 be the triangulated disc associated with the outward numbering 1, 2, . . . , 𝑖. We now consider the vertex 𝑣 numbered 𝑖 + 1 in the outward numbering of 𝐺. As in the proof of Lemma 11.1, the set of vertices in 𝐺 𝑖 to which vertex 𝑣 is adjacent in 𝐺 is a set {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } of vertices occurring consecutively on the boundary of the outer face of 𝐺 𝑖 where 𝑘 ≥ 2, with the vertices indexed clockwise around the boundary. The boundary of the outer face of 𝐺 𝑖+1 is formed from the boundary of the outer face of 𝐺 𝑖 by deleting the path 𝑣 1 𝑣 2 . . . 𝑣 𝑘 and replacing it with the path 𝑣 1 𝑣 𝑣 𝑘 . To label the vertex 𝑣, we consider two cases. Suppose that 𝑣 1 and 𝑣 𝑘 are labeled differently in 𝐺 𝑖 (with labels from the set {𝑎, 𝑏, 𝑐}). In this case, we label the vertex 𝑣 with the label not used for 𝑣 1 or 𝑣 𝑘 . This preserves properties P1 and P2. Suppose next that 𝑣 1 and 𝑣 𝑘 are labeled the same in 𝐺 𝑖 (with labels from the set {𝑎, 𝑏, 𝑐}). Property P2 implies that in this case, the vertices 𝑣 1 and 𝑣 𝑘 are not adjacent, and therefore 𝑘 ≥ 3 and the vertex 𝑣 2 has a different label than 𝑣 1 . In this case, we label the vertex 𝑣 with the label not used for 𝑣 1 or 𝑣 2 . Once again, this preserves both properties P1 and P2. Upon completion of the labeling of the vertices (using the labels 𝑎, 𝑏, and 𝑐), by property P1 the set of vertices with the same label is a dominating set of 𝐺 and so dom(𝐺) ≥ 3. The following upper bound on the domination number of a triangulated disc is an immediate consequence of Theorem 11.2. Corollary 11.3 ([585]) If 𝐺 is a triangulated disc of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. Matheson and Tarjan [585] gave a linear algorithm for finding a dominating set of cardinality at most 𝑛3 in a triangulated disc, by modifying slightly the construction of the outward numbering in the proof of Lemma 11.1. That the bound of Corollary 11.3 is tight may be seen as follows. For 𝑘 ≥ 1, take 2𝑘 vertex-disjoint copies 𝑇1 , 𝑇2 , . . . , 𝑇2𝑘 of a triangle, where 𝑉 (𝑇𝑖 ) = {𝑢 𝑖 , 𝑣 𝑖 , 𝑤 𝑖 }, and add the edges 𝑢 2𝑖−1 𝑢 2𝑖 , 𝑢 2𝑖−1 𝑤 2𝑖 , and 𝑤 2𝑖−1 𝑤 2𝑖 for all 𝑖 ∈ [𝑘]. To complete the construction, for 𝑘 ≥ 2 add the edges 𝑤 2𝑖−1 𝑢 2𝑖+1 , 𝑤 2𝑖−1 𝑢 2𝑖+2 , and 𝑤 2𝑖 𝑢 2𝑖+2 for all 𝑖 ∈ [𝑘 − 1]. Let 𝐺 𝑘 denote the resulting triangulated disc of order 𝑛 = 6𝑘. The graph 𝐺 3 , for example, is illustrated in Figure 11.2. Let Gouterplanar = {𝐺 𝑘 : 𝑘 ≥ 1}.

Figure 11.2 The graph 𝐺 3 Each graph 𝐺 ∈ Gouterplanar of order 𝑛 is an outerplanar graph with 𝑛/3 vertices of degree 2. Every 𝛾-set of 𝐺 contains at least one vertex from the closed neighborhood of every vertex of degree 2. However, such sets form a partition of 𝑉 (𝐺), noting that

Section 11.2. Domination in Planar Graphs

329

no two vertices of degree 2 in 𝐺 have a common neighbor. Therefore, 𝛾(𝐺) ≥ 31 𝑛. The set consisting of the 𝑛/3 vertices of degree 2 is an independent dominating set of 𝐺, and so 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 13 𝑛. Consequently, 𝛾(𝐺) = 𝑖(𝐺) = 13 𝑛. In 2010 by Honjo et al. [500] generalized the result of Corollary 11.3 and showed that every triangulation on the projective plane, the torus, and the Klein bottle of order 𝑛 has a dominating set of cardinality at most 13 𝑛. They showed that the same conclusion holds for other surfaces. The reader is referred to [500] for more details and definitions. Corollary 11.3 given by Matheson and Tarjan [585] and its generalization given by Honjo et al. in [500] follow from a more general recent result (see, Theorem 10.70 in Chapter 10) that if every vertex of a graph 𝐺 of order 𝑛 belongs to a triangle, then 𝛾(𝐺) ≤ 13 𝑛. However, Honjo et al. [500] proved a stronger result, which generalizes Theorem 11.2. Theorem 11.4 ([500]) If 𝐺 is a triangulation on the sphere, the projective plane, the torus, or the Klein bottle, then dom(𝐺) ≥ 3. Moreover, this bound is best possible. Since the class of triangulated discs is a superclass of planar triangulations, as an immediate consequence of Corollary 11.3, we have the following result. Corollary 11.5 ([585]) If 𝐺 is a planar triangulation of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. The bound in Corollary 11.5 is achieved, for example, by the planar triangulations 𝐾3 and the octahedron 𝐾2,2,2 , which we illustrate in Figure 11.3.

𝐾3

𝐾2,2,2

Figure 11.3 The planar triangulations 𝐾3 and the octahedron 𝐾2,2,2 A natural problem is to characterize the planar triangulations that achieve equality in the upper bound in Corollary 11.5. Moreover, if there are only finitely many graphs achieving equality in the bound, it would be interesting to determine if there is an improved upper bound when the order is sufficiently large, or whether the 13 -bound is asymptotically best possible. In 2020 Špacapan [688] made a breakthrough on these problems and questions when he showed that the triangle 𝐾3 and the octahedron 𝐾2,2,2 are the only planar triangulations achieving equality in the bound of Corollary 11.5. Moreover, he improved the upper bound in Corollary 11.5 for planar triangulations of order 𝑛 > 6. For this purpose, he used properties of weak triangulated discs, also referred to as

330

Chapter 11. Domination in Planar Graphs

weak near-triangulations in [688], to establish the improved bounds. Recall that a weak triangulated disc is a plane graph (possibly disconnected) all of whose faces are triangles, except possibly for the outer face, and every vertex of 𝐺 is contained in a triangle. By definition, every triangulated disc is a weak triangulated disc, but not conversely. An example of a weak triangulated disc in given in Figure 11.4.

Figure 11.4 A weak triangulated disc

The following selected properties of weak triangulated discs, given in [688], follow readily from the definition of a weak triangulated disc and properties of plane graphs. Recall that a block is a maximal nonseparable subgraph, or a maximal subgraph having no cut-vertices, and an endblock is a block containing at most one cut-vertex of 𝐺. Lemma 11.6 ([688]) If 𝐺 is a weak triangulated disc, then each of the following holds: (a) Every block of 𝐺 is either a triangulated disc or a copy of 𝐾2 . (b) Every endblock of 𝐺 is a triangulated disc. (c) The graph 𝐺 has a (vertex) 3-coloring such that each color class is a dominating set of 𝐺. As an immediate consequence of Lemma 11.6, we have the following bound on the domination number of a weak triangulated disc. Corollary 11.7 ([688]) If 𝐺 is a weak triangulated disc of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. Špacapan [688] defined a weak triangulated disc 𝐺 to be reducible if there exists a set 𝐷 ⊆ 𝑉 (𝐺) and a vertex 𝑣 ∈ 𝐷 such that all of the following three properties hold: (a) 𝐷 ⊆ N𝐺 [𝑣], (b) |𝐷| ≥ 4, (c) 𝐺 − 𝐷 is a weak triangulated disc. A weak triangulated disc is irreducible if it is not reducible. The key result of Špacapan [688] is the following theorem. Theorem 11.8 ([688]) If a weak triangulated disc 𝐺 has a block that is not an outerplanar graph and does not have order 6, then 𝐺 is reducible. As a consequence of Corollary 11.7 and Theorem 11.8, we have the following result.

Section 11.2. Domination in Planar Graphs

331

Corollary 11.9 ([688]) If 𝐺 is a triangulated disc of order 𝑛 ≠ 6 that is not an outerplanar graph, then 𝛾(𝐺) < 31 𝑛. Proof Let 𝐺 be a triangulated disc of order 𝑛 ≠ 6 that is not an outerplanar graph. By Theorem 11.8, the graph 𝐺 is reducible. Therefore, there exists a set 𝐷 ⊆ 𝑉 (𝐺) and a vertex 𝑣 ∈ 𝐷 that dominates 𝐷 such that |𝐷| ≥ 4 and 𝐺
− 𝐷 is a weak triangulated disc. By Corollary 11.7, we have 𝛾(𝐺 − 𝐷) ≤ 13 𝑛 − |𝐷 | . Every dominating set of 𝐺 − 𝐷 can be extended to a dominating set of 𝐺 by adding the vertex 𝑣 to it and so
𝛾(𝐺) ≤ 𝛾(𝐺 − 𝐷) + 1 ≤ 13 𝑛 − |𝐷 | + 1 ≤ 13 (𝑛 − 4) + 1 < 13 𝑛. 1 3𝑛

As a consequence of Corollary 11.7 and Theorem 11.8, the upper bound 𝛾(𝐺) ≤ in Corollary 11.5 for planar triangulations of order 𝑛 can be improved if 𝑛 > 6.

Theorem 11.10 ([688]) If 𝐺 is a planar triangulation of order 𝑛 > 6, then 𝛾(𝐺) ≤ 17 53 𝑛. Proof Let 𝐺 be a planar triangulation of order 𝑛 > 6 and consider an embedding of 𝐺 in the plane. By definition, all faces (including the outer face) of 𝐺 are triangles. If 𝐺 is irreducible, then let 𝐺 ′ = 𝐺. If 𝐺 is reducible, then let 𝐺 ′ be obtained from 𝐺 by repeatedly applying Theorem 11.8 until the resulting graph 𝐺 ′ is irreducible. Thus, in this case, there exist vertex-disjoint sets 𝐷 1 , 𝐷 2 , . . . , 𝐷 𝑘 of vertices of 𝐺 such that 𝐺 1 = 𝐺 and where 𝑣 𝑖 ∈ 𝐷 𝑖 , 𝐷 ⊆ N𝐺 [𝑣 𝑖 ], |𝐷 𝑖 | ≥ 4, and 𝐺 𝑖+1 = 𝐺 𝑖 − 𝐷 𝑖 is a weak triangulated disc for all 𝑖 ∈ [𝑘]. Moreover, 𝐺 ′ = 𝐺 𝑘+1 = 𝐺 𝑘 − 𝐷 𝑘 and the graph 𝐺 ′ is irreducible. By Theorem 11.8, every block of 𝐺 ′ is outerplanar or has order 6. Let 𝐺 ′ have order 𝑛′ , and so 𝑛′ ≤ 𝑛 − 4𝑘. The set 𝐷 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } dominates all vertices in 𝐷 1 ∪ 𝐷 2 ∪ · · · ∪ 𝐷 𝑘 and has cardinality |𝐷 | = 𝑘 ≤ 14 (𝑛 − 𝑛′ ), implying that the vertices in 𝑉 (𝐺) \ 𝑉 (𝐺 ′ ) are dominated by a set of cardinality at most 1 ′ 4 (𝑛 − 𝑛 ). Claim 11.10.1 If 𝐺 ′ contains all three external vertices of 𝐺, then 𝛾(𝐺)
0, there exists a constant 𝑐 = 𝑐(𝑆, 𝑡, 𝜀), dependent on 𝑆, 𝑡 and 𝜀, such that if 𝐺 is a triangulation on 𝑆 with at most 𝑡 vertices of degree other than 6, then
𝛾(𝐺) ≤ 16 + 𝜀 𝑛 + 𝑐. Liu and Pelsmajer [567] proved Theorem 11.14 with the constant 𝑐 = 𝑐(𝑆, 𝑡, 𝜀)
given by 𝑐 = O (𝑔 3 + 𝑔𝑡 2 )/𝜀 , where 𝑔 is the genus of 𝑆. Since 𝑛 is a trivial upper bound for the domination number, as remarked in√[567], Theorem 11.14 is only of
interest when 𝑐(𝑆, 𝑡, 𝜀) < 𝑛, in which case 𝑡 = O 𝑛 .

11.2.2

Domination in Outerplanar Graphs

Recall that mop is the abbreviation for maximal outerplanar graph. The class of triangulated discs is a superclass of mops. Hence, the following corollary is an

Chapter 11. Domination in Planar Graphs

334

immediate consequence of Corollary 11.3. We note that this corollary appeared as far back as 1977 in Mitchell’s PhD dissertation [592, 594]. Corollary 11.15 ([585]) If 𝐺 is a mop of order 𝑛 ≥ 3, then 𝛾(𝐺) ≤ 31 𝑛. The family Gouterplanar = {𝐺 𝑘 : 𝑘 ≥ 1} constructed earlier (see Figure 11.2 for an illustration of a graph in the family) of triangulated discs are in fact mops. Hence, the upper bound for mops given in Corollary 11.15 is best possible. The first results on the topic of domination in outerplanar graphs date back to at least 1978 when Fisk [304] produced an elegant proof to the Art Gallery Problem, a celebrated problem in the field of computational geometry, that was posed by Victor Klee to Chvátal in 1973. Problem 11.16 (Art Gallery Problem) Determine the minimum number of guards that need to be placed in an art gallery so that each point in the interior of the gallery is within the line of sight of at least one guard. The Art Gallery Problem can be placed in a mathematical context by modeling the art gallery by a polygon representing its plane view. More formally, let 𝑛 ≥ 3 be an integer. A set 𝑆 of points within a simple plane polygon P𝑛 with 𝑛 vertices (or corner points) guards P𝑛 if, for every point 𝑝 in the interior of P𝑛 , there exists some point 𝑞 ∈ 𝑆 such that the straight line segment between 𝑝 and 𝑞 lies inside the polygon P𝑛 . In 1975 Chvátal [174] solved the Art Gallery Problem.   Theorem 11.17 (Chvátal’s Watchman Theorem [174]) For 𝑛 ≥ 3, 𝑛3 guards are always sufficient and sometimes necessary to guard a simple plane polygon with 𝑛 corner points. That the 13 -upper bound in the Chvátal’s Watchman Theorem 11.17 is tight may be seen as follows. Consider a comb-shaped art museum 𝑀𝑘 with 𝑛 = 3𝑘 walls. For example, the museum 𝑀5 is illustrated in Figure 11.6(a). For 𝑘 ≥ 2 and for each 𝑖 ∈ [𝑘], the point 𝑝 𝑖 illustrated in Figure 11.6(a) can only be seen by a guard positioned in the shaded triangle containing 𝑝 𝑖 as illustrated in Figure 11.6(b). Since these 𝑘 triangles are disjoint, at least 𝑘 guards are needed. Placing one guard on the leftmost point of the bottom line and one guard on the rightmost point of the bottom line, and placing one guard at each of the points 𝑝 2 , 𝑝 3 , . . . , 𝑝 𝑘−1 , yields a placement of 𝑘 guards so that each point in the interior of the gallery is within the line of sight of at least one guard. Therefore, 𝑛3 guards are sufficient and necessary to guard a comb-shaped museum 𝑀𝑘 with 𝑛 = 3𝑘 walls. 𝑝1

𝑝2

𝑝3

(a)

𝑝4

𝑝5

𝑝1

𝑝2

𝑝3

(b)

Figure 11.6 A comb-shaped museum 𝑀5

𝑝4

𝑝5

Section 11.2. Domination in Planar Graphs

335

In 1978 Fisk [304] provided a proof of the Chvátal’s Watchman Theorem. Fisk’s proof appeared in the celebrated “Proofs from The Book” by Aigner and Ziegler [10], reserved for only the most elegant mathematical proofs. We present Fisk’s proof below. Proof of Theorem 11.17 View the polygon P𝑛 of the art gallery as a graph whose vertices are the corner points of the polygon and whose edges are the sides of the polygon. We now triangulate the part of this (planar) graph corresponding to the interior of the polygon P𝑛 to form a new graph by repeatedly adding 𝑛−3 non-crossing edges on the inside of the polygon between nonadjacent vertices until all faces of the interior are bound by triangles. The resulting graph is a mop. To illustrate this construction of a mop from a polygon P𝑛 , consider for example the mop in Figure 11.7 constructed from the polygon given in Figure 11.6(a).

Figure 11.7 A mop constructed from the polygon 𝑀5 in Figure 11.6(a) We prove by induction on 𝑛 ≥ 3 that the resulting mop 𝐺 is 3-colorable, that is, we can color the vertices with three colors so that no two adjacent vertices of 𝐺 receive the same color. For 𝑛 = 3, the graph 𝐺 = 𝐾3 and the result is immediate. This establishes the base case. Let 𝑛 ≥ 4, and assume that every mop of order 𝑛′ , where 3 ≤ 𝑛′ < 𝑛, is 3-colorable. Let 𝐺 be a mop of order 𝑛 and let 𝑢𝑣 be an arbitrary edge added to the polygon P𝑛 when constructing 𝐺. The edge 𝑢𝑣 splits the graph into two smaller mops 𝐺 1 and 𝐺 2 both containing the edge 𝑢𝑣. By induction, we may color each graph 𝐺 1 and 𝐺 2 with three colors 1, 2, and 3. Renaming colors if necessary, we may choose color 1 for the vertex 𝑢 and color 2 for the vertex 𝑣 in each coloring. Combining these colorings yields a 3-coloring of the original graph 𝐺, as desired. For example, the mop in Figure 11.7 can be 3-colored with the colors 1 (red), 2 (blue), and 3 (green) as illustrated in Figure 11.8.

Figure 11.8 A 3-coloring of the mop in Figure 11.7

Every triangle in 𝐺 contains a vertex of each color 1, 2, and 3. Hence, each color class of 𝐺 is a dominating set of 𝐺. Since there are 𝑛 vertices, it follows by the Pigeonhole Principle that  at least one of the color classes, say the vertices colored 1, contains at most 𝑛3 vertices. We now place our guards at all vertices colored with color 1, thereby guarding the whole art gallery with at most 𝑛3 vertices.

Chapter 11. Domination in Planar Graphs

336

For example, each color class of the mop in Figure 11.8 has cardinality 𝑛/3 = 5, and so we can place our guards at any of the three color classes. We can therefore place our guards at all vertices colored with color 1 (red), as illustrated by the black vertices in Figure 11.9(a). The associated guards in the original polygon 𝑀5 in Figure 11.6 are given by the black points in Figure 11.9(b). This completes the proof of Theorem 11.17.

(a)

(b)

Figure 11.9 A placement of guards in the mop in Figure 11.7

Implicit in the above proof of Chvátal’s Watchman Theorem 11.17 by Fisk [304] is that the domination number for a mop of order 𝑛 ≥ 3 is at most 𝑛/3. Thus, the result of Corollary 11.15 was implicitly proved in 1978 by Fisk [304]. It also follows, trivially, that the independent domination number of a mop of order 𝑛 is at most 𝑛/3. In 2013 Campos and Wakabayashi [130] and independently Tokunaga [712] established an improved upper bound on the domination number of a mop by using fundamental properties of mops. Let 𝑓 be an inner face of a mop 𝐺 (that is embedded in the plane). The face 𝑓 is called an inner triangle if none of its edges are incident to the outer face. The following upper bound on the domination number of a mop is given in [130, 712]. Theorem 11.18 ([130, 712]) If 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑘 ≥ 0 internal triangles, then 𝛾(𝐺) ≤ 14 (𝑛 + 𝑘 + 2). The authors in [130] first established results for mops without any internal triangles. Thereafter, they proved by induction the main result in Theorem 11.18 by utilizing chords which are edges of internal triangles. As observed in [130], if 𝐺 is a mop of order 𝑛 ≥ 4 having 𝑘 internal triangles, then 𝐺 has 𝑘 +2 vertices of degree 2.As an immediate consequence of Theorem 11.18 and our earlier observations, we have the following result. Corollary 11.19 ([130]) If 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 ≥ 2 vertices of degree 2, then 𝛾(𝐺) ≤ 14 (𝑛 + 𝑛2 ). The bound in Corollary 11.19 is tight. For example, consider the family Gouterplanar constructed earlier (and illustrated in Figure 11.2). If 𝐺 ∈ Gouterplanar , then 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1 and 𝐺 has order 𝑛 = 6𝑘 with 𝑛2 = 2𝑘 vertices of degree 2. As observed earlier, 𝛾(𝐺) = 13 𝑛. Thus, 𝛾(𝐺) = 2𝑘 = 14 (𝑛 + 𝑛2 ). As observed in [242], every mop 𝐺 of order 𝑛 ≥ 4 has at least two but not more than 𝑛/2 vertices of degree 2, that is, if 𝐺 has 𝑛2 vertices of degree 2, then 2 ≤ 𝑛2 ≤ 12 𝑛.

Section 11.2. Domination in Planar Graphs

337

In 2017 Dorfling et al. [242] improved the upper bound in Corollary 11.19 in the case when 31 𝑛 < 𝑛2 ≤ 12 𝑛. Theorem 11.20 ([242]) If 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 vertices of degree 2 such that 𝑛2 > 13 𝑛, then    𝑛2 + 1   if 𝑛2 = 12 𝑛   2 𝛾(𝐺) ≤   𝑛 − 𝑛2  if 𝑛2 < 12 𝑛.  2 If 𝐺 ∈ Gouterplanar , then 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1 and 𝐺 has order 𝑛 = 6𝑘 with 𝑛2 = 2𝑘 = 13 𝑛 vertices of degree 2. Thus, 𝛾(𝐺) = 13 𝑛 = 𝑛2 = 12 (𝑛 − 𝑛2 ). Adding triangles on some outer edges of 𝐺 that do not contain vertices of degree 2 shows tightness of the bound in Theorem 11.20 for all 𝑛2 with 13 𝑛 < 𝑛2 ≤ 12 𝑛. In 2016 Li et al. [561] improved the upper bound of Campos and Wakabayashi [130] and Tokunaga [712] given in Corollary 11.19. Their proof leverages the local structure of mops when vertices of degree 2 are at distance at least 3 on the outer cycle in the plane embedding of a mop. Theorem 11.21 ([561]) If 𝐺 is a mop of order 𝑛 ≥ 3 with ℓ pairs of consecutive vertices of degree 2 at distance at least 3 apart on the outer cycle, then    𝑛   if ℓ = 0    4 𝛾(𝐺) ≤  𝑛+ℓ   if ℓ ≥ 1.   4 That the bound in Theorem 11.21 is tight may once again be seen by taking a graph 𝐺 ∈ Gouterplanar . If 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1, then 𝐺 has order 𝑛 = 6𝑘 with ℓ = 2𝑘 consecutive vertices of degree 2 at distance at least 3 on the outer cycle. Thus, 𝛾(𝐺) = 2𝑘 = 14 (𝑛 + ℓ). As a consequence of Theorem 11.21, the authors in [561] proved the following upper bound on the domination number for Hamiltonian maximal planar graphs. Theorem 11.22 ([561]) If 𝐺 is a Hamiltonian maximal planar graph of order 5 𝑛 ≥ 7, then 𝛾(𝐺) ≤ 16 𝑛. In 1931 Whitney [750] proved that every 4-connected maximal planar graph is Hamiltonian. Combining this result with Theorem 11.22, we have the following result. Corollary 11.23 ([561]) If 𝐺 is a 4-connected maximal planar graph of order 5 𝑛 ≥ 7, then 𝛾(𝐺) ≤ 16 𝑛.

11.2.3

Domination in Planar Graphs with Small Diameter

In 1996 MacGillivray and Seyffarth [577] initiated the study of domination in graphs of bounded diameter. They remarked that the restriction of bounding the diameter

338

Chapter 11. Domination in Planar Graphs

on the domination number of a planar graph is reasonable to impose because planar graphs with small diameter are often important in applications, as explained, for example, in the 1995 paper by Fellows et al. [289]. A tree with diameter 4 can have arbitrarily large domination number. The interesting question is what happens when the diameter of a planar graph is 2 or 3. MacGillivray and Seyffarth [577] proved that planar graphs with diameter 2 or 3 have bounded domination numbers. In particular, this implies that the domination number of such a graph can be determined in polynomial time. The domination number of (general) graphs with diameter 2 can be arbitrarily number large. For example, recall that we showed in Chapter 7 that the domination √︁ of a random √︁ graph 𝐺 ∈ G(𝑛, 𝑝) on 𝑛 vertices, where 𝑝 ≈ (2 ln(𝑛))/𝑛, is of the order 𝑛 ln(𝑛) (see Theorem 7.21 for a more precise statement). In contrast, MacGillivray and Seyffarth [577] established that the domination number of planar graph with diameter 2 is at most 3. This bound was subsequently improved in 2002 by Goddard and Henning [349]. To state their result, let 𝐺 9 be the planar graph with diameter 2 and domination number 3 shown in Figure 11.10.

Figure 11.10 The planar graph 𝐺 9 with diameter 2 and domination number 3

Theorem 11.24 ([349]) If 𝐺 is a planar graph with diam(𝐺) = 2, then 𝛾(𝐺) ≤ 2, except for the graph 𝐺 = 𝐺 9 of Figure 11.10 for which 𝛾(𝐺 9 ) = 3. Proof Sketch We present here a sketch of the proof given in [349]. Suppose that 𝐺 is a planar graph with diam(𝐺) = 2 and 𝛾(𝐺) > 2. Since a cut-set dominates a diameter-2 graph, the graph 𝐺 is 3-connected. Therefore, 𝐺 has an essentially unique embedding in the plane (see [381]). We fix such an embedding of 𝐺 in the plane. From the Jordan Closed Curve Theorem, a cycle 𝐶 in 𝐺 separates the plane into two regions, called the sides of 𝐶. Vertices on different sides of 𝐶 are separated by 𝐶. The side of 𝐶 that consists of the unbounded face is called the outside of 𝐶, while the side of 𝐶 that consists of the bounded region is called the inside of 𝐶. If there are vertices inside 𝐶 and vertices outside 𝐶, then 𝐶 is called a cut-cycle. The theorem is now proven by a series of lemmas. The key lemma is to establish the existence of a 4-cycle with special properties, namely the 4-cycle is not both induced and dominating, nor both non-induced and dominating, and therefore not dominating. Thereafter, it is shown that 𝐺 is isomorphic to the graph 𝐺 9 shown in Figure 11.10. Therefore, if 𝐺 is a planar graph with diameter 2, then either 𝛾(𝐺) = 2 or 𝐺 = 𝐺 9 , in which case 𝛾(𝐺) = 3. In their 1996 paper, MacGillivray and Seyffarth [577] proved that planar graphs with diameter 3 have bounded domination numbers.

Section 11.2. Domination in Planar Graphs

339

Theorem 11.25 ([577]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 10. The upper bound in Theorem 11.25 in the case when the order of the planar graph is sufficiently large was improved in 2002 by Goddard and Henning [349]. For this purpose, they first proved the following result in the case when the radius is 2. Theorem 11.26 ([349]) If 𝐺 is a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, then 𝛾(𝐺) ≤ 6. Proof Sketch We present here a sketch of the proof given in [349]. Let 𝐺 be a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, and fix an embedding of 𝐺 in the plane. We adopt the notation in the proof of Theorem 11.24. Moreover, we define a basic cycle as a cut-cycle with certain special properties. More precisely, a basic cycle is an induced cycle 𝑥 𝑣 1 𝑣 2 . . . 𝑣 𝑟 𝑥 such that on both sides of the cycle there is a vertex whose neighbors on the cycle are a subset of the two consecutive vertices farthest from 𝑥, specifically 𝑣 (𝑟 −1)/2 and 𝑣 (𝑟+1)/2 if 𝑟 is odd and 𝑣 𝑟/2 and 𝑣 𝑟/2+1 if 𝑟 is even. The strategy of the proof is to establish special properties of cut-cycles and to show the existence of a basic cycle of length at most 5. Thereafter, the theorem is established by a series of lemmas to bound the domination number when there exists such basic cycles (of short length). Goddard and Henning [349] proved that if 𝐺 is a planar graph of sufficiently large order with radius and diameter 3, then the maximum domination number of such a planar graph is at most one more than the maximum for radius 2 and diameter 3. Their key lemma is the following result. Lemma 11.27 ([349]) For a sufficiently large planar graph 𝐺 with rad(𝐺) = diam(𝐺) = 3, there exists a planar graph 𝐺 ′ with rad(𝐺 ′ ) ≤ 2, diam(𝐺 ′ ) ≤ 3, and 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. As an immediate consequence of Theorem 11.26 and Lemma 11.27, we have the result of Goddard and Henning [349] that if 𝐺 is a planar graph of sufficiently large order with diameter 3, then 𝛾(𝐺) ≤ 7. In 2006 Dorfling et al. [239] improved the result in Theorem 11.26 and showed that every planar graph with diameter 3 and radius 2 has total domination number (and therefore domination number) at most 5. We remark that their proof used the same approach as that used to prove Theorem 11.26 in [349], but with more detailed analysis and with the use of a computer. Theorem 11.28 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, then 𝛾t (𝐺) ≤ 5. In order to handle the case when 𝐺 is a planar graph with diameter 3 and radius 3, the authors in [349] proved that the maximum domination number of such a planar graph is at most four more than the maximum for radius 2 and diameter 3. This, together with the result of Theorem 11.28, yielded the following slight improvement on the MacGillivray-Seyffarth result given in Theorem 11.25.

340

Chapter 11. Domination in Planar Graphs

Theorem 11.29 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 9. As an immediate consequence of Lemma 11.27 and Theorem 11.28, we have the following improved upper bound on the domination number of a planar graph of sufficiently large order. Theorem 11.30 ([239]) Every sufficiently large planar graph with diameter 3 has domination number at most 6. MacGillivray and Seyffarth [577] gave the example illustrated in Figure 11.11 of a planar graph with diameter 3 and domination number 6, where the six highlighted vertices form a 𝛾-set of the graph. By duplicating the vertices of degree 2, their example can be used to construct planar graphs of arbitrarily large order with diameter 3 and domination number 6. Hence, the upper bound in Theorem 11.30 is tight.

Figure 11.11 A planar graph with diameter 3 and domination number 6

MacGillivray and Seyffarth [577] showed that if we restrict the graphs in the statement of Theorem 11.25 to the class of outerplanar graphs, then the upper bound on the domination number can be decreased to 3. Theorem 11.31 ([577]) If 𝐺 is an outerplanar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 3. Analogous results on bounds on the domination number for other surfaces were presented in [349]. Theorem 11.32 ([349]) For each surface, there are finitely many graphs with diameter 2 and domination number more than 2. Theorem 11.33 ([349]) For each orientable surface, there is a maximum domination number of graphs with diameter 3.

Section 11.3. Total Domination in Planar Graphs

11.3

341

Total Domination in Planar Graphs

In this section, we present selected results on total domination in outplanar graphs and planar graphs with small diameter.

11.3.1

Total Domination in Outerplanar Graphs

In 2017 Dorfling et al. [242] proved an analogous result to Corollary 11.19 for total domination in mops. As observed earlier, if 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 vertices of degree 2, then 2 ≤ 𝑛2 ≤ 21 𝑛. Theorem 11.34 ([242]) If 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑛2 vertices of degree 2, then  2(𝑛 − 𝑛2 )   if 𝑛2 > 13 𝑛 and 𝑛 ≥ 5   3 𝛾t (𝐺) ≤ 𝑛 + 𝑛2   otherwise.   3 As shown in [242], both bounds in Theorem 11.34 hold, but which is the better bound depends on 𝑛2 , that is, if 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑛2 vertices of degree 2, then 𝛾t (𝐺) ≤ min 23 (𝑛 − 𝑛2 ), 13 (𝑛 + 𝑛2 ) . We note that if 𝑛2 > 13 𝑛, then it follows that 23 (𝑛 − 𝑛2 ) < 13 (𝑛 + 𝑛2 ). Several constructions showing tightness of the bounds in Theorem 11.34 are provided in [242]. We briefly present one such construction in the case when 𝛾t (𝐺) = 23 (𝑛 − 𝑛2 ). The family Gouterplanar constructed in Section 11.2.1 can be modified to a family Houterplanar = {𝐻 𝑘 : 𝑘 ≥ 1}, where each graph 𝐻 𝑘 has order 𝑛 = 10𝑘 with 𝑛2 = 4𝑘 vertices of degree 2 and total domination number 𝛾t (𝐻 𝑘 ) = 4𝑘 = 23 (𝑛 − 𝑛2 ). We omit the details of the construction and rather show the graph 𝐻3 ∈ Houterplanar in Figure 11.12 to illustrate the general construction.

Figure 11.12 The graph 𝐻3 Recall the basic result of Corollary 11.15 that if 𝐺 is a mop of order 𝑛 ≥ 3, then 𝛾(𝐺) ≤ 13 𝑛. A natural question is to determine an analogous upper bound (that depends only on 𝑛) for the total domination number of mops. This question was explored in 2016 by Dorfling et al. [241]. Let 𝐺 12.1 and 𝐺 12.2 be the two mops shown in Figure 11.13. Theorem 11.35 ([241]) If 𝐺 is a mop of order 𝑛 ≥ 5 and 𝐺 ∉ {𝐺 12.1 , 𝐺 12.2 }, then 𝛾t (𝐺) ≤ 25 𝑛.

Chapter 11. Domination in Planar Graphs

342

(a) 𝐺 12.1

(b) 𝐺 12.2

Figure 11.13 The mops 𝐺 12.1 and 𝐺 12.2

That the bound of Theorem 11.35 is tight may be seen as follows. Let 𝐹 be a copy of the mop illustrated in Figure 11.14(a), where we designate the two highlighted vertices as link vertices of 𝐹. For 𝑘 ≥ 1, take 𝑘 vertex-disjoint copies of the mop 𝐹 and add edges between the link vertices in such a way as to construct a mop. Let 𝐹𝑘 denote the resulting graph (which is not unique). We call each of the 𝑘 copies of the graph 𝐹 used to build the graph 𝐹𝑘 a unit of 𝐹𝑘 . A graph 𝐹3 , for example, constructed in this way is illustrated in Figure 11.14(b). Let Fouterplanar consist of all such graphs 𝐹𝑘 , where 𝑘 ≥ 1, that can be constructed in this way. Let 𝐺 ∈ Fouterplanar have order 𝑛 and so 𝐺 = 𝐹𝑘 for some 𝑘 ≥ 1 and 𝑛 = 5𝑘. Every TD-set in 𝐺 must contain at least two vertices from each of the 𝑘 units in 𝐺, implying that 𝛾t (𝐺) ≥ 2𝑘. The set consisting of the 2𝑘 vertices of 𝐺 that are neither link vertices nor vertices of degree 2 in 𝐺 form a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 2𝑘. Consequently, 𝛾t (𝐺) = 2𝑘 = 25 𝑛.

(a) 𝐹

(b) 𝐹3

Figure 11.14 The mop 𝐹 used to build a mop 𝐹3 ∈ Fouterplanar We remark that the proof of Theorem 11.35 is quite different than the proof of Corollary 11.3. Recall that the proof given by Matheson and Tarjan [585] followed from labeling arguments (and also follows from the more general result given in Theorem 10.70 in Chapter 10). However, the proof of Theorem 11.35 relied on a detailed case analysis due in part to the existence the two exceptional graphs 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. Subsequently, in 2017 Lemańska et al. [558] presented an alternate proof to that given in [241] and they discussed a relation between total domination in mops and the concept of watched guards in simple polygons. As observed by O’Rourke [620] and others, every mop has a unique Hamiltonian cycle. Following the notation of Lemańska et al. [558], we refer to an edge that belongs to the Hamiltonian cycle of a mop as a Hamiltonian edge and to every other edge of the mop as a diagonal.

Section 11.3. Total Domination in Planar Graphs

343

Key properties of mops used by Lemańska et al. [558] are the following results of O’Rourke [619]. Lemma 11.36 ([619]) If 𝐺 is a mop of order 𝑛 ≥ 10, then there exists a diagonal edge 𝑑 of 𝐺 such that 𝑑 partitions 𝐺 into two mops sharing the common edge 𝑑, one of which contains exactly 5, 6, 7, or 8 Hamiltonian edges of 𝐺. Lemma 11.37 ([619]) If 𝐺 is a mop of order 𝑛 ≥ 4 and 𝑒 is a Hamiltonian edge of 𝐺, then the graph resulting from contraction of the edge 𝑒 is a mop of order 𝑛 − 1. Lemańska et al. [558] used the special diagonal with the property in the statement of the Lemma 11.36, together with the edge contraction property of mops from Lemma 11.37 that allowed them to prove the desired result by induction on the order of a mop. An application of Theorem 11.35 is the following modification of the Art Gallery Problem 11.16, where now we require the additional condition that each guard is watched by at least one other guard. Problem 11.38 (Watched Art Gallery Problem) Determine the minimum number of guards that need to be placed in an art gallery so that the following hold: (a) Every point in the interior of the gallery is within the line of sight of at least one guard. (b) Every guard is within the line of sight of some other guard. Recall that for 𝑛 ≥ 3, a set of points 𝑆 within a simple plane polygon P𝑛 with 𝑛 corner points, guards P𝑛 if for every point 𝑝 in the interior of P𝑛 , there exists some point 𝑞 ∈ 𝑆 such that the straight line segment between 𝑝 and 𝑞 lies inside the polygon P𝑛 . If the set 𝑆 of points has the additional property that every point 𝑝 ∈ 𝑆 is seen by some other point 𝑞 ∈ 𝑆 (that is, the straight line segment between points 𝑝 and 𝑞 lies inside the polygon P𝑛 ), then we call the set 𝑆 a watched guard set in P𝑛 (called a guarded guard set in [590]). In 2003 Michael and Pinciu [590] solved Problem 11.38. set Theorem 11.39 ([590]) For 𝑛 ≥ 5, a minimum watched guard   in a simple plane polygon P𝑛 with 𝑛 corner points has cardinality at most 3𝑛−1 , and this bound is 7 best possible. The authors in [590] constructed a polygon 𝑃𝑛 with 𝑛 = 12 corner points for which a minimum watched guard set has cardinality exactly (3𝑛 − 1)/7 = 5. Triangulating the part of the (planar) graph corresponding to the interior of their polygon P12 (as explained in the proof of Theorem 11.17) produces one of the two mops 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. As observed by Lemańska et al. [558], if we modify the Watched Art Gallery Problem 11.38 by relaxing property (a) to require that only the 𝑛 corner points of polygon P𝑛 be guarded, while maintaining property (b) (that every guard is watched by at least one other guard), then as an immediate consequence of Theorem 11.35, we have the following result.

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Corollary 11.40 ([558]) For 𝑛 ≥ 5, the vertices of a simple plane polygon P𝑛 with 𝑛 corner points can be guarded by at most 52 𝑛 watched guards, apart from some 12-vertex polygons that require five watched guards. In 2018 Alvarado et al. [21] presented a unified proof of the results of Corollary 11.15 and Theorem 11.35. For 𝑘 a positive integer, they defined a 𝑘-component dominating set of a graph as a dominating set 𝐷 of 𝐺 with the additional property that every component of the subgraph 𝐺 [𝐷] of 𝐺 induced by 𝐷 has order at least 𝑘. We denote the minimum cardinality of a 𝑘-component dominating set of 𝐺 by 𝛾 𝑘,comp (𝐺). We note that 𝛾(𝐺) = 𝛾1,comp (𝐺) and 𝛾t (𝐺) = 𝛾2,comp (𝐺). For every positive integer 𝑘, Alvarado et al. [21] constructed a set H𝑘 of graphs, each member of which has order at least 4𝑘 + 4 and at most 4𝑘 2 − 2𝑘. For 𝑘 = 1, we have 4𝑘 + 4 > 4𝑘 2 − 2𝑘, and so H1 is necessarily empty. Moreover, H2 consists exactly of the two exceptional mops 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. The main result in [21] is the following. Theorem 11.41 ([21]) If 𝑘 and 𝑛 are positive integers with 𝑛 ≥ 2𝑘 + 1 and 𝐺 is a mop of order 𝑛, then l 𝑘𝑛 m    if 𝐺 ∈ H𝑘    2𝑘 + 1 𝛾 𝑘,comp (𝐺) ≤ j  𝑘𝑛 k   otherwise.   2𝑘 + 1 Since H1 = ∅, Theorem 11.41 implies Corollary 11.15. Furthermore, since H2 = {𝐺 12.1 , 𝐺 12.2 }, Theorem 11.41 implies Theorem 11.35.

11.3.2

Total Domination in Planar Graphs with Small Diameter

Recall that in Section 11.2.3, we discussed results on domination in planar graphs with small diameter. In this section, we present results on total domination in planar graphs with small diameter. One can readily deduce from Theorem 11.24 that the total domination number of a planar graph with diameter 2 is at most 3, as first observed in 2006 by Dorfling et al. [239]. Theorem 11.42 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 2, then 𝛾t (𝐺) ≤ 3. Proof Let 𝐺 be a planar graph with diam(𝐺) = 2. If 𝐺 = 𝐺 9 , where 𝐺 9 is the graph of Figure 11.10, then 𝛾t (𝐺) = 3. If 𝐺 ≠ 𝐺 9 , then by Theorem 11.24, we have 𝛾(𝐺) ≤ 2. Let {𝑢, 𝑣} be a 𝛾-set in 𝐺. If 𝑢 and 𝑣 are adjacent, then {𝑢, 𝑣} is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ 2. If 𝑢 and 𝑣 are not adjacent, then since diam(𝐺) = 2, there is a common neighbor 𝑤 of 𝑢 and 𝑣. In this case, {𝑢, 𝑣, 𝑤} is a TD-set in 𝐺 and so 𝛾t (𝐺) ≤ 3. There are infinitely many planar graphs 𝐺 with diam(𝐺) = 2 and 𝛾t (𝐺) = 3. An infinite family of such graphs can be obtained, for example, from a 5-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 by replacing the vertices 𝑣 2 and 𝑣 5 with nonempty independent sets 𝑉2 and 𝑉5 , respectively, and adding all edges between vertices in 𝑉2 and {𝑣 1 , 𝑣 3 } and adding all

Section 11.3. Total Domination in Planar Graphs

345

Figure 11.15 A planar graph with diameter 2 and total domination number 3

edges between vertices in 𝑉5 and {𝑣 1 , 𝑣 4 }. As an illustration, a graph in such a family with |𝑉2 | = |𝑉5 | = 4 is shown in Figure 11.15. It remains and open problem to characterize planar graphs with diameter 2 and total domination number 3. In 2009 Henning and McCoy [471] obtained a characterization in the special case when the planar graphs with diameter 2 have certain structural properties. More specifically, they define a graph 𝐺 to satisfy the domination-cycle property if there is some 𝛾-set of 𝐺 that is not contained in any induced 5-cycle. The authors in [471] characterized the planar graphs with diameter 2 and total domination number 3 that satisfy the domination-cycle property and showed that there are exactly 34 such planar graphs. In 2020 Goddard and Henning [355] showed that if we restrict the graphs in the statement of Theorem 11.42 to the class of outerplanar graphs, then the upper bound on the total domination number can be improved slightly. Theorem 11.43 ([355]) If 𝐺 is an outerplanar graph with diam(𝐺) = 2, then 𝛾t (𝐺) = 2, unless 𝐺 = 𝐶5 . Proof Let 𝐺 be an outerplanar graph with diam(𝐺) = 2. If 𝐺 has a cut-vertex, then this vertex dominates the graph, and 𝛾(𝐺) = 1 and 𝛾t (𝐺) = 2. Hence, we may assume that 𝐺 is 2-connected. Thus, we can embed 𝐺 in the plane so that all vertices of 𝐺 are external vertices on a Hamiltonian cycle 𝐶 and all edges lie within 𝐶. If there is a chord, then the ends of the chord form a TD-set of 𝐺, since there must be path between vertices on either side of the chord, and so 𝛾t (𝐺) = 2. If there is no chord, then the graph 𝐺 is a cycle of length at most 5. If 𝐺 has length at most 4, then 𝛾t (𝐺) = 2. If 𝐺 has length 5, then 𝐺 = 𝐶5 . Planar graphs with diameter 3 also have bounded total domination numbers. The main result in the 2006 paper by Dorfling et al. [239] is Theorem 11.28 that every planar graph with diameter 3 and radius 2 has total domination number at most 5. As a consequence of this result, the authors in [239] established the following upper bound on the total domination number of a planar graph with diameter 3. Theorem 11.44 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾t (𝐺) ≤ 𝛿(𝐺) + 5. Proof Let 𝐺 be a planar graph with diam(𝐺) = 3. If rad(𝐺) = 2, then 𝛾t (𝐺) ≤ 5 by Theorem 11.28. Hence, we may assume that rad(𝐺) = 3. Thus, every vertex has eccentricity 3, that is, ecc𝐺 (𝑣) = 3 for every vertex 𝑣 in 𝐺. Let 𝑣 be a vertex of minimum degree 𝛿(𝐺) in 𝐺. Let 𝐺 𝑣 be the planar graph obtained from 𝐺 by deleting

346

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the vertices in N(𝑣) and adding a new vertex 𝑥 that is adjacent to 𝑣 and to all vertices at distance 2 from 𝑣 in 𝐺. The distance between every pair of vertices in 𝑉 (𝐺 𝑣 ) \ {𝑥, 𝑣} is at most their distance in 𝐺. The vertex 𝑥 is within distance 2 from every vertex in 𝐺 𝑣 and so rad(𝐺 𝑣 ) ≤ 2. Moreover, every vertex at distance 2 or 3 from 𝑣 in 𝐺 remains the same distance from 𝑣 in 𝐺 𝑣 , and so ecc𝐺𝑣 (𝑣) = ecc𝐺 (𝑣) = 3, implying that diam(𝐺 𝑣 ) = 3 and rad(𝐺 𝑣 ) = 2. Applying Theorem 11.28 to the graph 𝐺 𝑣 , we have that 𝛾t (𝐺 𝑣 ) ≤ 5. Let 𝑆 𝑣 be a 𝛾t -set of 𝐺 𝑣 .
The vertex 𝑥 belongs to 𝑆 𝑣 in order to totally dominate the vertex 𝑣. Since 𝑆 𝑣 \ {𝑥} ∪ N𝐺 [𝑣] is a TD-set in 𝐺, we have that 𝛾t (𝐺) ≤ |𝑆 𝑣 | + 𝛿(𝐺) = 5 + 𝛿(𝐺). Every planar graph has minimum degree at most 5. Hence, as an immediate consequence of Theorem 11.44, we have the following upper bound on the total domination number of a planar graph with diameter 3. Corollary 11.45 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾t (𝐺) ≤ 10. The authors in [239] proved that if 𝐺 is a planar graph of sufficiently large order with radius and diameter 3, then the total domination number of 𝐺 is at most two more than the maximum total domination number of a planar graph with radius 2 and diameter 3. Lemma 11.46 ([239]) For a sufficiently large planar graph 𝐺 with rad(𝐺) = diam(𝐺) = 3, there exists a planar graph 𝐺 ′ with rad(𝐺 ′ ) ≤ 2, diam(𝐺 ′ ) ≤ 3, and 𝛾t (𝐺) ≤ 𝛾t (𝐺 ′ ) + 2. As an immediate consequence of Theorem 11.28 and Lemma 11.46, we have the following upper bound on the total domination number of a planar graph of sufficiently large order with diameter 3. Theorem 11.47 ([239]) Every sufficiently large planar graph with diameter 3 has total domination number at most 7. It is not known if the upper bound of Theorem 11.47 is best possible. The planar graph illustrated in Figure 11.11 has diameter 3 and total domination number 6, where the six highlighted vertices form a 𝛾t -set of the graph. By duplicating the vertices of degree 2, this example can be used to construct planar graphs of arbitrarily large order with diameter 3 and total domination number 6. Hence, the best possible upper bound we can hope for on the total domination number of a planar graph with diameter 3 is 6. In 2020 Goddard and Henning [355] showed that if we restrict the graphs in the statement of Corollary 11.45 to the class of outerplanar graphs, then the upper bound on the total domination number can be improved. Theorem 11.48 ([355]) If 𝐺 is an outerplanar graph with diam(𝐺) = 3, then 𝛾t (𝐺) ≤ 4. Furthermore, the largest such graph achieving this value has order 10. An outerplanar graph of largest possible order 10 with diameter 3 and total domination number 4 is shown in Figure 11.16.

Section 11.4. Independent Domination in Planar Graphs

347

Figure 11.16 An outerplanar graph with diameter 3 and total domination number 4

The authors in [355] also showed that if we restrict the graphs in the statement of Theorem 11.29 and Corollary 11.45 to the class of bipartite graphs, then the upper bound on the (total) domination number can be improved. Theorem 11.49 ([355]) If 𝐺 is a bipartite planar graph with diam(𝐺) = 3 and bipartition (𝑋, 𝑌 ), then there exist two vertices of 𝑋 that dominate 𝑌 , and vice versa. In particular, 𝛾t (𝐺) ≤ 4. As remarked in [355], the bound in Theorem 11.49 on the (total) domination number is tight. They gave the example illustrated in Theorem 11.17 of a bipartite planar graph 𝐺 with diameter 3 (where the bipartition is given by the different colored vertices). In this example, the inner trio of vertices of degree 2 need at least two vertices to dominate them, as do the outer trio of vertices of degree 2, implying that 𝛾(𝐺) ≥ 4. By Theorem 11.49, 𝛾(𝐺) ≤ 4. Consequently, 𝛾(𝐺) = 𝛾t (𝐺) = 4. By duplicating the vertices of degree 2, their example can be used to construct bipartite planar graphs of arbitrarily large order with diameter 3 and (total) domination number 4.

Figure 11.17 A bipartite planar graph with diameter 3 and (total) domination number 4

11.4

Independent Domination in Planar Graphs

In 2004 MacGillivray and Seyffarth [578] established a tight upper bound on the independent domination number of a graph in terms of the order of the graph and its chromatic number. We present here a short proof in [352] of their result using probabilistic methods.

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348

Theorem 11.50 ([578]) If 𝐺 is a connected graph of order 𝑛 with chromatic number 𝜒(𝐺) = 𝑘 ≥ 3, then   𝑘 −1 𝑖(𝐺) ≤ 𝑛 − (𝑘 − 2), 𝑘 and this bound is tight. Proof Let 𝐺 be a connected graph of order 𝑛 with 𝜒(𝐺) = 𝑘 ≥ 3, and let C be a 𝑘-coloring of the vertices of 𝐺 using colors 1, 2, . . . , 𝑘. We construct a maximal independent set 𝑆 as follows. For a color chosen at random, take all vertices of that color and extend the color class to a maximal independent set. For a vertex 𝑣 to belong to the set 𝑆, it is necessary that none of its neighbors has the chosen color. Thus, if 𝐷 (𝑣) is the number of different colors in N(𝑣), then there are 𝑘 − 𝐷 (𝑣) colors not
in N(𝑣). The probability that 𝑣 is chosen in the set 𝑆 is therefore at most 𝑘 − 𝐷 (𝑣) /𝑘; that is 𝑘 − 𝐷 (𝑣) Pr(𝑣 ∈ 𝑆) ≤ . 𝑘 Since the graph 𝐺 is not (𝑘 − 1)-colorable, we cannot eliminate any color. It follows that for each color 𝑐 ∈ [𝑘], there is a vertex 𝑣 𝑐 of color 𝑐 that has neighbors of every other color. For each color 𝑐 ∈ [𝑘], select one such vertex 𝑣 𝑐 and let 𝑋 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 }. Hence, for each 𝑣 ∈ 𝑋, we have 𝐷 (𝑣) = 𝑘 − 1 and Pr(𝑣 ∈ 𝑆) = 𝑘1 . For each 𝑣 ∈ 𝑉 \ 𝑋, we have 𝐷 (𝑣) ≥ 1 and Pr(𝑣 ∈ 𝑆) ≤

𝑘 − 𝐷 (𝑣) 𝑘 −1 ≤ . 𝑘 𝑘

Thus, by linearity of expectation, we can therefore bound the expected size of 𝑆 by ∑︁
∑︁ E |𝑆| = Pr(𝑣 ∈ 𝑆) + Pr(𝑣 ∈ 𝑆) 𝑣 ∈𝑋

𝑣 ∈𝑉\𝑋

    1 𝑘 −1 ≤𝑘 + (𝑛 − 𝑘) 𝑘 𝑘   𝑘 −1 = 𝑛 − (𝑘 − 2). 𝑘 Since expectation
is an average value, there is a maximal independent set 𝑆 in 𝐺 such that |𝑆| ≤ 𝑘−1 𝑘 𝑛 − (𝑘 − 2). That the bound of Theorem 11.50 is tight for every 𝑘 ≥ 3 may be seen by taking the generalized corona 𝐺 = cor(𝐾 𝑘 , 𝑟), where 𝑟 ≥ 1 is an arbitrary integer. In this case, the graph 𝐺 has order 𝑛 = 𝑘 (𝑟 + 1), 𝜒(𝐺) = 𝑘, and   𝑘 −1 𝑖(𝐺) = 1 + (𝑘 − 1)𝑟 = 𝑛 − (𝑘 − 2). 𝑘 Combining Theorem 11.50 with the Four Color Theorem, we have the following upper bound on the independent domination number of a planar graph in terms of its order.

Section 11.4. Independent Domination in Planar Graphs

349

Theorem 11.51 ([578]) If 𝐺 is a connected planar graph of order 𝑛 ≥ 10, then 𝑖(𝐺) ≤ 43 𝑛 − 2. The generalized corona cor(𝐾4 , 𝑟) for 𝑟 ≥ 1 has diameter 3 and attains the bound in Theorem 11.51. MacGillivray and Seyffarth [578] showed that if planar graphs are restricted to those with diameter 2, then the upper bound in Theorem 11.51 can be improved. Theorem 11.52 ([578]) If 𝐺 is a planar graph of order 𝑛 ≥ 6 with diam(𝐺) = 2, then 𝑖(𝐺) ≤ 13 𝑛. The graphs achieving equality in the upper bound in Theorem 11.52 are characterized in [578]. A simplest example of a planar graph 𝐺 achieving equality in the bound in Theorem 11.52 is when 𝐺 is the Hajós graph of order 𝑛 = 6, diam(𝐺) = 2, and 𝑖(𝐺) = 𝑛3 = 2. Also, a family of planar graphs 𝐺 of order 𝑛 ≥ 6, where 𝑛 ≡ 0 (mod 3), with diam(𝐺) = 2 and 𝑖(𝐺) = 13 𝑛 is constructed in [578]. An example of a graph in their family is illustrated in Figure 11.18, where the highlighted vertices form an 𝑖-set of the graph. By duplicating the vertices of degree 2, this example can readily be extended to planar graphs 𝐺 of arbitrarily large order 𝑛 with diam(𝐺) = 3 and 𝑖(𝐺) = 13 𝑛. Hence, the upper bound in Theorem 11.52 is tight.

Figure 11.18 A planar graph 𝐺 of order 𝑛 = 12 with diam(𝐺) = 2 and 𝑖(𝐺) = 13 𝑛 = 4 The lower bound on the independent domination number in Theorem 11.51 can be improved for planar graphs with minimum degree at least 2. An 𝑟-dynamic coloring of a graph 𝐺 is a proper coloring of the vertices of 𝐺 such that every vertex 𝑣 has at least min deg𝐺 (𝑣), 𝑟 colors in its neighborhood. In 2013 Kim et al. [524] showed that every connected planar graph has a 2-dynamic coloring using at most four colors, except for 𝐶5 . Using this key result from [524], in 2020 Goddard and Henning [357] established the following upper bound on the independent domination number of a planar graph with minimum degree at least 2. Theorem 11.53 ([357]) If 𝐺 is a planar graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝑖(𝐺) ≤ 12 𝑛. The authors in [357] constructed the following infinite family of connected planar graphs 𝐺 of order 𝑛 with minimum degree 2 and 𝑖(𝐺) = 12 𝑛. For 𝑠 ≥ 2, let 𝐻𝑠 be

Chapter 11. Domination in Planar Graphs

350

the graph of order 𝑛 = 2𝑠 + 4 obtained from a 4-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 1 by adding 𝑠 new vertices whose neighbors are the pair {𝑣 𝑖 , 𝑣 𝑖+2 } for each 𝑖 ∈ [2]. The graph 𝐻4 , for example, is illustrated in Figure 11.19. Let 𝑉𝑖,𝑖+2 be the set of 𝑠 common neighbors of degree 2 of 𝑣 𝑖 and 𝑣 𝑖+2 for 𝑖 ∈ [2]. The graph 𝐻𝑠 has three maximal independent sets, namely the sets 𝑉1,3 ∪ {𝑣 2 , 𝑣 4 }, 𝑉2,4 ∪ {𝑣 1 , 𝑣 3 }, and 𝑉1,3 ∪ 𝑉2,4 , implying that 𝑖(𝐻𝑠 ) = 𝑠 + 2 = 21 𝑛. Thus, the bound in Theorem 11.53 is tight. 𝑣1

𝑣2

𝑣3

𝑣4

Figure 11.19 The planar graph 𝐻4 of order 𝑛 = 12 with 𝑖(𝐻4 ) = 12 𝑛 = 6 We close this section with a discussion of the independent domination number of a planar cubic graph. Recall that Conjecture 6.95 in Chapter 6 states that if 𝐺 ∉ {𝐾3,3 , 𝐶5 □ 𝐾2 } is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. Recall that Theorem 10.52 in Chapter 10 states that if 𝐺 ≠ 𝐶5 □ 𝐾2 is a connected cubic graph of order 𝑛 that does not have a subgraph isomorphic to 𝐾2,3 , then 𝑖(𝐺) ≤ 38 𝑛. As an application of this theorem, in 2019 Abrishami et al. [5] proved that Conjecture 6.95 is true when 𝐺 is 2-connected and planar. Theorem 11.54 ([5]) If 𝐺 ≠ 𝐶5 □ 𝐾2 is a 2-connected planar cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. The following infinite family of 2-connected planar cubic graphs that achieve equality in the bound of Theorem 11.54 is constructed in [5]. Let (𝐶5 □ 𝐾2 ) − , illustrated in Figure 11.20, denote the graph obtained from a 5-prism by deleting an edge that does not belong to a 5-cycle.

Figure 11.20 The graph (𝐶5 □ 𝐾2 ) − Let 𝐹 and 𝐻 be two vertex-disjoint copies of (𝐶5 □ 𝐾2 ) − , let 𝑟 1 and 𝑠1 be the two vertices of degree 2 in 𝐹, and let 𝑝 1 and 𝑞 1 be the two vertices of degree 2 in 𝐻. For 𝑘 ≥ 1, we construct a graph 𝐺 𝑘 as follows. Consider two copies of the path 𝑃4𝑘+2 with respective vertex sequences 𝑐 0 𝑑0 𝑎 1 𝑏 1 𝑐 1 𝑑1 . . . 𝑎 𝑘 𝑏 𝑘 𝑐 𝑘 𝑑 𝑘 and 𝑦 0 𝑧0 𝑤 1 𝑥1 𝑦 1 𝑧1 . . . 𝑤 𝑘 𝑥 𝑘 𝑦 𝑘 𝑧 𝑘 . Join 𝑐 0 to 𝑧0 , join 𝑑0 to 𝑦 0 , and for each 𝑖 ∈ [𝑘], join 𝑎 𝑖 to 𝑤 𝑖 , 𝑏 𝑖 to 𝑥𝑖 , 𝑐 𝑖 to 𝑧𝑖 , and 𝑑𝑖 to 𝑦 𝑖 . To complete 𝐺 𝑘 , add a disjoint copy of 𝐹 and 𝐻, and

Section 11.4. Independent Domination in Planar Graphs

351

join 𝑐 0 to 𝑟 1 , 𝑦 0 to 𝑠1 , 𝑑 𝑘 to 𝑝 1 , and 𝑧 𝑘 to 𝑞 1 . We note that the graph 𝐺 𝑘 has order 8𝑘 + 24. Let Gplanar = {𝐺 𝑘 : 𝑘 ≥ 1}. The planar (not plane) graph 𝐺 2 ∈ Gplanar (of order 40) is illustrated in Figure 11.21.

Figure 11.21 The planar graph 𝐺 2 in the family Gplanar Abrishami et al. [5] showed that every graph in the family Gplanar achieves equality in the upper bound of Theorem 11.54. Proposition 11.55 ([5]) If 𝐺 ∈ Gplanar has order 𝑛, then 𝑖(𝐺) = 38 𝑛. The authors in [5] conjectured that the 2-connectivity property in the statement of Theorem 11.54 can be omitted, provided the planar graph contains no (𝐶5 □ 𝐾2 )component. Conjecture 11.56 ([5]) If 𝐺 is a planar cubic graph of order 𝑛 with no (𝐶5 □ 𝐾2 )component, then 𝑖(𝐺) ≤ 38 𝑛. If one considers bipartite planar cubic graphs, then it is conjectured in [5] that the upper bound on the independent domination number in Theorem 11.54 can be improved. Conjecture 11.57 ([5]) If 𝐺 is a bipartite planar cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 13 𝑛. An infinite family Fplanar of bipartite planar cubic graphs can be constructed as follows. For 𝑘 ≥ 2, define the graph 𝐹𝑘 as described below. Consider two copies of the cycle 𝐶2𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑎 2 𝑏 2 . . . 𝑎 𝑘 𝑏 𝑘 𝑎 1 and 𝑐 1 𝑑1 𝑐 2 𝑑2 . . . 𝑐 𝑘 𝑑 𝑘 𝑐 1 . To complete 𝐹𝑘 , add 2𝑘 new vertices 𝑒 1 , 𝑒 2 , . . . , 𝑒 𝑘 and 𝑓1 , 𝑓2 , . . . , 𝑓 𝑘 , and for each 𝑖 ∈ [𝑘], join 𝑒 𝑖 to 𝑎 𝑖 , 𝑐 𝑖 , and 𝑓𝑖 , and join 𝑓𝑖 to 𝑏 𝑖 and 𝑑𝑖 . We note that the graph 𝐹𝑘 has order 6𝑘. Let Fplanar = {𝐹𝑘 : 𝑘 ≥ 2}. The graph 𝐹5 (of order 𝑛 = 30) in the family Fplanar is illustrated in Figure 11.22, where the highlighted vertices are an example of an 𝑖-set in 𝐹5 of cardinality 10 = 𝑛/3.

Figure 11.22 The bipartite planar cubic graph 𝐹5 ∈ Fplanar

352

Chapter 11. Domination in Planar Graphs

It was also shown in [5] that every graph 𝐺 in the family Fplanar has independent domination number one-third its order. Thus, if Conjecture 11.57 is true, then the bound is best possible. Proposition 11.58 ([5]) If 𝐺 ∈ Fplanar has order 𝑛, then 𝑖(𝐺) = 31 𝑛.

Chapter 12

Domination Partitions 12.1

Introduction

Let 𝐺 be a graph and let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be a partition of Ð order 𝑘 of the vertex 𝑘 𝑉𝑖 = 𝑉, and for set 𝑉 of 𝐺, where by definition each set 𝑉𝑖 ∈ 𝜋 is nonempty, 𝑖=1 all 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, 𝑉𝑖 ∩ 𝑉 𝑗 = ∅. Let P denote some property of a set 𝑆 ⊆ 𝑉, such as the property of being an independent set, being a dominating set, inducing a connected subgraph 𝐺 [𝑆], etc. In this case, we say that 𝑆 is a P-set and a partition of 𝑉 into P-sets is called a P-partition. For example, if P denotes the property of being an independent set, then a P-partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } is a proper coloring. Given a graph 𝐺, the minimum order of a partition of the vertex set of 𝐺 into independent sets is the chromatic number of 𝐺 and is denoted 𝜒(𝐺). In this chapter, we will focus largely on the property P that the set 𝑆 is a dominating set. In the first sections of the chapter, we present results on the domatic number, that is, the maximum order of a partition of the vertex set of a graph into dominating sets. We also consider the same type of partitions for total domination and independent domination. In the final section of this chapter, we consider graphs whose vertex set can be partitioned into two specified types of dominating sets, for example a dominating set and a total dominating set.

12.2

Domatic Numbers

Domatic numbers were introduced by Cockayne and Hedetniemi, first in 1975 [192] and soon thereafter in 1977 [194]. A domatic 𝑘-partition is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } into 𝑘 dominating sets. In general, a domatic partition is a domatic 𝑘-partition for some unspecified integer 𝑘, and the domatic number dom(𝐺) is the maximum order 𝑘 of any domatic 𝑘-partition of 𝐺. We note here that elsewhere in the literature the notation for domatic number is 𝑑 (𝐺). We chose to use dom(𝐺) since the symbol 𝑑 is often used for domination number, diameter, degree, distance, and domatic number. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_12

353

Chapter 12. Domination Partitions

354

It can be seen that a domatic partition 𝜋 of order 𝑘 is a vertex partition into 𝑘 pairwise disjoint subsets such that every vertex 𝑣 ∈ 𝑉 is adjacent to at least one vertex in every other subset of 𝜋 than its own subset. Since every set 𝑉𝑖 in a domatic partition satisfies |𝑉𝑖 | ≥ 𝛾(𝐺), we note that if 𝐺 has order 𝑛 and {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } is a domatic partition of maximum order 𝑘 = dom(𝐺), then 𝑛=

𝑘 ∑︁

|𝑉𝑖 | ≥

𝑖=1

𝑘 ∑︁

𝛾(𝐺) = 𝑘𝛾(𝐺) = dom(𝐺) · 𝛾(𝐺),

𝑖=1

yielding the following upper bound on the domatic number. Theorem 12.1 If 𝐺 is a graph of order 𝑛, then dom(𝐺) ≤

𝑛 𝛾 (𝐺) .

The classic 1962 result by Ore [622], given in Chapter 2 and restated here, shows that any isolate-free graph has domatic number at least 2. Lemma 12.2 ([622]) If 𝐺 is an isolate-free graph, then the complement 𝑉 \ 𝑆 of any minimal dominating set 𝑆 is a dominating set of 𝐺. Corollary 12.3 If 𝐺 is an isolate-free graph, then dom(𝐺) ≥ 2. In their first paper to define the domatic number, Cockayne and Hedetniemi [192] determined a lower bound on the number of edges of
a graph in terms of its domatic number, where it is understood that if 𝑎 < 𝑏, then 𝑏𝑎 = 0. Theorem 12.4 ([192]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with dom(𝐺) = 𝑑 and 𝑛 = 𝑘 𝑑 + 𝑟, where 0 ≤ 𝑟 < 𝑑, then     𝑑 𝑑 −𝑟 𝑚≥ (𝑘 + 1) − . 2 2 Proof Let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑑 } be a domatic partition of 𝐺 into 𝑑 = dom(𝐺) sets, each of which is a dominating set of 𝐺. Let 𝑡𝑖 = |𝑉𝑖 | for 𝑖 ∈ [𝑑], and let the sets in the partition 𝜋 be arranged in nondecreasing order so that 0 < 𝑡1 ≤ 𝑡 2 ≤ · · · ≤ 𝑡 𝑑 . The set 𝑉𝑖 dominates 𝑉 𝑗 , implying that there are at least 𝑡 𝑗 edges between 𝑉𝑖 and 𝑉 𝑗 for all 𝑖 and 𝑗, where 1 ≤ 𝑖 < 𝑗 ≤ 𝑑. Hence, if [𝑉𝑖 , 𝑉 𝑗 ] denotes the set of edges between 𝑉𝑖 and 𝑉 𝑗 , then for 𝑖 ∈ [𝑑] 𝑑 ∑︁

| [𝑉𝑖 , 𝑉 𝑗 ] | ≥

𝑗=𝑖+1

𝑑 ∑︁

(12.1)

𝑡𝑗.

𝑗=𝑖+1

Thus, by Inequality (12.1), 𝑚≥

∑︁

| [𝑉𝑖 , 𝑉 𝑗 ] | =

1≤𝑖< 𝑗 ≤𝑑

𝑑−1 ∑︁ 𝑑 ∑︁ 𝑖=1

𝑗=𝑖+1

! | [𝑉𝑖 , 𝑉 𝑗 ] | ≥

𝑑−1 ∑︁ 𝑑 ∑︁ 𝑖=1

𝑗=𝑖+1

! 𝑡𝑗 =

𝑑 ∑︁

(𝑖 − 1)𝑡 𝑖 . (12.2)

𝑖=1

Among all partitions of the integer 𝑛 into 𝑑 strictly positive nondecreasing parts, let 𝜋1 = {𝑛1 , 𝑛2 , . . . , 𝑛 𝑑 } be chosen so that

Section 12.2. Domatic Numbers

355 𝑑 ∑︁

(𝑖 − 1)𝑛𝑖

𝑖=1

is minimized. By definition, we have 𝑛 = 𝑛1 +𝑛2 +· · ·+𝑛 𝑑 and 1 < 𝑛1 ≤ 𝑛2 ≤ · · · ≤ 𝑛 𝑑 . We show that 𝑛 𝑑 ≤ 𝑛1 + 1. Suppose, to the contrary, that 𝑛 𝑑 ≥ 𝑛1 + 2. Let ℓ1 = max{𝑖 : 𝑛𝑖 = 𝑛1 } and let ℓ2 = min{𝑖 : 𝑛𝑖 = 𝑛 𝑑 }. By supposition, 𝑛ℓ2 = 𝑛 𝑑 ≥ 𝑛1 + 2 = 𝑛ℓ1 + 2. We now consider the partition of 𝑛 given by 𝜋2 = {𝑛1′ , 𝑛2′ , . . . , 𝑛′𝑑 } where 𝑛ℓ′ 1 = 𝑛ℓ1 +1, 𝑛ℓ′ 2 = 𝑛ℓ2 − 1, and 𝑛𝑖′ = 𝑛𝑖 for all 𝑖 ∈ [𝑑] \ {ℓ1 , ℓ2 }, that is, 𝜋2 is obtained from 𝜋1 by replacing 𝑛ℓ1 with 𝑛ℓ1 + 1 and replacing 𝑛ℓ2 with 𝑛ℓ2 − 1, and leaving all other terms in the partition unchanged. By definition of ℓ1 and ℓ2 , we have 𝑛 = 𝑛1′ + 𝑛2′ + · · · + 𝑛′𝑑 and 1 < 𝑛1′ ≤ 𝑛2′ ≤ · · · ≤ 𝑛′𝑑 . Moreover, −1 𝑑 ∑︁

(𝑖 − 1)𝑛𝑖 −

𝑖=1

𝑑 ∑︁

1

z }| { z }| {

(𝑖 − 1)𝑛𝑖′ = (ℓ1 − 1) 𝑛ℓ1 − 𝑛ℓ′ 1 + (ℓ2 − 1) 𝑛ℓ2 − 𝑛ℓ′ 2 = ℓ2 − ℓ1 > 0.

𝑖=1

This contradicts the choice of the partition 𝜋1 . Hence, 𝑛 𝑑 ≤ 𝑛1 + 1. Recall that 𝑛 = 𝑡1 + 𝑡 2 + · · · + 𝑡 𝑑 and 1 < 𝑡1 ≤ 𝑡2 ≤ · · · ≤ 𝑡 𝑑 . The sum on the right hand side of Inequality (12.2) is therefore minimized if 𝑡 𝑑 ≤ 𝑡 1 + 1. In this case, either 𝑟 = 0 and 𝑡 𝑖 = 𝑘 for all 𝑖 ∈ [𝑑], or 0 < 𝑟 < 𝑑 and ( 𝑘 if 1 ≤ 𝑖 ≤ 𝑑 − 𝑟 𝑡𝑖 = 𝑘 + 1 if 𝑑 − 𝑟 + 1 ≤ 𝑖 ≤ 𝑑. If 𝑟 = 0, then 𝑑 ∑︁

𝑑 ∑︁

      𝑑 𝑑 −𝑟 𝑑 (𝑘 + 1) − = (𝑖 − 1) = 𝑘 , (𝑖 − 1)𝑡 𝑖 = 𝑘 2 2 2 𝑖=1 𝑖=1

as desired. If 𝑟 ≥ 1, then elementary algebra shows that 𝑑 ∑︁ 𝑖=1

(𝑖 − 1)𝑡𝑖 = 𝑘

𝑑−𝑟 ∑︁ 𝑖=1

(𝑖 − 1) + (𝑘 + 1)

𝑑 ∑︁

(𝑖 − 1) =

𝑖=𝑑−𝑟+1

    𝑑 −𝑟 𝑑 (𝑘 + 1) − . 2 2

As remarked in [192], the lower bound in Theorem 12.4 is tight, that is, given any positive integers 𝑛 and 𝑑, where 𝑛 = 𝑘 𝑑 + 𝑟 for 0 ≤ 𝑟 < 𝑑, we can construct a graph 𝐺 of order
𝑛 with domatic
number dom(𝐺) = 𝑑 and minimum possible size, namely 𝑚 = 𝑑2 (𝑘 + 1) − 𝑑−𝑟 2 . For example, let 𝐺 be a 𝑑-partite graph with partite sets 𝑉1 , 𝑉2 , . . . , 𝑉𝑑 , where 𝑡𝑖 = |𝑉𝑖 | = 𝑘 for 1 ≤ 𝑖 ≤ 𝑑 − 𝑟 and 𝑡 𝑖 = |𝑉𝑖 | = 𝑘 + 1 for 𝑑 − 𝑟 + 1 ≤ 𝑖 ≤ 𝑑. If 𝑡 𝑖 = 𝑡 𝑗 for some 𝑖, 𝑗 where 1 ≤ 𝑖 < 𝑗 ≤ 𝑑, then we add a perfect matching between the vertices in 𝑉𝑖 and 𝑉 𝑗 . If 𝑡 𝑖 = 𝑡 𝑗 − 1 for some 𝑖, 𝑗 where 1 ≤ 𝑖 < 𝑗 ≤ 𝑑, then we add a matching between 𝑡 𝑖 vertices in 𝑉𝑖 and 𝑉 𝑗 and we add one additional edge from the resulting unmatched vertex in 𝑉 𝑗 to an arbitrary vertex of 𝑉𝑖 . In both cases, | [𝑉𝑖 , 𝑉 𝑗 ] | = 𝑡 𝑗 for 1 ≤ 𝑖 < 𝑗 ≤ 𝑑. Further, the set 𝑉𝑖 dominates 𝑉 𝑗 and the set 𝑉 𝑗 dominates the set 𝑉𝑖 . The resulting graph 𝐺 has order 𝑛, domatic number dom(𝐺) = 𝑑, and size

𝑚 = 𝑑2 (𝑘 + 1) − 𝑑−𝑟 2 .

Chapter 12. Domination Partitions

356

As an illustration of this construction for 𝑛 = 9 and 𝑑 = 4, the graph
𝐺 shown
in Figure 12.1 has order 𝑛 = 9, dom(𝐺) = 𝑑 = 4, and minimum size 𝑚 = 42 · 3 − 32 = 15. In this example, 𝑘 = 2 and 𝑟 = 1 and the four dominating sets are the four partite sets of 𝐺 indicated by the shaded groupings labeled 𝑉1 , 𝑉2 , 𝑉3 , 𝑉4 , where |𝑉1 | = |𝑉2 | = |𝑉3 | = 2 and |𝑉4 | = 3. 𝑉4

𝑉3

𝑉1

𝑉2

Figure 12.1 A graph 𝐺 of order 𝑛 = 9 with domatic number dom(𝐺) = 4

12.2.1

Domatically Full Graphs

Domatic 𝑘-partitions can be stated in terms of coloring the vertices with 𝑘 colors. A vertex 𝑣 is called colorful if it is adjacent to vertices of every other color, except possibly for the color that is used to color 𝑣, that is, all 𝑘 colors appear in the closed neighborhood N[𝑣] of 𝑣. A domatic 𝑘-partition is, therefore, a vertex partition in which every vertex is colorful. This means that every vertex has degree at least 𝑘 − 1, giving rise to the following upper bound on the domatic number in terms of minimum degree. Observation 12.5 ([194]) For any graph 𝐺, dom(𝐺) ≤ 𝛿(𝐺) + 1. The next corollary is a direct consequence of Corollary 12.3 and Observation 12.5. Corollary 12.6 A graph 𝐺 has dom(𝐺) = 1 if and only if 𝐺 has an isolated vertex. A graph 𝐺 is called domatically full if its domatic number achieves the bound of Observation 12.5, that is, if dom(𝐺) = 𝛿(𝐺) + 1. For example, complete graphs 𝐾𝑛 have dom(𝐾𝑛 ) = 𝑛 = 𝛿(𝐾𝑛 ) + 1 and as mentioned in Chapter 11, maximal outerplanar graphs 𝐺 have dom(𝐺) = 3 = 𝛿(𝐺) + 1. Other examples, given in [194], of domatically full graphs include empty graphs 𝐾 𝑛 , cycles 𝐶3𝑘 for any positive integer 𝑘, and nontrivial trees. Note that not all cycles are domatically full as dom(𝐶3𝑘+1 ) = dom(𝐶3𝑘+2 ) = 2. In 1984 Farber [269] algorithmically proved that strongly chordal graphs are domatically full by establishing a close relationship between such graphs and totally balanced matrices. In 1990 Lu et al. [575] showed that interval graphs are domatically full. A graph whose blocks are either cycles or complete is called a block-cactus graph. In 1998 Rautenbach and Volkmann [652] determined the domatic numbers of block-cactus graphs and characterized the domatically full block-cactus graphs.

Section 12.2. Domatic Numbers

357

In 1994 Chang [143] determined the domatic numbers of grid graphs showing that with two small exceptions grid graphs are domatically full. His proof makes use of the following observation, where the disjoint union of graphs 𝐺 1 and 𝐺 2 is denoted by 𝐺 1 ∪ 𝐺 2 . Observation 12.7 The following hold:  (a) If 𝐺 = 𝐺 1 ∪ 𝐺 2 , then dom(𝐺 1 ∪ 𝐺 2 ) = min dom(𝐺 1 ), dom(𝐺 2 ) . (b) If 𝐻 is a spanning subgraph of a graph 𝐺, then dom(𝐻) ≤ dom(𝐺). To present Chang’s proof, we recall from Chapter 9 the notation used for grids 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 , for integers 𝑚, 𝑛 ≥ 1. For a fixed value of 𝑖, the set of vertices of the form (𝑖, 𝑗), 𝑗 ∈ [𝑛], is called the 𝑖 th row of 𝐺 𝑚,𝑛 , and for a fixed value of 𝑗, the set of vertices of the form (𝑖, 𝑗), 𝑖 ∈ [𝑚], is called the 𝑗 th column of 𝐺 𝑚,𝑛 . Thus, the subgraph induced by the vertices in any given row 𝑖 is a copy of the path 𝑃𝑛 , while the subgraph induced by the vertices in any given column 𝑗 is a copy of the path 𝑃𝑚 . Theorem 12.8 ([143]) For integers 𝑚, 𝑛 such that 2 ≤ 𝑚 ≤ 𝑛, dom(𝑃𝑚 □ 𝑃𝑛 ) = 3, except dom(𝑃2 □ 𝑃2 ) = dom(𝑃2 □ 𝑃4 ) = 2. Proof It is straightforward to check that dom(𝑃2 □ 𝑃2 ) = dom(𝑃2 □ 𝑃4 ) = 2. Assume that 2 ≤ 𝑚 ≤ 𝑛. Since 𝛿(𝑃𝑚 □ 𝑃𝑛 ) = 2, by Observation 12.5, we have dom(𝑃𝑚 □ 𝑃𝑛 ) ≤ 3. Suppose, firstly, that 𝑚 is odd. In this case, it can be seen that {𝑆1 , 𝑆2 , 𝑆3 } is a domatic partition of 𝑃𝑚 □ 𝑃𝑛 , where these sets are defined as follows:  𝑆1 = (𝑖, 𝑗) : 𝑖 ≡ 0 (mod 2), 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛]  𝑆2 = (𝑖, 𝑗) : 𝑖 ≡ 1 (mod 4), 𝑗 ≡ 1 (mod 2), 𝑖 ∈ [𝑚],  ∪ (𝑖, 𝑗) : 𝑖 ≡ 3 (mod 4), 𝑗 ≡ 0 (mod 2), 𝑖 ∈  𝑆3 = (𝑖, 𝑗) : 𝑖 ≡ 1 (mod 4), 𝑗 ≡ 0 (mod 2), 𝑖 ∈ [𝑚],  ∪ (𝑖, 𝑗) : 𝑖 ≡ 3 (mod 4), 𝑗 ≡ 1 (mod 2), 𝑖 ∈

𝑗 ∈ [𝑛]



[𝑚], 𝑗 ∈ [𝑛] 𝑗 ∈ [𝑛]



[𝑚], 𝑗 ∈ [𝑛] .

Figure 12.2 illustrates this domatic 3-partition for the grid 𝑃5 □ 𝑃7 , where the vertices in 𝑆1 , 𝑆2 , and 𝑆3 are highlighted in red, green, and blue, respectively.

Figure 12.2 Domatic 3-partition of 𝑃5 □ 𝑃7

Chapter 12. Domination Partitions

358

Similarly, a domatic partition can be defined if 𝑛 is odd by considering the grid 𝑃𝑛 □ 𝑃𝑚 . Figure 12.3 illustrates this domatic 3-partition for 𝑃2 □ 𝑃7 . Hence, if 𝑚 or 𝑛 is odd, then dom(𝑃𝑚 □ 𝑃𝑛 ) ≥ 3 and so dom(𝑃𝑚 □ 𝑃𝑛 ) = 3.

Figure 12.3 Domatic 3-partition of 𝑃2 □ 𝑃7 Assume that both 𝑚 and 𝑛 are even. Then either 𝑚 = 2 and 𝑛 ≥ 6, or 𝑛 ≥ 𝑚 ≥ 4. The partition of the grid 𝑃4 □ 𝑃4 given in Figure 12.4 shows that dom(𝑃4 □ 𝑃4 ) ≥ 3, and so dom(𝑃4 □ 𝑃4 ) = 3. Hence, we may assume that 𝑛 ≥ 6. If 𝑚 ≥ 6, then let 𝐺 1 be the subgraph of 𝑃𝑚 □ 𝑃𝑛 induced by all vertices in the first three rows, and let 𝐺 2 be the subgraph induced by the remaining vertices in 𝑃𝑚 □ 𝑃𝑛 . Thus, 𝐺 1 = 𝑃3 □ 𝑃𝑛 and 𝐺 2 = 𝑃𝑚−3 □ 𝑃𝑛 , where each of 𝐺 1 and 𝐺 2 has an odd number of rows. By our previous argument, dom(𝐺 1 ) = dom(𝐺 2 ) = 3. If 𝑚 < 6, then the rows and columns of 𝑃𝑚 □ 𝑃𝑛 can be interchanged to form 𝐺 1 and 𝐺 2 , that is, consider the subgraphs 𝐺 1 and 𝐺 2 defined on 𝑃𝑛 □ 𝑃𝑚 . In both cases, the union 𝐺 1 ∪ 𝐺 2 is a spanning subgraph of 𝑃𝑚 □ 𝑃𝑛 . By  Observation 12.7, dom(𝑃𝑚 □ 𝑃𝑛 ) ≥ dom(𝐺 1 ∪ 𝐺 2 ) = min dom(𝐺 1 ), dom(𝐺 2 ) = 3.

Figure 12.4 Domatic 3-partition of 𝑃4 □ 𝑃4

12.2.2

Lower Bounds

In 1983 Zelinka [772] established the following lower bound on the domatic number of a graph.   Theorem 12.9 ([772]) If 𝐺 is a graph of order 𝑛, then dom(𝐺) ≥ 𝑛− 𝛿𝑛(𝐺) . Proof Let 𝑆 ⊆ 𝑉 be any set of cardinality at least 𝑛 − 𝛿(𝐺). If 𝑣 ∈ 𝑉 \ 𝑆, then since deg(𝑣) ≥ 𝛿(𝐺),  it follows that
 N(𝑣) ∩ 𝑆 ≠ ∅, and so 𝑆 is a dominating set of 𝐺. Thus, one can form 𝑛/ 𝑛 − 𝛿(𝐺) disjoint subsets of cardinality 𝑛 − 𝛿(𝐺), each of which is a dominating set. Adding any remaining vertices
to one of these subsets creates a  domatic partition of 𝐺 having order 𝑛/ 𝑛 − 𝛿(𝐺) .

Section 12.2. Domatic Numbers

359

Considering the bound of Theorem 12.9, intuitively it might seem that graphs with large minimum degree will have a large domatic number. However, Zelinka [771] showed that this is not necessarily true by constructing graphs with arbitrarily large minimum degree and domatic number 2. Theorem 12.10 ([772]) No minimum degree is sufficient to guarantee the existence of a partition into three dominating sets. Proof Construct the bipartite graph 𝐺 𝑛𝑘 from a set 𝑋 of 𝑛 vertices and a set 𝑌 , whose vertices are all the 𝑘-element subsets of 𝑋, by adding edges from each
vertex 𝑥 ∈ 𝑋 to the subsets of 𝑌 containing 𝑥 as a member. Thus, |𝑋 | = 𝑛, |𝑌 | = 𝑛𝑘 and the resulting graph 𝐺 𝑛𝑘 has minimum degree 𝑘. If 𝑛 ≥ 3𝑘 − 2, then in any 3-coloring of the vertices in 𝑋, at least 𝑘 vertices must receive the same color. Such a set of 𝑘 vertices in 𝑋 is the neighborhood of some vertex 𝑣 ∈ 𝑌 . However, such a vertex 𝑣 has all its neighbors of the same color and is therefore not dominated by at least one of the three color classes. Hence, if 𝑛 ≥ 3𝑘 − 2, then no 3-coloring of the vertices of 𝐺 𝑛𝑘 is a colorful coloring; that is, at least one of the three colors is missing from the closed neighborhood of some vertex in 𝐺 𝑛𝑘 . It follows that no minimum degree is sufficient to guarantee the existence of a domatic 3-partition, implying that dom(𝐺) ≤ 2. Consequently, by Corollary 12.3, dom(𝐺) = 2. In contrast to the result of Theorem 12.10, in 2004 Dankelmann and Calkin [214] showed that for regular graphs, a large minimum degree does guarantee a large domatic number by establishing the following lower bound. Theorem 12.11 ([214]) If 𝐺 is an 𝑟-regular graph 𝐺, then dom(𝐺) ≥

𝑟+1 3 ln(𝑟+1) .

Also in [214], the authors proved that the domatic number of a random 𝑟-regular graph is almost surely at most 𝑟, and that for random cubic graphs, the domatic number is almost surely equal to 3.

12.2.3

Generalizations of the Domatic Number

To conclude this section, we briefly mention some generalizations of the domatic number. Transitivity and Upper Domatic Number Given two disjoint sets 𝑅 and 𝑆 of vertices of a graph 𝐺, we say that 𝑅 dominates 𝑆, if every vertex in 𝑆 is adjacent to at least one vertex in 𝑅. In 2018 Hedetniemi and Hedetniemi [447] introduced the transitivity Tr(𝐺) of a graph 𝐺 as the maximum order 𝑘 of a partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉 such that for every 𝑖, 𝑗, with 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, the set 𝑉𝑖 dominates the set 𝑉 𝑗 . Later in 2020, Haynes et al. [404] extended this concept to define the upper domatic number of a graph as follows. A vertex partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of a graph 𝐺 such that for every 𝑖, 𝑗 with 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, either 𝑉𝑖 dominates 𝑉 𝑗 or 𝑉 𝑗 dominates 𝑉𝑖 , or both, is called an upper domatic partition of 𝐺.

360

Chapter 12. Domination Partitions

The upper domatic number Dom(𝐺) equals the maximum order of an upper domatic partition of 𝐺. From the definitions, it can be observed that for any graph 𝐺 of order 𝑛, dom(𝐺) ≤ Tr(𝐺) ≤ Dom(𝐺) ≤ 𝑛.

(12.3)

To illustrate this concept, we note that the upper bound on the domatic number of 𝛿(𝐺) + 1 does not hold for Dom(𝐺). Consider, for example, the graph 𝐺 formed by adding a new vertex 𝑢 adjacent to exactly one vertex 𝑣 of a complete graph 𝐾𝑛 . The graph 𝐺 satisfies dom(𝐺) = 2 = 𝛿(𝐺) +1, while Dom(𝐺) = 𝑛 since the set containing {𝑢, 𝑣} along with 𝑛 − 1 singleton sets, each containing a vertex of 𝐾𝑛 − {𝑢, 𝑣}, is an upper domatic partition of order Dom(𝐺). Not surprisingly, the difference between the domatic number dom(𝐺) and the upper domatic number Dom(𝐺) can be made arbitrarily large. Proposition 12.12 ([404]) For any positive integer 𝑘, there
is a graph 𝐺 such that Dom(𝐺) − dom(𝐺) ≥ 𝑘 and such that Dom(𝐺) − 𝛿(𝐺) + 1 ≥ 𝑘. Although the bound of 𝛿(𝐺) + 1 does not hold for Dom(𝐺), replacing minimum degree with maximum degree gives an upper bound. Theorem 12.13 ([404]) For any graph 𝐺, Dom(𝐺) ≤ Δ(𝐺) + 1. Coalition Partitions Coalition partitions were introduced in 2020 by Haynes et al. [405–409]. A coalition in a graph 𝐺 = (𝑉, 𝐸) consists of two disjoint sets of vertices 𝑉1 and 𝑉2 , neither of which is a dominating set but whose union 𝑉1 ∪ 𝑉2 is a dominating set. The sets 𝑉1 and 𝑉2 are said to form a coalition. A coalition partition, abbreviated 𝑐-partition, in a graph 𝐺 is a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } such that every set 𝑉𝑖 of 𝜋 is either a singleton set containing a dominating vertex or is not a dominating set but forms a coalition with another set 𝑉 𝑗 in 𝜋. The coalition number 𝐶 (𝐺) equals the maximum order 𝑘 of a 𝑐-partition of 𝐺. A set that does not dominate 𝐺 is called a non-dominating set. Notice that if a graph 𝐺 has no dominating vertex, then every set 𝑉𝑖 in a 𝑐-partition is a non-dominating set and hence must form a coalition with another set 𝑉 𝑗 in the partition. Theorem 12.14 ([405]) Every graph 𝐺 has a 𝑐-partition. Proof Let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be a domatic partition of a graph 𝐺, where 𝑘 = dom(𝐺). We can assume that 𝑉1 , 𝑉2 , . . . , 𝑉𝑘−1 are minimal dominating sets of 𝐺; if any set 𝑉𝑖 , for 𝑖 ∈ [𝑘 − 1], is not a minimal dominating set, let 𝑉𝑖′ ⊂ 𝑉𝑖 be a minimal dominating set contained in 𝑉𝑖 and add the vertices in 𝑉𝑖 \ 𝑉𝑖′ to 𝑉𝑘 . Notice that any partition of a non-singleton, minimal dominating set into two nonempty sets creates two non-dominating sets whose union forms a coalition. Therefore, for every 𝑖 ∈ [𝑘 − 1], every non-singleton set 𝑉𝑖 can be partitioned into two sets 𝑉𝑖,1 and 𝑉𝑖,2 , which form a coalition. Doing this for each 𝑉𝑖 , for 𝑖 ∈ [𝑘 − 1], we obtain a collection 𝜋 ′ of sets, each of which is either a singleton dominating set or is a non-dominating set that forms a coalition with another non-dominating set in 𝜋 ′ .

Section 12.3. Idomatic Number

361

Now consider the dominating set 𝑉𝑘 . If 𝑉𝑘 is a singleton dominating set, then adding 𝑉𝑘 to 𝜋 ′ gives a 𝑐-partition of 𝐺 of order at least 𝑘 = dom(𝐺). If 𝑉𝑘 is a non-singleton minimal dominating set, then we can partition it into two nondominating sets, add these two sets to 𝜋 ′ , and create a 𝑐-partition of 𝐺 of order at least 𝑘 + 1 > dom(𝐺). If 𝑉𝑘 is not a minimal dominating set, let 𝑉𝑘′ ⊂ 𝑉𝑘 be a minimal dominating set contained in 𝑉𝑘 and let 𝑊 𝑘 = 𝑉𝑘 \ 𝑉𝑘′ . Further, partition of 𝑉𝑘′ into two nonempty ′ and 𝑉 ′ and add the sets 𝑉 ′ and 𝑉 ′ to 𝜋 ′ . non-dominating sets 𝑉𝑘,1 𝑘,2 𝑘,2 𝑘,1 It follows that 𝑊 𝑘 is not a dominating set, else there are at least 𝑘 + 1 disjoint dominating sets in 𝐺, a contradiction since 𝑘 = dom(𝐺). If 𝑊 𝑘 forms a coalition with any non-dominating set, then adding 𝑊 𝑘 to 𝜋 ′ , we have a 𝑐-partition of 𝐺 of order at least 𝑘 + 2 > dom(𝐺). However, if 𝑊 𝑘 does not form a coalition with any set ′ from 𝜋 ′ and add the set 𝑉 ′ ∪ 𝑊 to 𝜋 ′ , thereby creating a in 𝜋 ′ , then remove 𝑉𝑘,2 𝑘 𝑘,2 𝑐-partition of 𝐺 of order at least 𝑘 + 1 > dom(𝐺). We give, as examples, the coalition numbers of paths and cycles. Theorem 12.15 ([405]) For the path 𝑃𝑛 ,

𝐶 (𝑃𝑛 ) =

 𝑛      4  5    6 

if if if if

𝑛≤4 𝑛=5 6≤𝑛≤9 𝑛 ≥ 10.

Theorem 12.16 ([405]) For the cycle 𝐶𝑛 ,   𝑛    𝐶 (𝐶𝑛 ) = 5   6 

if 𝑛 ≤ 6 if 𝑛 = 7 if 𝑛 ≥ 8.

For graphs 𝐺 with no isolated vertex and no dominating vertex, Haynes et al. [405] showed that the coalition number 𝐶 (𝐺) is at least twice the domatic number dom(𝐺). Corollary 12.17 ([405]) If 𝐺 is a graph of order 𝑛 with 2 ≤ 𝛿(𝐺) ≤ 𝑛 − 2, then 4 ≤ 2 dom(𝐺) ≤ 𝐶 (𝐺). In [407] the authors determined the following two upper bounds on the coalition number.
2 Theorem 12.18 ([407]) For any graph 𝐺, 𝐶 (𝐺) ≤ Δ(𝐺) + 3 /4. Theorem 12.19 ([407]) If 𝐺 is
a graph with no dominating vertex and 𝛿(𝐺) <
Δ(𝐺)/2, then 𝐶 (𝐺) ≤ 𝛿(𝐺) + 1 Δ(𝐺) − 𝛿(𝐺) + 2 .

12.3

Idomatic Number

We adopt our earlier notation in the book and abbreviate an independent dominating set by ID-set. A graph 𝐺 is said to be idomatic if it has a domatic partition 𝜋 = {𝑉1 , 𝑉2 ,

362

Chapter 12. Domination Partitions

. . . , 𝑉𝑘 } in which every subset 𝑉𝑖 , for 𝑖 ∈ [𝑘], is an ID-set, in which case 𝜋 is called an independent domatic partition. Thus, an independent domatic partition of 𝐺 is a collection of ID-sets and is also a proper coloring of 𝐺. The idomatic number idom(𝐺) equals the maximum order of an independent domatic partition of 𝐺. If a graph 𝐺 does not have an independent domatic partition, then we define idom(𝐺) = 0. For example, idom(𝐶5 ) = 0. It follows from the definitions, however, that for every idomatic graph 𝐺, 𝜒(𝐺) ≤ idom(𝐺) ≤ dom(𝐺). Cockayne and Hedetniemi [193] showed, for example, that the following families of graphs are idomatic: • complete graphs and their complements • connected bipartite graphs • complete 𝑘-partite graphs 𝐾𝑛1 ,𝑛2 ,...,𝑛𝑘 • cycles 𝐶𝑛 , where 𝑛 is even or 𝑛 is congruent to 0 modulo 3 • uniquely 𝑘-colorable graphs • maximal outerplanar graphs • 𝑛-cubes 𝑄 𝑛 • 𝑘-trees • complete graphs 𝐾2𝑛 minus a 1-factor. We note that triangulated discs and weak triangulated discs, which are discussed in Chapter 11, are two more examples of idomatic graphs. Cockayne and Hedetniemi [193] also presented three classes of graphs which are not idomatic: (i) graphs, other than 𝐾 𝑛 , with an isolated vertex, (ii) graphs for which 𝜒(𝐺) > dom(𝐺), and (iii) graphs for which 𝜒(𝐺) > 𝛿(𝐺) + 1. In 1983 Zelinka [770] established the following result. Theorem 12.20 ([770]) For any integers 𝑎 and 𝑏 with 2 ≤ 𝑎 ≤ 𝑏, there exists a graph 𝐺 such that idom(𝐺) = 𝑎 and dom(𝐺) = 𝑏. Proof Let 𝑎 and 𝑏 be integers such that 2 ≤ 𝑎 ≤ 𝑏 and let the vertex set of 𝐺 be the union of three sets: 𝑋 = {𝑥1 , 𝑥2 , . . . , 𝑥 𝑏−𝑎+2 }, 𝑌 = {𝑦 1 , 𝑦 2 , . . . , 𝑦 𝑏−𝑎+2 }, and 𝑍 = {𝑧1 , 𝑧2 , . . . , 𝑧 𝑎−2 }. Two vertices of 𝐺 are adjacent if and only if they are not both in 𝑋 or not both in 𝑌 . Stated equivalently, 𝑋 and 𝑌 are independent sets, the subgraph induced by 𝑍 is a complete graph, and any two vertices in two different sets are adjacent. Then the partition  𝜋1 = 𝑋, 𝑌 , {𝑧1 }, {𝑧2 }, . . . , {𝑧 𝑎−2 } is an idomatic partition of order 𝑎, and so idom(𝐺) ≥ 𝑎. There can be no idomatic partition having more sets than 𝜋 since all independent sets are either singleton vertices from 𝑍, or are a subset of either 𝑋 or 𝑌 and no proper subset of 𝑋 or 𝑌 is an ID-set. Thus, idom(𝐺) ≤ 𝑎 and hence idom(𝐺) = 𝑎. Similarly, the partition  𝜋2 = {𝑥1 , 𝑦 1 }, {𝑥 2 , 𝑦 2 }, . . . , {𝑥 𝑏−𝑎+2 , 𝑦 𝑏−𝑎+2 }, {𝑧1 }, {𝑧2 }, . . . , {𝑧 𝑎−2 }

Section 12.4. Total Domatic Number

363

is a domatic partition of order 𝑏. Thus, dom(𝐺) ≥ 𝑏. But any domatic partition of 𝐺 with more than 𝑏 sets must contain an additional singleton set, and the only singleton dominating sets are those consisting of a single vertex in 𝑍. Thus, dom(𝐺) ≤ 𝑏, and so dom(𝐺) = 𝑏. Notice that if 𝑆 is a maximal clique in a graph 𝐺, then 𝑆 is a dominating set in the complement 𝐺. Thus, if 𝑐(𝐺) denotes the minimum order of a maximal clique in 𝐺, then 𝛾(𝐺) ≤ 𝑐(𝐺). But it can be further noticed that if 𝑆 is a maximal clique in a graph 𝐺, then 𝑆 is an ID-set in 𝐺 and so 𝑖(𝐺) ≤ 𝑐(𝐺). Moreover, since every minimum ID-set in 𝐺 is a maximal clique in 𝐺, it follows that 𝑐(𝐺) ≤ 𝑖(𝐺). Consequently, 𝑖(𝐺) = 𝑐(𝐺). It can be shown that 𝑐(𝐺) and dom(𝐺) are in general incomparable. But for idomatic graphs we can say the following. Proposition 12.21 ([194]) If 𝐺 is an idomatic graph, then 𝛾(𝐺) ≤ 𝑖(𝐺) = 𝑐(𝐺) ≤ 𝜒(𝐺) ≤ idom(𝐺) ≤ dom(𝐺) ≤ Tr(𝐺) ≤ Dom(𝐺) ≤ Δ(𝐺) + 1. Proof As observed above, 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝑐(𝐺). By definition, 𝑐(𝐺) ≤ 𝜒(𝐺). And as previously observed, for every idomatic graph 𝐺, 𝜒(𝐺) ≤ idom(𝐺) ≤ dom(𝐺). By Inequality (12.3), dom(𝐺) ≤ Tr(𝐺) ≤ Dom(𝐺). And by Theorem 12.13, we have Dom(𝐺) ≤ Δ(𝐺) + 1. Not only is 𝑐(𝐺) ≤ dom(𝐺) for idomatic graphs 𝐺, but it is true for graphs whose complement is idomatic. Proposition 12.22 ([194]) If the complement 𝐺 of a graph 𝐺 is idomatic, then 𝑐(𝐺) ≤ dom(𝐺). Proof Let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be an independent domatic partition of the idomatic complement𝐺 of 𝐺. Since each set 𝑉𝑖 is a maximal clique in 𝐺, it follows that 𝑐(𝐺) ≤ min |𝑉𝑖 | : 𝑖 ∈ [𝑘] . But any set 𝑆 containing one vertex from each set 𝑉𝑖  of 𝜋 is a dominating set of 𝐺 and there are min |𝑉𝑖 | : 𝑖 ∈ [𝑘] disjoint sets  with this property. Hence, dom(𝐺) ≥ min |𝑉𝑖 | : 𝑖 ∈ [𝑘] ≥ 𝑐(𝐺), and therefore 𝑐(𝐺) ≤ dom(𝐺). Related topics not covered here concern the maximum number of vertex-disjoint dominating sets of a specific type in a graph 𝐺. For example, Cockayne and Hedetniemi [193] investigated the maximum number of vertex-disjoint ID-sets in a graph 𝐺, which they denoted by 𝑏(𝐺). This number is not the same as idom(𝐺) since a partition is not required in order to achieve the value 𝑏(𝐺). Indeed, 𝑏(𝐺) ≥ 1 for every graph 𝐺 since every graph 𝐺 has at least one ID-set. For example, idom(𝐶5 ) = 0 while 𝑏(𝐶5 ) = 2.

12.4

Total Domatic Number

A total domatic 𝑘-partition of a graph 𝐺 = (𝑉, 𝐸) is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉 into 𝑘 TD-sets. In general, a total domatic partition is a total domatic 𝑘-partition

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for some unspecified integer 𝑘. It can be seen that a total domatic 𝑘-partition is a vertex partition into 𝑘 pairwise disjoint subsets such that every vertex 𝑣 ∈ 𝑉 is adjacent to at least one vertex in every subset including its own subset. The total domatic number tdom(𝐺) of a graph 𝐺 equals the maximum order of a total domatic partition of 𝐺. Thus, tdom(𝐺) is the maximum number of TD-sets into which the vertex set of 𝐺 can be partitioned. This concept was introduced by Cockayne et al. [182] in 1980. Note that the total domatic number of 𝐺 only makes sense if 𝐺 is an isolate-free graph since the total domination number is not defined for graphs with isolated vertices. The parameter tdom(𝐺) is equivalent to the maximum number of colors in a (not necessarily proper) coloring of the vertices of a graph where every color appears in every open neighborhood. In 2015 Chen et al. [154] called this the coupon coloring problem. Let 𝐺 be an isolate-free graph of order 𝑛. Since every vertex in a total domatic partition of 𝐺 must have a neighbor in every set including its own set, we have tdom(𝐺) ≤ 𝛿(𝐺). Every set in a total domatic partition of 𝐺 is a TD-set, implying that tdom(𝐺) ≤ 𝑛/𝛾t (𝐺). Since 𝑛 ≥ 𝛾t (𝐺) ≥ 2, this implies that tdom(𝐺) ≤ ⌊𝑛/2⌋, as first observed by Zelinka [776]. This bound is achievable by taking, for example, the complete bipartite graph 𝐾 𝑝, 𝑝 of order 𝑛 = 2𝑝 where tdom(𝐺) = 𝑝 = 𝑛/2. Since every TD-set is a dominating set of a graph 𝐺, every partition of 𝑉 into TD-sets is a partition of 𝑉 into dominating sets and so dom(𝐺) ≥ tdom(𝐺). If 𝐺 is a graph whose domatic number dom(𝐺) is even, then combining pairs of dominating sets in a domatic partition of 𝑉 yields a partition of 𝑉 into 21 dom(𝐺) TD-sets, implying that tdom(𝐺) ≥ 12 dom(𝐺). Further, if dom(𝐺) ≥ 3 is odd, then we can combine three dominating sets into one set and pair off the remaining dominating
sets in the partition yielding a partition of 𝑉 into 12 dom(𝐺) − 1 TD-sets. The above observations yield the following bounds for the total domatic number. Theorem 12.23 The following hold in an isolate-free graph 𝐺: (a) tdom(𝐺) ≤ 𝛿(𝐺). (b) tdom(𝐺) ≤ 𝑛/𝛾t (𝐺).   (c) tdom(𝐺) ≤ 12 𝑛 . (d) tdom(𝐺) ≤ dom(𝐺) ≤ 2 tdom(𝐺) + 1. We remark that equality can occur in the upper bound dom(𝐺) ≤ 2 tdom(𝐺) + 1 in Theorem 12.23(d). For example, if 𝐺 = 𝐶𝑛 , where 𝑛 ≡ 0 (mod 3) but 𝑛 . 0 (mod 4), then dom(𝐺) = 3 but tdom(𝐺) = 1. In 1989 Zelinka [776] showed that there exist graphs with arbitrarily large minimum degree and total domatic number 1. Theorem 12.24 ([776]) No minimum degree is sufficient to guarantee the existence of a partition into two total dominating sets. Proof Let 𝐺 𝑛𝑘 be the bipartite graph with partite sets 𝑋 and
𝑌 constructed in the proof of Theorem 12.10. Recall that |𝑋 | = 𝑛 and |𝑌 | = 𝑛𝑘 and that graph 𝐺 𝑛𝑘 has minimum degree 𝑘. If 𝑛 ≥ 2𝑘 − 1, then in any 2-coloring of the vertices of 𝑋, at least 𝑘 vertices must receive the same color. Such a set of 𝑘 vertices in 𝑋 is the

Section 12.4. Total Domatic Number

365

neighborhood of some vertex 𝑣 ∈ 𝑌 . However, such a vertex 𝑣 has all its neighbors of the same color and is therefore not totally dominated by one of the color classes. Hence, in this case when 𝑛 ≥ 2𝑘 − 1, no 2-coloring of the vertices of 𝐺 𝑛𝑘 has the property that every vertex has a neighbor of both colors. Equivalently, the vertex set of graph 𝐺 𝑛𝑘 cannot be partitioned into two total dominating sets, implying that tdom(𝐺 𝑛𝑘 ) = 1. As first observed by Zelinka [776], if 𝐺 is an isolate-free graph and 𝐷 is an arbitrary set of 𝑛 − 𝛿(𝐺) + 1 vertices, then every vertex has at least one neighbor in 𝐷, and so 𝐷 is a TD-set of 𝐺. Hence, a partition of 𝑉 into classes each of cardinality 𝑛 − 𝛿(𝐺) + 1, with the exception of at most one class which would have cardinality at least 𝑛 − 𝛿(𝐺) + 1 is a total domatic partition of 𝐺. This yields the following lower bound on the total domatic number. Proposition 12.25 ([776]) If 𝐺 is a graph of order 𝑛, then   𝑛 tdom(𝐺) ≥ . 𝑛 − 𝛿(𝐺) + 1

12.4.1

Total Domatic Number in Graph Families

For examples, we list the total domatic numbers of some well-known families of graphs. In what follows, the wheel 𝑊𝑛 is the graph built from the cycle 𝐶𝑛 by adding a new vertex adjacent to every vertex on the cycle, and the 𝑛-dimensional cube 𝑄 𝑛 is the graph defined recursively by 𝑄 1 = 𝐾2 and 𝑄 𝑛 = 𝑄 𝑛−1 □ 𝐾2 . Proposition 12.26 (a) ([182]) For 𝑛 (b) ([182]) For 1 (c) ([182]) For 𝑘 (d) ([780]) For 𝑛 (e) ([769]) For 𝑘

For integers 𝑚, 𝑛,and  𝑘, the following hold: ≥ 2, tdom(𝐾𝑛 ) = 𝑛2 . ≤ 𝑚 ≤ 𝑛, tdom(𝐾𝑚,𝑛 ) = 𝑚. ≥ 1, tdom(𝐶4𝑘 ) = 2. ≥ 3, tdom(𝑊𝑛 ) = 2. ≥ 1, tdom(𝑄 2𝑘 ) = 2 𝑘 .

In 2012 Aram et al. [34] showed that the total domatic number of a random 𝑟-regular graph is almost surely at most 𝑟 − 1. They also gave a lower bound on the 𝑟 total domatic number of an 𝑟-regular graph 𝐺, namely, 3 ln(𝑟 ) ≤ tdom(𝐺). In 2018 Akbari et al. [11] established a condition for a cubic graph to have total domatic number at least two.

12.4.2

Total Domatic Number in Planar Graphs

In 2018 Goddard and Henning [354] established best possible upper bounds on the total domatic number of a planar graph and a toroidal graph. Since the minimum degree in a planar graph is at most 5, an immediate upper bound on the total domatic number of a planar graph is 5. This trivial upper bound was improved in [354]. Theorem 12.27 ([354]) If 𝐺 is a planar graph, then tdom(𝐺) ≤ 4.

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As remarked in [354], there are planar graphs 𝐺 with tdom(𝐺) = 4. The example they provided is a graph 𝐺 obtained from a truncated tetrahedron (which has four regular hexagonal faces and four equilateral triangles) by adding a vertex inside each hexagonal face and joining it to each of the six vertices on that face. The resulting graph 𝐺, illustrated in Figure 12.5, where the dashed edges are the new edges added to the truncated tetrahedron, is a planar graph of order 16 satisfying tdom(𝐺) = 4 with the four color classes comprising four vertex-disjoint TD-sets in 𝐺 (or equivalently, a coupon coloring using four colors).

Figure 12.5 A planar graph with tdom(𝐺) = 4

An analogous result on bounds on the total domatic number for toroidal graphs was presented in [354]. We note that here a toroidal graph is a graph that can be embedded on the surface of a torus, while in other chapters, the Cartesian product 𝐶𝑚 □ 𝐶𝑛 is called a toroidal graph or torus. Theorem 12.28 ([354]) If 𝐺 is a toroidal graph, then tdom(𝐺) ≤ 5, and this bound is best possible. As remarked in [354], if 𝐺 is the toroidal graph illustrated in Figure 12.6, where the top and bottom dotted lines should be identified and similarly with the left and right dotted lines, then tdom(𝐺) = 5, where the vertices of the same color form a TD-set of 𝐺.

12.5

Results of Zelinka on Domatic Numbers

Perhaps no one has investigated properties of the domatic, idomatic, and total domatic numbers, and other variants of the domatic number not mentioned in this chapter, more than Bohdan Zelinka (1940–2005). In recognition of his pioneering work in this area, we present a representative sample of his results involving the core parameters. For more of his work, see [766, 767, 773, 775, 778, 779, 781–784]. We note that the chapter titled Domatic Numbers of Graphs in the book [416] was written by Professor Zelinka [780]. Zelinka studied domatic numbers of several families of

Section 12.5. Results of Zelinka on Domatic Numbers

367

Figure 12.6 A toroidal graph 𝐺 with tdom(𝐺) = 5

graphs. It will be helpful to define the following well-studied classes of graphs. For two integers 𝑘 and 𝑛 with 2 ≤ 𝑘 ≤ 𝑛, the Kneser graph 𝐾 (𝑛, 𝑘) is the graph with vertex set corresponding one-to-one with the 𝑘-element subsets of an 𝑛-element set, where two vertices are adjacent if and only if their corresponding subsets are disjoint. These graphs were introduced by Kneser [531] in 1978. For any positive integer 𝑛, the lattice graph 𝐿 𝑛 of dimension 𝑛 is the graph whose vertex set is the set of all 𝑛-dimensional vectors with integer coordinates, where two vertices are adjacent if and only if their corresponding vectors differ in precisely one coordinate by a value of one. Theorem 12.29 ([504]) For the Kneser graph 𝐾 (𝑛, 2), the following hold: (a) If 𝑛 = 5, then dom(𝐾 (5, 2)) = 2.   (b) If 𝑛 ≥ 3 and 𝑛 ≠ 5, then dom(𝐾 (𝑛, 2)) = 16 𝑛(𝑛 − 1) .   (c) If 𝑛 ≥ 6, then tdom(𝐾 (𝑛, 2)) = 16 𝑛(𝑛 − 1) . Theorem 12.30 ([774, 777]) The following hold: (a) For any finite nontrivial cactus 𝐺, either dom(𝐺) = 2 or dom(𝐺) = 3. (b) For the lattice graph 𝐿 𝑛 , dom(𝐿 𝑛 ) = 2𝑛 + 1 and tdom(𝐿 𝑛 ) = 2𝑛. Zelinka [768] showed the existence of graphs 𝐺 having given vertex connectivity 𝜅(𝐺) (edge connectivity 𝜆(𝐺)) and domatic number. Theorem 12.31 ([768]) The following hold: (a) For any positive integers 𝑝 < 𝑞, there exists a graph 𝐺 with 𝜅(𝐺) = 𝑝 and dom(𝐺) = 𝑞. (b) For any positive integers 𝑝 < 𝑞, there exists a graph 𝐺 with 𝜆(𝐺) = 𝑝 and dom(𝐺) = 𝑞. (c) For any positive integer ℎ, there exists a graph 𝐺 with 𝜅(𝐺) − dom(𝐺) = 𝜆(𝐺) − dom(𝐺) = ℎ. We note that a graph consisting of two clique subgraphs with 𝑞 vertices each, having exactly 𝑝 vertices in common, satisfies the conditions of Theorem 12.31(a), while a graph consisting of two disjoint cliques with 𝑞 vertices each and 𝑝 independent

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edges joining vertices of the distinct cliques has the properties of Theorem 12.31(b). For Theorem 12.31(c), the Cartesian product 𝐾 ℎ+2 □ 𝐾 ℎ+2 has the desired values.

12.6

Dominating Bipartitions of Graphs

In this section, we study graphs whose vertex set can be partitioned into two specified types of dominating sets. Recall that Ore’s Lemma (see Chapter 2) states that if 𝐺 is an isolate-free graph, then the complement 𝑉 \ 𝐷 of any minimal dominating set 𝐷 is a dominating set. Thus, the vertex set of any isolate-free graph can be partitioned into two dominating sets. Since any maximal independent set is also a minimal dominating set, we have the following corollary. Corollary 12.32 ([622]) If 𝐺 is an isolate-free graph, then the vertices of 𝐺 can be partitioned into an ID-set and a dominating set. Although Ore’s Lemma shows that every graph with minimum degree 1 can be partitioned into two dominating sets, this is not true in general when other types of dominating sets are considered. Not all graphs can be partitioned into two TD-sets. For example, if 𝐺 is a cycle 𝐶𝑛 where 𝑛 . 0 (mod 4), then tdom(𝐺) = 1. Indeed, as stated in Theorem 12.24, no minimum degree is sufficient to guarantee the existence of two disjoint TD-sets. A simple example of a graph that does not have a vertex bipartition into two ID-sets is the cycle 𝐶5 .

12.6.1 Dominating and Total Dominating Set Partition In 2008 Henning and Southey [478] established the following sufficient condition for graphs whose vertex set can be partitioned into a dominating set and a TD-set. To present their proof, we need the following terminology. A vertex 𝑣 is called an 𝑆-bad vertex if N[𝑣] ⊆ 𝑆. Elsewhere in this book, as originally defined by Slater, an 𝑆-bad vertex is called an enclave of 𝑆. Further, a vertex 𝑢 ∈ 𝑆 is called an 𝑆-weak vertex if 𝑢 has degree 1 in 𝐺 [𝑆] and its neighbor in 𝑆 is an 𝑆-bad vertex. A component of a graph 𝐺 that is isomorphic to a graph 𝐹 we call an 𝐹-component of 𝐺. Theorem 12.33 ([478]) If 𝐺 is a graph with no 𝐶5 -component and 𝛿(𝐺) ≥ 2, then the vertices of 𝐺 can be partitioned into a dominating set and a TD-set. Proof Among all TD-sets of 𝐺, let 𝑆 be chosen so that (a) the number of 𝑆-bad vertices is minimized, and (b) subject to (a), the number of 𝑆-weak vertices is minimized. Assume that there is at least one 𝑆-bad vertex 𝑣. If 𝑣 has no 𝑆-weak neighbor, then 𝑆 ′ = 𝑆 \ {𝑣} is a TD-set of 𝐺 with fewer 𝑆 ′ -bad vertices than 𝑆-bad vertices, contradicting our choice of 𝑆. Hence, we may assume that every 𝑆-bad vertex has at least one 𝑆-weak neighbor. Let 𝑤 be an 𝑆-weak vertex adjacent to 𝑣. Since 𝛿(𝐺) ≥ 2, the vertex 𝑤 is adjacent to at least one vertex in 𝑉 \ 𝑆. If epn(𝑤, 𝑆) = ∅, then 𝑆 ′ = 𝑆 \ {𝑤} is a TD-set of 𝐺 with fewer 𝑆 ′ -bad vertices than 𝑆-bad vertices (since 𝑣 is not an 𝑆 ′ -bad

Section 12.6. Dominating Bipartitions of Graphs

369

vertex), contradicting our choice of 𝑆. Hence, |epn(𝑤, 𝑆)| ≥ 1. Let 𝑤 ′ ∈ epn(𝑤, 𝑆). Since 𝛿(𝐺) ≥ 2, the vertex 𝑤 ′ is adjacent to at least one vertex in 𝑉 \ 𝑆 and N[𝑤 ′ ] \ {𝑤} ⊆ 𝑉 \ 𝑆. We show next that 𝑤 has degree 2 in 𝐺. Suppose, to the contrary, that deg(𝑤) ≥ 3. Then, 𝑆 ′ = 𝑆 ∪ {𝑤 ′ } is a TD-set of 𝐺 that satisfies condition (a), but with fewer 𝑆 ′ weak vertices than 𝑆-weak vertices, contradicting our choice of 𝑆. Hence, deg(𝑤) = 2 and since 𝑣 and 𝑤 are arbitrary 𝑆-bad and 𝑆-weak vertices, respectively, every 𝑆-weak vertex has degree 2. For 𝑘 ≥ 1, let 𝑊 = {𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑘 } be the set of all 𝑆-weak neighbors of 𝑣. Then, N(𝑤 𝑖 ) = {𝑣, 𝑤 𝑖′ } for 𝑖 ∈ [𝑘]. Let 𝑊 ′ = {𝑤 1′ , 𝑤 2′ , . . . , 𝑤 ′𝑘 }. If every vertex in 𝑊 ′ is adjacent to a vertex in 𝑉 \ (𝑆 ∪ 𝑊 ′ ), then 𝑆 ′ = (𝑆 ∪ 𝑊 ′ ) \ {𝑣} is a TD-set of 𝐺 with fewer 𝑆 ′ -bad vertices than 𝑆-bad vertices, contradicting our choice of 𝑆. Hence, renaming vertices if necessary, we may assume that N[𝑤 1′ ] ⊆ 𝑊 ′ ∪ {𝑤 1 } and that 𝑤 1′ 𝑤 2′ is an edge of 𝐺.
If deg(𝑣) ≥ 3, then 𝑆 ′ = 𝑆 ∪ {𝑤 1′ , 𝑤 2′ } \ {𝑤 1 , 𝑤 2 } is a TD-set of 𝐺 with fewer 𝑆 ′ -bad vertices than 𝑆-bad vertices, contradicting our choice of 𝑆. Hence, each of 𝑣, 𝑤 1 , and 𝑤 2 has degree 2 in 𝐺 and 𝐶 : 𝑣 𝑤 1 𝑤 1′ 𝑤 2′ 𝑤 2 𝑣 is an induced 5-cycle in 𝐺. Since 𝐺 contains no 𝐶5 -component, the vertex 𝑤 1′ or the vertex 𝑤 2′ , say
𝑤 2′ , is adjacent to some vertex not in the 5-cycle 𝐶. But then 𝑆 ′ = 𝑆 ∪ {𝑤 1′ , 𝑤 2′ } \ {𝑣, 𝑤 1 } is a TD-set of 𝐺 with fewer 𝑆 ′ -bad vertices than 𝑆-bad vertices, contradicting our choice of 𝑆. We deduce, therefore, that the TD-set 𝑆 contains no 𝑆-bad vertices. Hence, 𝑉 \ 𝑆 is a dominating set of 𝐺, and the result holds.

As noted in [478], the minimum degree condition of Theorem 12.33 cannot be relaxed to 𝛿(𝐺) ≥ 1. For example, the vertex set of a 2-corona 𝐻 ◦ 𝑃2 cannot be partitioned into a dominating set and a TD-set. By Theorem 12.33, every connected graph with minimum degree at least two that is not a cycle on five vertices has a disjoint dominating set and a TD-set. Further, Henning and Southey [478] presented a constructive characterization of connected graphs of order at least 4 that have this property. It follows from Theorem 12.33 that every connected graph 𝐺 of minimum degree at least three has a dominating set 𝐷 and a TD-set 𝑇 that are disjoint. Thus, 𝐺 has a vertex bipartition {𝐷 ′ , 𝑇 }, where 𝐷 ′ is a superset of 𝐷 and hence is a dominating set of 𝐺. Henning et al. [467] proved that the Petersen graph is the only such graph for which 𝐷 ′ is necessarily equal to the set 𝐷, that is, where 𝐷 ∪ 𝑇 necessarily contains all vertices of the graph. Recall that as a special case of Theorem 6.52 from Chapter 6, the total domination number of a cubic graph is at most half its order; that is, if 𝐺 is a cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 12 𝑛. Southey and Henning [685] showed the following. Theorem 12.34 ([685]) If 𝐺 is a cubic graph of order 𝑛, then the vertices of 𝐺 can be partitioned into a dominating set and a TD-set, where the TD-set has cardinality at most 12 (𝑛 + 2).

370

12.6.2

Chapter 12. Domination Partitions

Total Dominating Set and Independent Dominating Set Partition

In 2019 Delgado et al. [224] studied graphs whose vertex set can be partitioned into a TD-set and an ID-set. The paths and cycles having such a partition were observed in [224]. Proposition 12.35 ([224]) The following hold: (a) The vertices of a cycle 𝐶𝑛 can be partitioned into a TD-set and an ID-set if and only if 𝑛 ≡ 0 (mod 3). (b) The vertices of a nontrivial path 𝑃𝑛 can be partitioned into a TD-set and an ID-set if and only if 𝑛 ≡ 1 (mod 3). Characterizing graphs with a vertex bipartition into a TD-set and an ID-set, in general, seems to be quite difficult. We say that a graph 𝐺 is a TI-graph if its vertex set can be partitioned into a TD-set and an ID-set. It was shown in [224] that there is no forbidden subgraph characterization of TI-graphs. Although the authors of [224] were not able to characterize TI-graphs, they gave a constructive characterization of TI-trees and characterized the TI-graphs of diameter 2. Proposition 12.36 ([224]) A diameter-2 graph 𝐺 is a TI-graph if and only if 𝐺 has a maximal independent set that is not the open neighborhood of some vertex. Several sufficient conditions for a graph to be a TI-graph were also given in [224]. Theorem 12.37 ([224]) Any of the following conditions is sufficient for 𝐺 of order 𝑛 to be TI-graph: (a) 𝛿(𝐺) > 21 𝑛. (b) 𝛾(𝐺) ≥ 3. (c) 𝐺 is claw-free and 𝛿(𝐺) ≥ 3. (d) 𝑖(𝐺) < 𝛿(𝐺).

12.6.3

Partitions into Two Total Dominating Sets

In this section, we consider the graphs whose vertex set can be partitioned into two TD-sets. We call a graph having such a partition a total dominating partitionable graph, abbreviated TDP-graph. Observe that a graph 𝐺 is a TDP-graph if and only if its total domatic number tdom(𝐺) ≥ 2. Note also that total domination is defined only for isolate-free graphs and that 𝛾t (𝐺) ≥ 2 when defined. Hence, the following trivial conditions are necessary for a graph to be a TDP-graph. Observation 12.38 If 𝐺 is a TDP-graph of order 𝑛, then 𝑛 ≥ 4 and 𝛿(𝐺) ≥ 2. Recall from Theorem 12.24 that no minimum degree is sufficient to guarantee the existence of a partition into two total dominating sets. In 1998 Heggernes and Telle [451] showed that the decision problem to decide for a given graph if it has two disjoint TD-sets is NP-complete, even for bipartite graphs. In 2016 Delgado

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et al. [223] showed that there is no forbidden subgraph characterization of TDPgraphs. On the other hand, in 2002 Feige et al. [287] and in 2004 Dankelmann and Calkin [214] independently showed that if the maximum degree of a graph is not too large relative to the minimum degree, then a sufficiently large minimum degree is sufficient to guarantee that a graph is a TDP-graph. The problem of determining the minimum number of edges necessary to add to a graph in order to guarantee it is TDP-graph was studied by Broere et al. [121] in 2004 and by Dorfling et al. [238] in 2005. Broere et al. [121] observed the following about cycles. Proposition 12.39 ([121]) A cycle 𝐶𝑛 is a TDP-graph if and only if 𝑛 ≡ 0 (mod 4). Delgado et al. [223] proved that the Cartesian product of any two isolate-free graphs is a TDP-graph. Moreover, they provided the following class of TDP-graphs. Theorem 12.40 ([223]) If 𝐺 ≠ 𝐶5 is a self-complementary graph with 𝛿(𝐺) ≥ 2, then 𝐺 is a TDP-graph. The authors of [223] also obtained several sufficient conditions to guarantee that a graph be a TDP-graph, which we summarize in the next theorem. Theorem 12.41 ([223]) If 𝐺 is a graph of order 𝑛, then any one of the following conditions is sufficient for 𝐺 to be a TDP-graph:   (a) 𝛿(𝐺) ≥ 𝑛2 + 1. (b) 𝛾t (𝐺) ≤ 𝛿(𝐺) − 1. (c) 𝛾(𝐺) ≥ 4. (d) 𝛾(𝐺) = 3 and 𝛾t (𝐺) ≠ 𝛿(𝐺). As noted in [223], there exist graphs 𝐺 for which 𝛾(𝐺) = 3 and 𝛾t (𝐺) = 𝛿(𝐺) that are not TDP-graphs. The smallest such examples are the complete graph 𝐾3 and the bowtie graph illustrated in Figure 12.7. In both cases, 𝛾(𝐺) = 3 and 𝛾t (𝐺) = 2 = 𝛿(𝐺), but 𝐺 is not a TDP-graph.

Figure 12.7 The bowtie graph

Using a link between hypergraphs and regular graphs established in [710], Henning and Yeo [488] proved that every 𝑟-regular graph, for 𝑟 ≥ 4, has two disjoint TD-sets. We note that if a graph 𝐺 has two disjoint TD-sets 𝐷 1 and 𝐷 2 , then {𝐷 1 , 𝑉 \ 𝐷 1 } is a vertex bipartition into two TD-sets. Hence, we have the following result. Theorem 12.42 ([488, 710]) Every 𝑟-regular graph, for 𝑟 ≥ 4, is a TDP-graph.

372

Chapter 12. Domination Partitions

In 1988 Alon and Bregman [18] were the first to observe that the hypergraph equivalent of Theorem 12.42 is not true when 𝑟 = 3. There are, in fact, infinitely many examples of this result being false for 𝑟 = 3 (see [229, 488]). Thus, there are infinitely many cubic graphs that are not TDP-graphs. Recall that for 𝑘 ≥ 3 an integer, a graph 𝐺 is 𝑘-chordal if it does not have an induced cycle of length more than 𝑘. Chordal graphs coincide with 3-chordal graphs. The only connected chordal cubic graph is the complete graph 𝐾4 , which trivially has a vertex bipartition into two TD-sets. In 2017 Desormeaux et al. [229] noted that examples of 6-chordal cubic graphs which are not TDP-graphs abound. The simplest example of a 6-chordal cubic graph that is not a TDP-graph is the Heawood graph 𝐺 14 shown in Figure 12.8.

Figure 12.8 The Heawood graph 𝐺 14 , which is 6-chordal, is not a TDP-graph Every 6-chordal graph is, by definition, a 𝑘-chordal graph for every 𝑘 ≥ 6. Desormeaux et al. [229] gave the following construction to build 𝑘-chordal connected cubic graphs whose vertex sets cannot be partitioned into two TD-sets. Let B𝐺 be the family of all cubic graphs 𝐺 that can be obtained from two cubic graphs 𝐺 1 and 𝐺 2 , where neither 𝐺 1 nor 𝐺 2 is a TDP-graph, as follows. Let 𝑢 ∈ 𝑉 (𝐺 1 ) and let 𝑣 ∈ 𝑉 (𝐺 2 ). Let 𝑢 1 , 𝑢 2 , and 𝑢 3 be the neighbors of 𝑢 in 𝐺 1 and let 𝑣 1 , 𝑣 2 , and 𝑣 3 be the neighbors of 𝑣 in 𝐺 2 . Let 𝐺 be obtained from the disjoint union of 𝐺 1 − 𝑢 and 𝐺 2 − 𝑣 by adding the three edges 𝑢 1 𝑣 1 , 𝑢 2 𝑣 2 , and 𝑢 3 𝑣 3 . By construction, 𝐺 is a connected cubic graph that is not a TDP-graph. For example, if we let 𝐺 1 and 𝐺 2 be two disjoint copies of the Heawood graph, then the resulting graph 𝐺 ∈ B𝐺 , shown in Figure 12.9, is a 10-chordal cubic graph that does not have vertex bipartition into two TD-sets. Using this construction, one can construct for sufficiently large 𝑘, infinitely many examples of 𝑘-chordal connected cubic graphs that are not TDP-graphs. In view of the above observations, we have the following result. Proposition 12.43 ([229]) There exist 6-chordal connected cubic graphs that are not TDP-graphs. Moreover, for sufficiently large 𝑘, there are infinitely many examples of 𝑘-chordal connected cubic graphs that are not TDP-graphs. On the other hand, Desormeaux et al. [229] showed that for 𝑘 ∈ {4, 5}, the 𝑘-chordal connected cubic graphs are TDP-graphs. Theorem 12.44 ([229]) If 𝐺 is a connected 𝑘-chordal cubic graph, for 𝑘 ∈ {4, 5}, then 𝐺 is a TDP-graph.

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Figure 12.9 A 10-chordal cubic graph that is not a TDP-graph

It was also proven in [229] that cubic graphs containing a diamond as a subgraph are TDP-graphs, where a diamond is the complete graph 𝐾4 minus an arbitrary edge. Theorem 12.45 ([229]) Every connected cubic graph containing a diamond is a TDP-graph. We conclude this section with a proof from [229] that shows that connected claw-free cubic graphs are TDP-graphs. Let 𝐵 denote the set of vertices colored black and 𝑊 the set of vertices colored white. The proof uses the three following lemmas. The first of these lemmas, due to Bollobás and Cockayne [84] in 1979, is stated in Chapter 4 and repeated here. The second lemma is due to Southey and Henning [685] in 2011 and the third is due to Henning and Löwenstein [466] in 2012. Lemma 12.46 ([84]) Every isolate-free graph 𝐺 contains a minimum dominating set 𝐷 in which epn(𝑣, 𝐷) ≠ ∅ for every vertex 𝑣 ∈ 𝐷. Lemma 12.47 ([685]) If 𝐺 is a connected non-bipartite cubic graph, then for every vertex 𝑣 ∈ 𝑉, there exists a bipartition {𝐷, 𝑇 } of 𝑉 such that 𝑇 is a TD-set and every vertex in 𝑉 \ {𝑣} has a neighbor in 𝐷. Lemma 12.48 ([466]) The vertex set of a connected claw-free cubic graph 𝐺 ≠ 𝐾4 can be uniquely partitioned into sets each of which induces a triangle or a diamond in 𝐺. By Lemma 12.48, the vertex set 𝑉 of a connected claw-free cubic graph 𝐺 ≠ 𝐾4 can be uniquely partitioned into sets each of which induces a triangle or a diamond in 𝐺. Using notation adopted from [466], we refer to such a partition as a trianglediamond partition of 𝐺, abbreviated Δ-D-partition. Every triangle and diamond induced by a set in a Δ-D-partition is called a unit of the partition. A unit that is a triangle is called a triangle-unit and a unit that is a diamond is called a diamond-unit. Two units in the Δ-D-partition are said to be adjacent if there is an edge between a vertex in one unit and a vertex in the other unit. We are now in position to present the result of Desormeaux et al. [229]. Theorem 12.49 ([229]) Every connected claw-free cubic graph is a TDP-graph. Proof If 𝐺 = 𝐾4 , then it is immediate that 𝐺 is a TDP-graph. Assume then that 𝐺 ≠ 𝐾4 . By Lemma 12.48, 𝐺 has a triangle-diamond partition. We may assume that

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every unit of 𝐺 is a triangle-unit, otherwise 𝐺 is a TDP-graph by Theorem 12.45. We proceed with the following claim. Claim 12.49.1 Two triangle-units are joined by at most one edge. Proof If two triangle-units are joined by three edges, then 𝐺 is a prism 𝐾3 □ 𝐾2 , which is a TDP-graph. Suppose two triangle-units 𝑇1 and 𝑇2 are joined by exactly two edges. Let 𝑉 (𝑇𝑖 ) = {𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 }, where 𝑎 1 𝑎 2 ∈ 𝐸 and 𝑏 1 𝑏 2 ∈ 𝐸. Thus, 𝑐 1 𝑐 2 ∉ 𝐸. Let 𝑑𝑖 be the neighbor of 𝑐 𝑖 not in 𝑇𝑖 . Note that Lemma 12.48 implies that 𝑑1 ≠ 𝑑2 . Since 𝐺 is not bipartite, by Lemma 12.47, there exists a partition {𝐷, 𝑇 } of the vertices of 𝐺, where 𝑇 is a TD-set of 𝐺 and every vertex in 𝑉 \ {𝑎 1 } has a neighbor in 𝐷. Let the vertices of 𝐷 be colored white and those in 𝑇 black. If 𝑎 1 is adjacent to a vertex in 𝐷, then the partition {𝐷, 𝑇 } is a bipartition into two TD-sets of 𝐺, and we are finished. Thus, assume that each of 𝑏 1 , 𝑐 1 , and 𝑎 2 is in 𝑇 and is colored black. If 𝑎 1 is colored black, then 𝑏 2 , 𝑐 2 , and 𝑑1 are colored white since 𝐷 totally dominates 𝑉 \ {𝑎 1 }. But then if 𝑏 1 is recolored white, {𝐵, 𝑊 } is a vertex bipartition into two TD-sets of 𝐺. Thus, we may assume that 𝑎 1 is white. If 𝑑1 is black, then recoloring 𝑏 1 white gives the desired partition {𝐵, 𝑊 } of 𝐺. Hence, we may assume 𝑑1 is white. Further, at least one of 𝑏 2 and 𝑐 2 is black since 𝑇 totally dominates 𝑎 2 . If 𝑏 2 is white, then since 𝑏 2 has a white neighbor, 𝑐 2 is white, a contradiction. Hence, 𝑏 2 is black, and 𝑐 2 and 𝑑2 are white. But then 𝑎 2 can be recolored white, and again {𝐵, 𝑊 } is a vertex bipartition into two TD-sets of 𝐺. By Theorem 12.45, every unit in 𝐺 is a triangle-unit, and by Claim 12.49.1, two triangle-units are joined by at most one edge. We now construct a graph 𝐻 from 𝐺 as follows. For each triangle-unit in 𝐺, we associate a vertex of 𝐻. If two triangle-units in 𝐺 are joined by an edge, then add an edge between the corresponding vertices in 𝐻. For each vertex 𝑣 ∈ 𝑉 (𝐻), let 𝑇𝑣 be the associated triangle-unit in 𝐺. The graph 𝐻 is called a contraction graph of 𝐺. A claw-free cubic graph 𝐺 and its contraction graph 𝐻 are shown in Figure 12.10. 𝑣

𝑢′

𝑣′

𝑢 (a) 𝐺

(b) 𝐻

Figure 12.10 A graph 𝐺 and its associated contraction graph 𝐻 We now construct a vertex bipartition {𝑋, 𝑌 }, where 𝑋 and 𝑌 are TD-sets in 𝐺, as follows. By Lemma 12.46, there exists a minimum dominating set 𝐷 in 𝐻 such that epn 𝐻 (𝑣, 𝐷) ≠ ∅ for every vertex 𝑣 ∈ 𝐷. For each vertex 𝑣 ∈ 𝐷, select an arbitrary Ð vertex 𝑣 ′ ∈ epn 𝐻 (𝑣, 𝐷) and let 𝐷 ′ = 𝑣 ∈𝐷 {𝑣 ′ }.

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Initially, let 𝑋 = 𝑌 = ∅. For each 𝑣 ∈ 𝐷, consider the chosen vertex 𝑣 ′ ∈ epn 𝐻 (𝑣, 𝐷). Since 𝑣 and 𝑣 ′ are adjacent in 𝐻, there is an edge 𝑒 𝑣 joining a vertex in the triangle-unit associated with 𝑇𝑣 in 𝐺 to a vertex in the triangle-unit associated with 𝑇𝑣 ′ . We add the ends of 𝑒 𝑣 to the set 𝑋, the two vertices in 𝑇𝑣 not incident with 𝑒 𝑣 to the set 𝑌 , and the two vertices in 𝑇𝑣 ′ not incident with 𝑒 𝑣 to 𝑌 , and the two vertices in 𝑇𝑣 ′ not incident to 𝑒 𝑣 to the set 𝑌 . For the remaining triangle-units in 𝐺 that are not associated with a vertex in 𝐷 ∪ 𝐷 ′ , do the following. Each remaining triangle-unit in 𝐺 is associated with a vertex 𝑤 ′ ∈ 𝑉 (𝐻) \ 𝐷. Since 𝐷 is a dominating set, such a vertex 𝑤 ′ is adjacent to at least one vertex in 𝐷. For each such vertex 𝑤 ′ ∈ 𝑉 (𝐻) \ 𝐷, select one vertex 𝑤 in 𝐷 that is adjacent to 𝑤 ′ in 𝐻. Since 𝑤 and 𝑤 ′ are adjacent in 𝐻, there is an edge 𝑒 𝑤 joining a vertex in the triangle-unit associated with 𝑇𝑤 in 𝐺 to a vertex in the triangle-unit associated with 𝑇𝑤 ′ . Note that the end of 𝑒 𝑤 that belongs to 𝑇𝑤 already belongs to set 𝑌 . Next add the end of 𝑒 𝑤 that belongs to 𝑇𝑤 ′ to the set 𝑌 . Further, we add the two vertices in 𝑇𝑤 ′ not incident with 𝑒 𝑤 to the set 𝑋. The resulting partition {𝑋, 𝑌 } of 𝑉 (𝐺) is a bipartition of the vertex set of 𝐺 into two TD-sets. For an illustration of the construction of such a partition, consider for example the associated graph 𝐻 of the graph 𝐺 shown in Figure 12.10. Here, 𝐷 = {𝑢, 𝑣} is a minimum dominating set of 𝐻 and 𝑣 ′ ∈ epn 𝐻 (𝑣, 𝐷) and 𝑢 ′ ∈ epn 𝐻 (𝑢, 𝐷). Perhaps the most significant result for TDP-graphs is a constructive characterization given by Henning and Peterin [473] in 2019. The authors described a procedure to build TDP-graphs in terms of a 2-coloring of the vertices that indicate the role each vertex plays in the sets associated with the two disjoint TD-sets. They proved that the resulting family of graphs, starting from four initial base graphs and applying one of seventeen operations to extend graphs in the family to larger graphs, is precisely the class of all TDP-graphs. In other words, every graph generated by this method is a TDP-graph and every TDP-graph can be constructed in this way. Although the steps of the proof of this result actually define an algorithm to solve the decision problem to decide if a graph is a TDP-graph, Henning and Peterin remarked that the algorithm is far from polynomial time complexity. We present their characterization, albeit without proof. In order to formally present the characterization given in [473], the authors constructed a graph family G such that every graph in the family has two disjoint TD-sets. For this purpose, a 2-coloring of a graph 𝐺 is defined as a partition 𝑆 = (𝑆 𝑅 , 𝑆 𝐵 ) of 𝑉 (𝐺). The color color(𝑣) of a vertex 𝑣 is the letter 𝑋 ∈ {𝑅, 𝐵} such that 𝑣 ∈ 𝑆 𝑋 , where “𝑅” and “𝐵” here stand for red and blue, respectively. Thus, the 2-coloring of 𝐺 given in [473] is a coloring of the vertices of 𝐺, one color to each vertex, using the colors red and blue. We denote by 𝑋 the letter 𝑋 ∈ {𝑅, 𝐵} \ 𝑋, and we call 𝑋 the color different from 𝑋. Thus, if 𝑋 = 𝑅, then 𝑋 = 𝐵 while if 𝑋 = 𝐵, then 𝑋 = 𝑅. We denote by (𝐺, 𝑆) a graph 𝐺 with a given 2-coloring 𝑆. The authors in [473] described a procedure to build TDP-graphs in terms of 2-colorings. For 𝑖 ∈ [4], by a 2-colored graph 𝐺 𝑖 , we shall mean the graph 𝐺 𝑖 and its associated 2-coloring shown in Figure 12.11. Further, each 2-colored 𝐺 𝑖 is called a 2-colored base graph. Let G be the minimum family of 2-colored graphs that (i) contains the four 2-colored base graphs, and

Chapter 12. Domination Partitions

376 𝑋

𝑋

𝑋

𝑋 𝑋

𝑋

𝑋

𝑋

(a) 𝐺 1

𝑋

𝑋

𝑋 (b) 𝐺 2

𝑋

𝑋

𝑋 𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

(c) 𝐺 3

𝑋

𝑋

𝑋

𝑋

𝑋

(d) 𝐺 4

Figure 12.11 The four 2-colored base graphs 𝐺 1 , 𝐺 2 , 𝐺 3 , and 𝐺 4

(ii) is closed under Operations O1 through to O17 listed below, which extend a 2-colored graph (𝐺 ′ , 𝑆 ′ ) to a new 2-colored graph (𝐺, 𝑆). Operation O1 . (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding an edge between two nonadjacent vertices of the same color. See Figure 12.12(a). Operation O2 . (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding an edge between two nonadjacent vertices of different color. See Figure 12.12(b). 𝐺′

𝑋

𝑋

𝑋

𝑋 (a) O1

𝐺′

𝑋

𝑋

𝑋

𝑋 (b) O2

Figure 12.12 Operations O1 and O2 Operation O3 . If 𝑢 and 𝑣 are vertices of different color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a new vertex of any color adjacent to both 𝑢 and 𝑣. See Figure 12.13(a). Operation O4 . If 𝑢 and 𝑣 are distinct vertices of the same color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding adjacent vertices 𝑥 and 𝑦 and edges 𝑢𝑥 and 𝑣𝑦 with color(𝑥) = color(𝑦) ≠ color(𝑢). See Figure 12.13(b). Operation O5 . If 𝑢 and 𝑣 are vertices of different color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding adjacent vertices 𝑥 and 𝑦 and edges 𝑢𝑥 and 𝑣𝑦 with color(𝑥) = color(𝑢) ≠ color(𝑦). See Figure 12.13(c). Operation O6 . If 𝑢 and 𝑣 are distinct vertices of the same color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a path 𝑥 𝑦 𝑧 with color(𝑦) = color(𝑧) ≠ color(𝑥) = color(𝑢) and adding edges 𝑢𝑥 and 𝑣𝑧. See Figure 12.13(d).

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Operation O7 . If 𝑢 and 𝑣 are distinct vertices of the same color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a path 𝑥 𝑦 𝑧 𝑤 and edges 𝑢𝑥 and 𝑣𝑤 with color(𝑥) = color(𝑤) = color(𝑢) ≠ color(𝑦) = color(𝑧). See Figure 12.13(e). Operation O8 . If 𝑢 and 𝑣 are vertices of different color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a path 𝑥 𝑦 𝑧 𝑤 and edges 𝑢𝑥 and 𝑣𝑤 with color(𝑥) = color(𝑦) = color(𝑣) ≠ color(𝑧) = color(𝑤). See Figure 12.13(f). 𝐺′

𝐺′

𝑋

𝑋

𝑋

𝑋

𝑋

𝐺′

𝑋

𝑋

𝑋

𝑋

𝑋/𝑋 𝑋 (a) O3

(b) O4

(c) O5

𝑋 𝐺′

𝑋

𝑋

𝐺′

𝑋 𝐺′

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋 𝑋

𝑋

𝑋

𝑋 𝑋

(d) O6

𝑋

(e) O7

(f) O8

Figure 12.13 Operations O3 –O8 Operation O9 . If 𝑢 and 𝑣 are adjacent vertices of different color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by subdividing 𝑢𝑣 with four consecutive vertices 𝑥, 𝑦, 𝑧, 𝑤, where 𝑥 is adjacent to 𝑢 and color(𝑢) = color(𝑧) = color(𝑤) ≠ color(𝑥) = color(𝑦). See Figure 12.14(a). Operation O10 . If 𝑢 and 𝑣 are adjacent vertices of the same color from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by subdividing 𝑢𝑣 with four consecutive vertices 𝑥, 𝑦, 𝑧, 𝑤, where 𝑥 is adjacent to 𝑢 and color(𝑢) = color(𝑥) = color(𝑤) ≠ color(𝑦) = color(𝑧). See Figure 12.14(b). Operation O11 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from ′ (𝐺 , 𝑆 ′ ) by adding an edge 𝑥𝑦 together with the edges 𝑣𝑥 and 𝑣𝑦, where color(𝑥) = color(𝑦) ≠ color(𝑣). See Figure 12.15(a). Operation O12 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from ′ (𝐺 , 𝑆 ′ ) by adding a path 𝑥 𝑦 𝑧 together with the edges 𝑣𝑥 and 𝑣𝑧, where color(𝑥) = color(𝑦) ≠ color(𝑧) = color(𝑣). See Figure 12.15(b). Operation O13 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from ′ (𝐺 , 𝑆 ′ ) by adding a path 𝑥 𝑦 𝑧 𝑤 together with the edges 𝑣𝑥 and 𝑣𝑤, where color(𝑥) = color(𝑤) = color(𝑣) ≠ color(𝑦) = color(𝑧). See Figure 12.15(c). Operation O14 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from ′ (𝐺 , 𝑆 ′ ) by adding a 3-cycle 𝑥 𝑦 𝑧 𝑥, together with the edge 𝑣𝑥, where color(𝑥) = color(𝑣) ≠ color(𝑦) = color(𝑧). See Figure 12.16(a). Operation O15 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ) of any color, then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a 4-cycle 𝑥 𝑦 𝑧 𝑤 𝑥, together with the edge 𝑣𝑥, where

Chapter 12. Domination Partitions

378 𝐺′

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

(a) O9

𝐺′

𝑋

𝑋

𝑋

𝑋 (b) O10

Figure 12.14 Operations O9 and O10 𝑋 𝐺′

𝑋

𝐺′ 𝑋

𝑋

𝑋

𝑋

𝑋

𝑋

𝐺′ 𝑋

𝑋

𝑋 𝑋 (a) O11

(b) O12

(c) O13

Figure 12.15 Operations O11 , O12 , and O13

color(𝑥) = color(𝑦) ≠ color(𝑧) = color(𝑤). See Figure 12.16(b), where the notation 𝑋/𝑋 means that the vertex can have any color. Operation O16 . If 𝑣 is a vertex from (𝐺 ′ , 𝑆 ′ ), then (𝐺, 𝑆) is obtained from (𝐺 ′ , 𝑆 ′ ) by adding a 5-cycle 𝑥 𝑦 𝑧 𝑤 𝑡 𝑥, together with the edge 𝑣𝑥, where color(𝑥) = color(𝑦) = color(𝑡) ≠ color(𝑧) = color(𝑤) = color(𝑣). See Figure 12.16(c). Operation O17 . If 𝑢 is a cut-vertex in (𝐺 ′ , 𝑆 ′ ) with associated subgraphs 𝐻1𝑢 and 𝐻2𝑢 , and in N 𝐻1𝑢 (𝑢 ′ ) there exists a vertex of the same color as 𝑢 and in N 𝐻2𝑢 (𝑢 ′′ ) there exists a vertex of different color as 𝑢, then (𝐺, 𝑆) is obtained from 𝐻1𝑢 and 𝐻2𝑢 by adding a new vertex 𝑣 and the edges 𝑢 ′ 𝑣 and 𝑣𝑢 ′′ . The color of all vertices from 𝐻1𝑢 remains the same as in 𝐺 ′ , color(𝑣) = color(𝑢 ′′ ) ≠ color(𝑢 ′ ) = color(𝑢) and the color of all vertices from 𝐻2𝑢 is exchanged with respect to their color in 𝐺 ′ . See Figure 12.17, where the notation 𝐴 means that the color of all vertices from the set 𝐴 in (𝐺, 𝑆) is changed in (𝐺 ′ , 𝑆 ′ ). We are now in a position to present the constructive characterization of the graphs that have two disjoint TD-sets. Theorem 12.50 ([473]) A graph 𝐺 is a TDP-graph if and only if every component of (𝐺, 𝑆) belongs to the family G for some 2-coloring 𝑆. Further, if (𝐺, 𝑆) ∈ G, then 𝑆 = (𝑆 𝑅 , 𝑆 𝐵 ) is a partition of 𝑉 (𝐺) into two TD-sets of 𝐺.

Section 12.6. Dominating Bipartitions of Graphs

379

𝑋

𝑋

𝐺′

𝐺′ 𝑋

𝑋

𝑋

𝑋

𝑋/𝑋 𝑋

𝑋

(a) O14

(b) O15

𝑋

𝑋

𝑋

𝑋

𝐺′ 𝑋 𝑋

(c) O16

Figure 12.16 Operations O14 –O16 𝐺′ 𝐴

𝐴 𝑢 𝑋

𝑋

𝑋

𝑋

𝑢′

𝑣

𝑢 ′′

𝑋

𝑋

𝑋

𝑋

Figure 12.17 Operation O17

Thus, even though the characterization in Theorem 12.50 solves the long-standing problem of determining whether the vertex set of a graph can be partitioned into two TD-sets, it remains a challenging problem to determine in polynomial time if a given graph is a TDP-graph, even for some special graph classes.

Chapter 13

Domination Critical and Stable Graphs 13.1 Introduction It is often important to know the behavior of a graph parameter when changes are made to the graph. The effects of modifying a graph by removing or adding an edge, or by removing a vertex are of particular interest in many applications of graph theory. In network design, for example, fault tolerance is the ability of a network to continue functioning in the presence of faults, such as a link failure (removing an edge) or a node failure (removing a vertex). In other cases, network performance can be enhanced by adding certain links (edges). For instance, a file server is a computer node responsible for the central storage and management of data files so that other nodes can access these files. In a graph 𝐺 modeling a computer network, a dominating set of 𝐺 represents a set of file servers having the ability to communicate directly over one link to any other node in the network. The minimum number of file servers needed is the domination number of 𝐺. If it is essential that the number of file servers does not have to increase in the event of a node (or link) failure, then the fault-tolerance criterion is the domination number of 𝐺. In other words, a network that is fault-tolerant to node failure with respect to the domination number has the following properties: if a user’s node fails, that user can go to another node to continue working on the same files; if a file server fails, then it can be replaced by another node while maintaining the same minimum number of file servers in the network. Under other conditions, adding an edge might provide a solution if the desire is to decrease the minimum number of file servers needed to service the entire network. Studying a graph parameter’s behavior, either changing or remaining the same, upon some modification to the graph was first proposed by Harary [382] in 1982. In this chapter, we explore this concept for domination and total domination. We first consider the effects of three graph modifications on the domination number of a graph. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_13

381

382

Chapter 13. Domination Critical and Stable Graphs

Recall that for a graph 𝐺, 𝐺 − 𝑣 (respectively, 𝐺 − 𝑒) denotes the graph formed by removing vertex 𝑣 (respectively, edge 𝑒) from 𝐺. Further, 𝐺 + 𝑒 denotes the addition of an edge 𝑒 ∈ 𝐸 ( 𝐺) in the complement 𝐺 to 𝐺. We note that removing an edge cannot decrease the domination number, that is, removing an edge will either increase the domination number or leave it the same. Consider, for example, the path 𝐺 = 𝑃5 given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 . In this example, 𝛾(𝐺) = 2, while 𝛾(𝐺 − 𝑣 1 𝑣 2 ) = 3 > 𝛾(𝐺) and 𝛾(𝐺 − 𝑣 2 𝑣 3 ) = 2 = 𝛾(𝐺). Similarly, adding an edge cannot increase the domination number and will either decrease it or leave it the same. For example, consider the path 𝐺 = 𝑃4 given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 . In this example, 𝛾(𝐺) = 2, while 𝛾(𝐺 + 𝑣 1 𝑣 3 ) = 1 < 𝛾(𝐺) and 𝛾(𝐺 + 𝑣 1 𝑣 4 ) = 2 = 𝛾(𝐺). However, removing a vertex can affect the domination number in one of three ways, namely, increase it, decrease it, or leave it unchanged. In our 𝐺 = 𝑃4 example, we have 𝛾(𝐺 − 𝑣 1 ) = 1 < 𝛾(𝐺) and 𝛾(𝐺 − 𝑣 2 ) = 2 = 𝛾(𝐺). On the other hand, for a star 𝐺 = 𝐾1,𝑘 with 𝑘 ≥ 2 and central vertex 𝑥 (of degree 𝑘), we have 𝛾(𝐺 − 𝑥) = 𝑘 > 𝛾(𝐺) = 1. It is useful to form a weak partition of the vertices of 𝐺 into three sets according to how their removal affects 𝛾(𝐺). Recall that a weak partition of a set 𝑋 is a collection of vertex-disjoint subsets of 𝑋 whose union is 𝑋. We note that, in contrast to a partition, elements of a weak partition are allowed to be empty. Let {𝑉 0 , 𝑉 + , 𝑉 − } be a weak partition of 𝑉 (𝐺), where  𝑉 0 = 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) = 𝛾(𝐺)  𝑉 + = 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) > 𝛾(𝐺)  𝑉 − = 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) < 𝛾(𝐺) . Similarly, we form a weak partition {𝐸 0 , 𝐸 + } of 𝐸 (𝐺), where  𝐸 0 = 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 − 𝑢𝑣) = 𝛾(𝐺)  𝐸 + = 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 − 𝑢𝑣) > 𝛾(𝐺) . Finally, let {𝐸 0 , 𝐸 − } be a weak partition of the edge set of 𝐸 (𝐺), where  𝐸 0 = 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 + 𝑢𝑣) = 𝛾(𝐺)  𝐸 − = 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 + 𝑢𝑣) < 𝛾(𝐺) . An edge in a graph 𝐺 is called domination critical, or simply critical, if its removal from 𝐺 increases the domination number of 𝐺. Similarly, a vertex is called domination critical if its removal changes, either increases or decreases, the domination number. An added edge is also called critical if its addition to 𝐺 decreases the domination number of 𝐺. If the domination number is the same in the graph resulting from the modification, then the edge (respectively, vertex) is said to be domination stable, or simply stable. For more details, see [61, 138, 417, 661]. It is possible for a single graph 𝐺 to have all possible combinations of critical and stable edges and vertices. That is, there exists a partition {𝑉 0 , 𝑉 + , 𝑉 − } of 𝑉 (𝐺), a partition {𝐸 0 , 𝐸 + } of 𝐸 (𝐺), and a partition {𝐸 0 , 𝐸 + } of 𝐸 (𝐺). For example, consider the double star 𝑇𝑘 = 𝑆(1, 𝑘) for 𝑘 ≥ 2 illustrated in Figure 13.1. The set {𝑢, 𝑣} is

Section 13.2. The Six Graph Families

383

a 𝛾-set of 𝑇𝑘 and so 𝛾(𝑇𝑘 ) = 2. We note that 𝛾(𝑇𝑘 − 𝑢) = 𝛾(𝑇𝑘 − 𝑣 𝑖 ) = 2 = 𝛾(𝑇𝑘 ) for 𝑖 ∈ [𝑘]. Moreover, 𝛾(𝑇𝑘 − 𝑣) = 𝑘 + 1 > 𝛾(𝑇𝑘 ) and 𝛾(𝑇𝑘 − 𝑢 1 ) = 1 < 𝛾(𝑇𝑘 ), and so 𝑇𝑘 has all possible types of vertices. To see that 𝑇𝑘 has all possible types of edges, note that 𝛾(𝑇𝑘 − 𝑢𝑣) = 𝛾(𝑇𝑘 − 𝑢𝑢 1 ) = 2 = 𝛾(𝑇𝑘 ), while 𝛾(𝑇𝑘 − 𝑣𝑣 𝑖 ) = 3 > 𝛾(𝑇𝑘 ) for 𝑖 ∈ [𝑘]. Furthermore, 𝛾(𝑇𝑘 + 𝑣𝑢 1 ) = 1 < 𝛾(𝑇𝑘 ), while 𝛾(𝑇𝑘 + 𝑢𝑣 𝑖 ) = 𝛾(𝑇𝑘 + 𝑢 1 𝑣 𝑖 ) = 𝛾(𝑇𝑘 + 𝑣 𝑖 𝑣 𝑗 ) = 2 = 𝛾(𝑇𝑘 ) for 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗. Thus, 𝑉 0 = 𝑢, 𝑣 𝑖 : 𝑖 ∈ [𝑘] ,   𝑉 + = {𝑣}, 𝑉 − = {𝑢 1 }, 𝐸 0 = {𝑢𝑣, 𝑢𝑢 1 }, 𝐸 + = 𝑣𝑣 𝑖 : 𝑖 ∈ [𝑘] , 𝐸 0 = 𝑢𝑣 𝑖 , 𝑢 1 𝑣 𝑖 , 𝑣 𝑖 𝑣 𝑗 : 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗 , and 𝐸 − = {𝑣𝑢 1 }.
푣


푢

...
푢 1


푣 1


푣 2


푣 𝑘

Figure 13.1 A double star 𝑆(1, 𝑘)

Of particular interest are the graphs for which every edge (respectively, vertex) is critical or every edge (respectively, vertex) is stable under a given graph modification. Hence, the three graph modifications of vertex removal, edge removal, and edge addition give rise to three families of critical graphs and three families of stable graphs with respect to a given graph parameter. As we shall see in Section 13.2, if every vertex in a graph 𝐺 is critical, then the removal of any arbitrary vertex of 𝐺 decreases the domination number. In other words, a graph 𝐺 for which the removal of any vertex changes the domination number has no vertex in 𝑉 0 ∪ 𝑉 + . A graph is domination vertex (edge) critical if the domination number decreases upon the removal (respectively, addition) of any arbitrary vertex of 𝑉 (𝐺) (respectively, edge of 𝐸 (𝐺)). These two families are the most studied of the six families and will be the main focus of this chapter. We begin by discussing all six families and relationships among them in Section 13.2. Domination vertex-critical graphs are covered in Section 13.3, and domination edge-critical graphs in Section 13.4. In Section 13.5, we present total domination edge-critical graphs. Although in this chapter we consider only the three graph modifications of removing a vertex, removing an edge, and adding an edge, we note that the effects of other types of graph modifications on the domination number have been studied. For a sampling of such studies as well as related topics including criticality of independent and total domination numbers, see [33, 65, 120, 126, 127, 155, 163, 175, 225, 226, 231, 232, 250, 277, 298, 319, 320, 322, 394, 403, 410, 412, 418, 419, 461, 508, 534, 707, 708, 727].

13.2

The Six Graph Families

In this section, we consider the graphs for which every edge (respectively, vertex) is critical or every edge (respectively, vertex) is stable under a given graph modification.

384

Chapter 13. Domination Critical and Stable Graphs

As mentioned in the introduction, this defines the following six classes of graphs with respect to domination. We use acronyms to denote the following classes of graphs, where C represents changing; U unchanging; V vertex; E edge; R removal; A addition: (a) (CVR) 𝛾(𝐺 − 𝑣) ≠ 𝛾(𝐺) for all 𝑣 ∈ 𝑉 (𝐺). (b) (CER) 𝛾(𝐺 − 𝑒) ≠ 𝛾(𝐺) for all 𝑒 ∈ 𝐸 (𝐺). (c) (CEA) 𝛾(𝐺 + 𝑒) ≠ 𝛾(𝐺) for all 𝑒 ∈ 𝐸 ( 𝐺). (d) (UVR) 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) for all 𝑣 ∈ 𝑉 (𝐺). (e) (UER) 𝛾(𝐺 − 𝑒) = 𝛾(𝐺) for all 𝑒 ∈ 𝐸 (𝐺). (f) (UEA) 𝛾(𝐺 + 𝑒) = 𝛾(𝐺) for all 𝑒 ∈ 𝐸 (𝐺). These six graph families for domination were first suggested in 1991 by Carrington et al. in [138] and have been studied individually in the literature with varying terminology. In this section, we present properties of these graph families and relationships among the families. Recall that a star is a tree with at most one vertex of degree 2 or more and a galaxy is a union of stars. Note that this definition includes the trivial graph 𝐾1 in the family of stars.

13.2.1

CVR, CER, and CEA

By definition, a graph 𝐺 for which the domination number changes when an arbitrary vertex is removed, that is, a graph in CVR, has 𝑉 (𝐺) = 𝑉 − ∪ 𝑉 + . Carrington et al. [138] showed that for any graph 𝐺 in CVR, 𝛾(𝐺 − 𝑣) < 𝛾(𝐺) for all 𝑣 ∈ 𝑉, that is, for such a graph 𝐺, we have 𝑉 = 𝑉 − and 𝑉 + = ∅. Furthermore, in 1992 Sampathkumar and Neeralagi [662] gave a descriptive characterization of the vertices in 𝑉 − . Theorem 13.1 ([662]) A vertex 𝑣 is in 𝑉 − if and only if pn[𝑣, 𝑆] = {𝑣} for some 𝛾-set 𝑆 containing 𝑣. It follows from Theorem 13.1 that removing a vertex can decrease the domination number by at most one. Combining this with the fact from [138] that 𝑉 (𝐺) = 𝑉 − for any graph in CVR, it follows that if 𝛾(𝐺 − 𝑣) < 𝛾(𝐺) for all 𝑣 ∈ 𝑉, then 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1 for all 𝑣 ∈ 𝑉. Thus, the graphs in CVR are precisely the domination vertex-critical graphs introduced in 1988 by Brigham et al. [118]. These graphs will be covered in more detail in Section 13.3. As previously mentioned, deleting (respectively, adding) an edge cannot decrease (respectively, increase) the domination number. Further, we note that deleting an edge can increase the domination number by at most one, while adding an edge can decrease it by at most one. Hence, a graph in family CEA, that is, a graph for which the domination number changes when an arbitrary edge is added, has the property that 𝛾(𝐺 + 𝑒) = 𝛾(𝐺) − 1 for all 𝑒 ∈ 𝐸 (𝐺). Thus, the graphs in CEA are the domination edge-critical graphs introduced in 1983 by Sumner and Blitch [694]. These graphs will be covered in Section 13.4. Problems associated with the classes CVR and CEA seem to be inherently complex, so it is not surprising that neither class has been characterized for general graphs. In contrast, a straightforward characterization of the graphs in CER was

Section 13.2. The Six Graph Families

385

obtained by Walikar and Acharya [740] in 1979. We note that this result was independently obtained by Bauer et al. [61] in 1983. Theorem 13.2 ([740]) A graph 𝐺 ∈ CER if and only if 𝐺 is a galaxy. Proof The sufficiency is clear. Let 𝐺 ∈ CER and 𝑆 be a 𝛾-set of 𝐺. Hence, the removal of any edge increases the domination number. There can be no edge 𝑒 between two vertices in 𝑉 \ 𝑆 or between two vertices in 𝑆 since 𝑆 is a dominating set of 𝐺 − 𝑒. Thus, 𝑆 and 𝑉 \ 𝑆 are independent sets in 𝐺. Then N(𝑣) ⊆ 𝑆 for all 𝑣 ∈ 𝑉 \ 𝑆. Since 𝑆 is a dominating set, every vertex in 𝑉 \ 𝑆 has at least one neighbor in 𝑆. If 𝑣 ∈ 𝑉 \ 𝑆 has degree at least 2, then removing one of the edges incident to 𝑣 does not increase the domination number, a contradiction. Hence, every vertex in 𝑉 \ 𝑆 has exactly one neighbor in 𝑆 and has degree 1 in 𝐺, implying that 𝐺 is a galaxy.

13.2.2

UVR, UER, and UEA

Although no constructive characterizations of the graphs in the classes UVR, UER, and UEA have been obtained, descriptive characterizations have been given, which we present here. Suppose that the domination number of a graph 𝐺 is unchanged when an arbitrary vertex is removed, that is, suppose that 𝐺 ∈ UVR. In this case, 𝑉 = 𝑉 0 . For example, complete bipartite graphs 𝐾𝑟 ,𝑠 are in UVR when 3 ≤ 𝑟 ≤ 𝑠. In 1991 Carrington et al. [138] observed the following about the family UVR. Theorem 13.3 ([138]) A graph 𝐺 is in UVR if and only if 𝐺 is an isolate-free graph and for each vertex 𝑣 of 𝐺, either (a) or (b) holds: (a) There is a 𝛾-set of 𝐺 that does not contain 𝑣 and for each 𝛾-set 𝑆 of 𝐺 such that 𝑣 ∈ 𝑆, epn[𝑣, 𝑆] ≠ ∅. (b) The vertex 𝑣 is in every 𝛾-set of 𝐺 and there is a subset of 𝛾(𝐺) vertices in 𝑉 \ N[𝑣] that dominates 𝐺 − 𝑣. In 1979 Walikar and Acharya [740] observed the following about graphs in UER. Theorem 13.4 ([740]) A graph 𝐺 is in UER if and only if for each edge 𝑢𝑣 ∈ 𝐸, there exists a 𝛾-set 𝑆 of 𝐺 such that one of the following conditions holds: (a) 𝑢, 𝑣 ∈ 𝑆, (b) 𝑢, 𝑣 ∈ 𝑉 \ 𝑆, (c) 𝑢 ∈ 𝑆 and 𝑣 ∈ 𝑉 \ 𝑆 implies |N(𝑣) ∩ 𝑆| ≥ 2. The complete graphs 𝐾𝑛 for any 𝑛 ≥ 3 are in UER. We note that all cycles are in UER, as removing an edge from a cycle 𝐶𝑛 results in a path 𝑃𝑛 and 𝛾(𝐶𝑛 ) = 𝛾(𝑃𝑛 ). For another example of graphs in UER, consider the graph 𝐺 formed from the union of two complete graphs 𝐾𝑠 ∪ 𝐾𝑡 , where 2 ≤ 𝑠 ≤ 𝑡, by adding exactly one edge between a vertex in 𝐾𝑠 and a vertex in 𝐾𝑡 . The resulting graph 𝐺 has 𝛾(𝐺) = 2 and 𝛾(𝐺 − 𝑢𝑣) = 2 for any edge 𝑢𝑣. In 1992 Hartnell and Rall [393] gave a constructive characterization for all trees in UER, which we shall present in Section 13.2.3. In 1988 Dutton and Brigham [251] investigated the family UER, which they called

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𝛾-insensitive graphs, and determined the minimum number of edges in some of these graphs. Theorem 13.5 ([251]) If 𝐺 ∈ UER is a graph of order 𝑛 ≥ 3𝛾(𝐺) and size 𝑚 with 𝛾(𝐺) ≥ 2, then 𝑚 ≥ 2𝑛 − 3𝛾(𝐺). Cycles 𝐶𝑛 , where 𝑛 ≡ 0, 2 (mod 3), are examples of graphs for which the domination number is unchanged when an arbitrary edge is added. The UEA graphs were characterized in terms of their vertex sets by Carrington et al. [138]. Theorem 13.6 ([138]) A graph 𝐺 is in UEA if and only if 𝑉 − is empty. Proof Let 𝐺 ∈ UEA and suppose that 𝐺 has a vertex 𝑥 ∈ 𝑉 − . Let 𝑆 be a 𝛾-set of 𝐺 − 𝑥. Since 𝑥 ∈ 𝑉 − , we have |𝑆| = 𝛾(𝐺 − 𝑥) < 𝛾(𝐺). But then 𝑆 is a dominating set for 𝐺 + 𝑥𝑦, where 𝑦 is an arbitrary vertex of 𝑆. Thus, 𝛾(𝐺 + 𝑥𝑦) ≤ |𝑆| < 𝛾(𝐺), contrary to the fact that 𝐺 ∈ UEA. To prove the converse, suppose that 𝐺 has no vertices in 𝑉 − and that 𝛾(𝐺 + 𝑢𝑣) = 𝛾(𝐺) − 1 for some pair of nonadjacent vertices 𝑢 and 𝑣 in 𝐺. In this case, every 𝛾-set 𝑆 of 𝐺 + 𝑢𝑣 must contain exactly one of 𝑢 or 𝑣, say 𝑢. Hence, 𝑆 dominates 𝐺 − 𝑣, implying that 𝛾(𝐺 − 𝑣) ≤ |𝑆| = 𝛾(𝐺) − 1. Thus, 𝑣 ∈ 𝑉 − , a contradiction.

13.2.3

Relationships Among the Families

There are many interesting relationships among these six classes. For instance, the characterization of the UEA graphs given in Theorem 13.6 relates them to the graphs in CVR. In 2003 Haynes and Henning [422] established the relationships among the classes of graph given by the Venn diagram in Figure 13.2. In order to obtain the Venn diagram, they prohibited graphs that were in a class vacuously. For example, complete graphs 𝐾𝑛 are both in UEA and in CEA (vacuously) and so for the purpose of developing a Venn diagram, this case is not allowed. Hence, we assume that the complete graph 𝐾𝑛 , for 𝑛 ≥ 3, is in UER, but not in UEA ∪ CEA. Let G be the universal set of all graphs and let R be the set of all graphs in at least one of the graph six graph families, that is R = CEA ∪ CER ∪ CVR ∪ UEA ∪ UER ∪ UVR. We label the regions of the Venn diagram of Figure 13.2 as illustrated in Figure 13.3 and explained in Table 13.1. The following observation from [422] shows that no region of the Venn diagram in Figure 13.3 is empty, that is, none of the Regions R1–R14 defined in Table 13.1 are empty. To attach a pendant edge to a vertex 𝑣 in a graph 𝐺 means adding to the graph 𝐺 a new vertex 𝑢 and new edge 𝑢𝑣. Observation 13.7 ([422]) The following are examples of graphs in each of the regions R1–R14: • R1. Graph obtained from a complete graph 𝐾𝑛 , where 𝑛 ≥ 3 and 𝑣 is a vertex of 𝐾𝑛 , by attaching at least one pendant edge to 𝑣,

Section 13.2. The Six Graph Families

387 UEA

UER

UVR

CER

CVR CEA

Figure 13.2 Relationships among the six families Region

Equals

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14

UEA \ (UVR ∪ CER ∪ UER) UVR \ (UER ∪ CER) (UER ∩ UEA) \ UVR UER ∩ UEA ∩ UVR CER ∩ UEA ∩ UVR (CER ∩ UEA) \ UVR G\R UER \ (UEA ∪ CEA ∪ CVR) CER \ (UEA ∪ CEA) CVR \ CEA CVR ∩ CEA (UER ∩ CEA) \ CVR CEA \ (UER ∪ CER) CER ∩ CEA

Table 13.1 Relationships among the six families • R2. Graph obtained from the complete bipartite graph 𝐾2,𝑠 , for 𝑠 ≥ 3, by attaching a pendant edge to each vertex of degree 2, • R3. Graph obtained from the complete graph 𝐾4 by adding a new vertex 𝑣, joining it to three vertices of the 𝐾4 , and then subdividing each of the edges incident to 𝑣, • R4. Cycles 𝐶𝑛 , where 𝑛 ≡ 0, 2 (mod 3), • R5. Graph 𝑚𝐾2 for 𝑚 ≥ 2, • R6. Isolate-free galaxy with at least one star of order 3 or more, • R7. Double star 𝑆(1, 𝑘) shown in Figure 13.1, • R8. Graph obtained from the diamond 𝐾4 − 𝑢𝑣 by adding a pendant edge to 𝑣, • R9. Galaxy having at least one isolated vertex and at least two edges, • R10. Cycles 𝐶𝑛 , where 𝑛 ≡ 1 (mod 3) and 𝑛 ≥ 7,

Chapter 13. Domination Critical and Stable Graphs

388

UEA R1 UER R3 CVR R10

R4

R8

UVR R2

R7

CER R6

R5

R9

CEA R11

R12

R13

R14

Figure 13.3 Labeled regions • • • •

R11. Cycle 𝐶4 , R12. Corona 𝐾𝑛 ◦ 𝐾1 , for 𝑛 ≥ 3, R13. Graph 𝐶4 ∪ 𝐾2 , R14. Graphs of order 𝑛 ≥ 3 and size 𝑚 = 1.

It follows from Observation 13.7 that none of the regions of the Venn diagram in Figure 13.3 are empty. However, Haynes and Henning [422] showed that the regions R5, R9, R13, and R14 are empty for connected graphs, giving the Venn diagram in Figure 13.4. Furthermore, they showed that the regions R3, R4, R10, R11, and R12 shown in Figure 13.4 are empty for trees, reducing that Venn diagram to five regions, as shown in Figure 13.5. We note also that no region of this Venn diagram is empty. For example, region R6 consists of stars of order 𝑛 ≥ 3. The double star 𝑆(1, 𝑘) shown in Figure 13.1 is in R7, the double star 𝑆(𝑟, 𝑠), for 3 ≤ 𝑟 ≤ 𝑠, is in R1, the subdivided star 𝑆(𝐾1,𝑘 ) (obtained from the star 𝐾1,𝑘 by subdividing every edge exactly once), for 𝑘 ≥ 2, is in R2, and the tree 𝐹5 shown in Figure 13.6 is in R8. We next present characterizations given in [422] of the trees in the five regions in Figure 13.5. Recall that complete graphs are not considered as they would belong to several regions vacuously. Hence, the trees in this section have order at least 3. The first observation characterizes the trees in region R6. Proposition 13.8 ([422]) A tree 𝑇 is in R6 if and only if 𝑇 is a star of order 𝑛 ≥ 3. To state a characterization of the trees in R1, let H be the family of trees defined as follows. A tree 𝑇 is in H if 𝑇 can be obtained from a sequence of trees 𝑇1 , 𝑇2 , . . . , 𝑇 𝑗 , for 𝑗 ≥ 2, such that 𝑇1 is the star 𝐾1,𝑘 , where 𝑘 ≥ 2, 𝑇 = 𝑇 𝑗 , and for every 𝑖 ∈ [ 𝑗 − 1], 𝑇𝑖+1 can be obtained from 𝑇𝑖 by one of the following operations: Operation O1 . Add a star 𝑇𝑤 with center vertex 𝑤 and order at least 3, and add the edge 𝑤𝑦 where 𝑦 ∈ 𝑉 + (𝑇𝑖 ). Operation O2 . Add a nontrivial rooted tree 𝑇𝑤 with root 𝑤 in which every leaf of 𝑇𝑤 , except possibly 𝑤, is at distance 2 from 𝑤, and add the edge 𝑤𝑦, where 𝑦 ∈ 𝑉 (𝑇𝑖 ).

Section 13.2. The Six Graph Families

389 UEA R1

UER R3

R4

CVR R10

UVR R2

CER

∅

R7

R8

R6

∅

CEA R11

R12

∅

∅

Figure 13.4 Relationships for connected graphs UEA R1 UER ∅

∅

CVR ∅

R8

UVR R2

CER R6

∅

R7

∅

CEA ∅

∅

∅

∅

Figure 13.5 Relationships for trees

Theorem 13.9 ([422]) A tree 𝑇 of order 𝑛 ≥ 3 is in region R1 if and only if 𝑇 ∈ H . The family of UER trees was characterized in 1992 by Hartnell and Rall [393] as follows. Let 𝐹𝑚 , for 𝑚 ≥ 1, be a tree that can be obtained from 𝑚 ≥ 1 disjoint copies of 𝑃4 by adding a new vertex 𝑤 and 𝑚 new edges joining 𝑣 to a support vertex in each of the 𝑚 paths. See Figure 13.6 for the tree 𝐹5 . Define a family F of trees to consist of all trees 𝑇 where 𝑇 ∈ {𝐾1 } ∪ {𝐹𝑚 : 𝑚 ≥ 2} or 𝑇 can be obtained from a sequence of trees 𝑇1 , 𝑇2 , . . . , 𝑇 𝑗 , for 𝑗 ≥ 1, such that 𝑇1 is the path 𝑃4 , 𝑇 = 𝑇 𝑗 , and if 𝑗 ≥ 2, then for every 𝑖 ∈ [ 𝑗 − 1], 𝑇𝑖+1 can be obtained from 𝑇𝑖 by one of the following operations: Operation O1 . Add a path 𝑃2 and the edge 𝑤𝑦, where 𝑦 ∈ 𝑉 0 (𝑇𝑖 ) and 𝑦 belongs to at least one 𝛾-set of 𝑇𝑖 , and 𝑤 is a vertex of the 𝑃2 . Operation O2 . Add a path 𝑃3 and the edge 𝑤𝑦, where 𝑦 ∈ 𝑉 − (𝑇𝑖 ) and 𝑤 is a leaf of the added 𝑃3 .

Chapter 13. Domination Critical and Stable Graphs

390

Operation O3 . Add the tree 𝐹1 and the edge 𝑤𝑦, where 𝑦 belongs to at least one 𝛾-set of 𝑇𝑖 and 𝑤 is the labeled vertex in 𝐹1 . Operation O4 . Add the tree 𝐹𝑚 , for some 𝑚 ≥ 2, and the edge 𝑤𝑦, where 𝑦 ∈ 𝑉 (𝑇𝑖 ) and 𝑤 is the labeled vertex in 𝐹𝑚 . Theorem 13.10 ([393]) A tree 𝑇 is in UER if and only if 𝑇 ∈ F .

Figure 13.6 The tree 𝐹5 in R8 is in UER, but not in UEA, CEA, or CVR

Thus, Theorem 13.10 characterizes the trees in R8 of Figure 13.5. Corollary 13.11 ([393]) A tree 𝑇 is in region R8 if and only if 𝑇 ∈ F \ {𝐾1 }.

(a)

(b)

Figure 13.7 Examples of trees in R2 Recall that region R2 is the set UVR \ (UER ∪ CER). We note that UER ∩ UVR and CER ∩ UVR are empty for trees. The two trees shown in Figure 13.7 are examples of trees in R2. A constructive characterization of trees in region R2 is given in [422]. Thus, R7 is the class of all trees of order 𝑛 ≥ 3 not in one of the regions R1, R2, R6, and R8. We conclude this section with the characterization of the class of trees R2. We first state some necessary lemmas without proof. Lemma 13.12 ([422]) If a tree 𝑇 of order 𝑛 ≥ 3 is in region R2, then the following hold: (a) Every support vertex of 𝑇 is adjacent to exactly one leaf. (b) No support vertex of 𝑇 is in every 𝛾-set of 𝑇. (c) Every leaf in 𝑇 belongs to some 𝛾-set of 𝑇. (d) The set of support vertices in 𝑇 is an independent set.

Section 13.2. The Six Graph Families

391

To state the characterization, let T be the family of trees 𝑇 that can be obtained from a sequence of trees 𝑇1 , 𝑇2 , . . . , 𝑇 𝑗 , for 𝑗 ≥ 1, such that 𝑇1 is a subdivided star 𝑆(𝐾1,𝑘 ) for some 𝑘 ≥ 2, 𝑇 = 𝑇 𝑗 , and if 𝑗 ≥ 2, then for every 𝑖 ∈ [ 𝑗 − 1], 𝑇𝑖+1 can be obtained from 𝑇𝑖 by applying one of two operations defined below. Let 𝑊1 be the set containing the central vertex of 𝑇1 and 𝑆1 = 𝑉 (𝑇1 ) \ 𝑊1 . The set 𝑊𝑖 as constructed below contains the vertices of 𝑇𝑖 that are in no 𝛾-set of 𝑇𝑖 and 𝑆𝑖 = 𝑉 (𝑇𝑖 ) \ 𝑊𝑖 is the set of vertices that are in some 𝛾-set of 𝑇𝑖 . Let 𝑇𝑖+1 be obtained from 𝑇𝑖 by applying Operation T1 or Operation T2 : Operation T1 . Add a subdivided star 𝑆(𝐾1,𝑘 ), for 𝑘 ≥ 2, and the edge 𝑤𝑦, where 𝑤 is the center vertex of the subdivided star and 𝑦 is an arbitrary vertex in 𝑇𝑖 . Let 𝑊𝑖+1 = 𝑊𝑖 ∪ {𝑤}. Operation T2 . Add a path 𝑃3 and the edge 𝑤𝑦, where 𝑤 is an endvertex of the path 𝑃3 and 𝑦 ∈ 𝑆𝑖 . Let 𝑊𝑖+1 = 𝑊𝑖 ∪ {𝑤}. We note that if 𝑇 = 𝑇 𝑗 is rooted at vertex 𝑦, then 𝑇 𝑗 −1 = 𝑇 − 𝑇𝑤 . Before presenting a characterization of the trees in region R2, we need two lemmas from [422]. We use the terminology defined for the construction of the family T in the proofs. Lemma 13.13 ([422]) Let 𝑇 be a tree of order 𝑛 ≥ 3 with diam(𝑇) ≤ 5. Then 𝑇 is in R2 if and only if 𝑇 = 𝑇1 = 𝑆(𝐾1,𝑘 ), for 𝑘 ≥ 2, or 𝑇 can be formed from 𝑇1 by exactly one application of Operation T1 , and so 𝑇 ∈ T . By Lemma 13.13, a tree 𝑇 of order 𝑛 ≥ 3 with diameter at most 5 is in R2 if and only if 𝑇 can be obtained from two subdivided stars, each having order at least 5, by adding an edge between their centers. Lemma 13.14 ([422]) Let 𝑇 be a tree in T such that 𝑇 = 𝑇 𝑗 and 𝑇 is obtained from 𝑇 ′ = 𝑇 𝑗 −1 by applying Operation T1 or Operation T2 to attach a tree 𝑇𝑤 . Then 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 𝛾(𝑇𝑤 ) and every vertex in 𝑆 𝑗 is in some 𝛾-set of 𝑇. We are now in a position to present a characterization of the trees in region R2. Theorem 13.15 ([422]) A tree 𝑇 of order 𝑛 ≥ 3 is in R2 if and only if 𝑇 ∈ T . Proof Lemma 13.13 establishes the theorem if diam(𝑇) ≤ 5. From our previous discussion, we know that 𝑃𝑛 ∈ R2 if and only if 𝑛 ≥ 3 and 𝑛 ≡ 2 (mod 3). Since the paths 𝑃𝑛 for 𝑛 ≡ 2 (mod 3) can be obtained from the path 𝑃5 = 𝑆(𝐾1,2 ) by repeated applications of Operation T2 , the theorem holds for paths. Let 𝑇 be a tree of order 𝑛 in T . We proceed by induction on 𝑛. If diam(𝑇) ≤ 5 or 𝑇 is a path, then the result holds, establishing the base cases. Hence, we may assume that diam(𝑇) ≥ 6 and Δ(𝑇) ≥ 3, implying that 𝑛 ≥ 8. Assume that for all trees 𝑇 ′ with fewer than 𝑛 vertices, 𝑇 ′ ∈ R2 if and only if 𝑇 ′ ∈ T . Thus, 𝑇 = 𝑇 𝑗 and 𝑇 𝑗 is constructed from a subdivided star 𝑇1 = 𝑆(𝐾1,𝑘 ), for 𝑘 ≥ 2, by a finite sequence of trees 𝑇1 , 𝑇2 , . . . , 𝑇 𝑗 , for 𝑗 ≥ 2, using Operations T1 and T2 . We consider two possibilities depending on which operation is used to form 𝑇 𝑗 from 𝑇 𝑗 −1 . First suppose that 𝑇 = 𝑇 𝑗 is obtained from 𝑇 𝑗 −1 using Operation T1 . In this case, 𝑇 is the tree obtained from 𝑇 𝑗 −1 by adding a subdivided star 𝑆(𝐾1,𝑘 ) and

392

Chapter 13. Domination Critical and Stable Graphs

an edge 𝑦𝑤, where 𝑘 ≥ 2, 𝑤 is the center vertex of 𝑆(𝐾1,𝑘 ), and 𝑦 is a vertex of 𝑇 𝑗 −1 . We now root 𝑇 at vertex 𝑦. Since 𝑇𝑤 is a subdivided star, 𝑇𝑤 ∈ R2. By construction, 𝑇 𝑗 −1 ∈ T . We may apply our inductive hypothesis on 𝑇 𝑗 −1 to show that 𝑇 𝑗 −1 ∈ R2. Hence, removing any vertex of 𝑇 𝑗 −1 (respectively, 𝑇𝑤 ) does not change the domination number of 𝑇 𝑗 −1 (respectively, 𝑇𝑤 ). From Lemma 13.14, we know that 𝛾(𝑇) = 𝛾(𝑇 𝑗 −1 ) + 𝛾(𝑇𝑤 ). Thus, 𝑉 (𝑇) = 𝑉 0 . Since 𝑇𝑤 is a subdivided star, it follows that 𝛾(𝑇 − 𝑤𝑢) = 𝛾(𝑇), where 𝑢 is a support vertex of 𝑇𝑤 , and 𝛾(𝑇 − 𝑢𝑣) > 𝛾(𝑇), where 𝑣 is the leaf adjacent to 𝑢. Therefore, 𝑇 ∈ UVR and 𝑇 ∉ CER ∪ UER, and so 𝑇 ∈ R2. Next suppose that 𝑇 = 𝑇 𝑗 is obtained from 𝑇 𝑗 −1 using Operation T2 . In this case, 𝑇 is obtained from 𝑇 𝑗 −1 by adding the path 𝑃3 and edge 𝑤𝑦, where 𝑤 is a leaf of 𝑃3 and 𝑦 is in 𝑉 (𝑇 𝑗 −1 ) and 𝑦 ∈ 𝑆 𝑗 −1 . We now root 𝑇 at the vertex 𝑦. Applying our inductive hypothesis, 𝑇 𝑗 −1 ∈ R2. Since 𝑦 ∈ 𝑆 𝑗 −1 , 𝑦 is in some 𝛾-set of 𝑇 𝑗 −1 and Lemma 13.14 implies that 𝑦 is in some 𝛾-set of 𝑇. Thus, 𝑦 can dominate 𝑤 in 𝑇 − 𝑢, where 𝑢 is the support vertex adjacent to 𝑤. Then in 𝑇, the vertices of 𝑇𝑤 are in 𝑉 0 . Using a similar argument as before, it follows that 𝑇 ∈ R2. To prove the converse, we again use induction the order 𝑛. Let 𝑇 be a tree in R2 and root 𝑇 at an endvertex 𝑟 of a longest path in 𝑇. Let 𝑤 be a vertex at distance diam(𝑇) − 2 from 𝑟 on a longest path beginning at 𝑟. Lemma 13.12 implies that 𝑤 has no leaf as a neighbor. Hence, every leaf of 𝑇𝑤 , except possibly 𝑤 itself, is at a distance 2 from 𝑤. Let 𝑦 denote the parent of 𝑤 in 𝑇. Let 𝐷 ′ be a 𝛾-set of 𝑇 ′ = 𝑇 − 𝑇𝑤 and let 𝑆 be the set of support vertices in 𝑇𝑤 . The set 𝐷 ′ ∪ 𝑆 is a dominating set of 𝑇, and so 𝛾(𝑇) ≤ |𝐷 ′ | + |𝑆|. Every 𝛾-set of 𝑇 must include at least |𝑆| vertices from 𝑉 (𝑇𝑤 ) \ {𝑤} to dominate the leaves of 𝑇𝑤 . Thus, if 𝛾(𝑇) < |𝐷 ′ | + |𝑆|, then it follows that fewer than |𝐷 ′ | vertices can dominate 𝑇 ′ , contradicting the fact that 𝐷 ′ is a 𝛾-set of 𝑇 ′ . Hence, 𝛾(𝑇) ≥ |𝐷 ′ | + |𝑆|, and so 𝛾(𝑇) = |𝐷 ′ | + |𝑆| and 𝐷 ′ ∪ 𝑆 is a 𝛾-set of 𝑇. Since diam(𝑇) ≥ 6, the tree 𝑇 ′ has order at least 4. Since 𝑇 ′ is a tree, by our previous discussion, 𝑇 ′ ∈ R1 ∪ R2 ∪ R6 ∪ R7 ∪ R8 as shown in Figure 13.5. It follows that if 𝑇 ′ ∉ R2, then 𝑇 ′ ∉ UVR, that is, there exists a vertex 𝑣 of 𝑇 ′ such that 𝛾(𝑇 ′ − 𝑣) ≠ 𝛾(𝑇 ′ ). Let 𝐷 𝑣 be a 𝛾-set of 𝑇 ′ − 𝑣. Then 𝐷 𝑣 ∪ 𝑆 is a dominating set of 𝑇 − 𝑣, so 𝛾(𝑇 − 𝑣) ≤ |𝐷 𝑣 | + |𝑆|. As before, every 𝛾-set of 𝑇 − 𝑣 must include at least |𝑆| vertices from 𝑉 (𝑇𝑤 ) \ {𝑤} to dominate the leaves of 𝑇𝑤 . Thus, if 𝛾(𝑇 − 𝑣) < |𝐷 𝑣 | + |𝑆|, then it follows that fewer than |𝐷 𝑣 | vertices can dominate 𝑇 ′ − 𝑣, contradicting the fact that 𝐷 𝑣 is a 𝛾-set of 𝑇 ′ − 𝑣. Hence, 𝛾(𝑇 − 𝑣) ≥ |𝐷 𝑣 | + |𝑆|, and so 𝛾(𝑇 − 𝑣) = |𝐷 𝑣 | + |𝑆| ≠ |𝐷 ′ | + |𝑆| = 𝛾(𝑇), contradicting the fact that 𝑇 ∈ R2. Hence, 𝑇 ′ ∈ R2. We can apply our inductive hypothesis to assume that 𝑇 ′ ∈ T . If 𝑇𝑤 = 𝑆(𝐾1,𝑘 ) for 𝑘 ≥ 2, then 𝑇 can be constructed from 𝑇 ′ using Operation T1 , and so 𝑇 ∈ T . If 𝑇𝑤 = 𝑃3 , then 𝛾(𝑇) = |𝐷 ′ | + |𝑆| = |𝐷 ′ | + 1 = 𝛾(𝑇 ′ ) + 1, and so 𝛾(𝑇 ′ ) = 𝛾(𝑇) − 1. Consider 𝑇 − 𝑢, where 𝑢 is the support vertex adjacent to 𝑤. Since 𝑇 ∈ R2, 𝛾(𝑇 − 𝑢) = 𝛾(𝑇), implying that some set
𝑋 of 𝛾(𝑇) − 1 vertices dominates 𝑉 (𝑇 ′ ) ∪ {𝑤}. Note that if 𝑤 ∈ 𝑋, then 𝑋 \ {𝑤} ∪ {𝑦} is also a dominating set of 𝑉 (𝑇 ′ ) ∪ {𝑤}. It follows that there exists a 𝛾-set of 𝑇 ′ containing 𝑦. Therefore, 𝑇 can be constructed from 𝑇 ′ using Operation T2 to attach 𝑇𝑤 , and hence 𝑇 ∈ T .

Section 13.3. Domination Vertex-Critical Graphs (CVR)

13.3

393

Domination Vertex-Critical Graphs (CVR)

From our previous discussion, the graphs in CVR are precisely the domination vertexcritical graphs, or just vertex-critical graphs, first studied by Brigham et al. [118]. Hence, a vertex-critical graph 𝐺 has 𝑉 (𝐺) = 𝑉 − and 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1 for all 𝑣 ∈ 𝑉 (𝐺). A vertex-critical graph 𝐺 with 𝛾(𝐺) = 𝑘 is called 𝑘-vertex-critical. For example, the cycles 𝐶3𝑘+1 , for 𝑘 ≥ 1, are (𝑘 + 1)-vertex-critical. Brigham et al. [118] determined the 𝑘-critical graphs for small 𝑘 ∈ [2]. Proposition 13.16 ([118]) The following hold: (a) A graph 𝐺 is 1-vertex-critical if and only if 𝐺 is the trivial graph 𝐾1 . (b) A graph 𝐺 is 2-vertex-critical if and only if 𝐺 is a complete graph of even order with a 1-factor removed. While the 𝑘-vertex-critical graphs for 𝑘 ∈ [2] have straightforward characterizations, 𝑘-vertex-critical graphs for 𝑘 ≥ 3 have not yet been characterized.

13.3.1

Vertex-Critical Graphs

In 1983 Bauer et al. [61] showed that 𝑉 0 is never empty for a nontrivial tree; hence, no nontrivial tree is vertex-critical. Theorem 13.17 ([61]) Every nontrivial tree 𝑇 contains a vertex 𝑣 such that 𝛾(𝑇 − 𝑣) = 𝛾(𝑇). Proof The result is immediate if 𝑇 = 𝐾2 . Therefore, assume that 𝑇 has order at least 3. Let 𝑣 be a support vertex that is adjacent to at most one non-leaf in 𝑇. Let 𝑢 be a leaf neighbor of 𝑣. If 𝑣 is adjacent to two (or more) leaves, then 𝑣 is in every 𝛾-set of 𝑇 and 𝛾(𝑇 − 𝑢) = 𝛾(𝑇). If not, then 𝑣 is adjacent to exactly one leaf 𝑢 and deg𝑇 (𝑣) = 2. Since 𝑢 is a leaf of 𝑇, we have 𝛾(𝑇) − 1 ≤ 𝛾(𝑇 − 𝑢) ≤ 𝛾(𝑇). If 𝛾(𝑇 − 𝑢) = 𝛾(𝑇), then we have the desired result. Hence, we may assume that 𝛾(𝑇 − 𝑢) = 𝛾(𝑇) − 1. Let 𝑆𝑢 be a 𝛾-set of 𝑇 − 𝑢. Note that 𝑣 ∉ 𝑆𝑢 for otherwise 𝑆𝑢 is a dominating set of 𝑇 having fewer than 𝛾(𝑇) vertices. Also, 𝑣 is dominated by 𝑆𝑢 and 𝑆𝑢 ∪ {𝑢}
dominates 𝑇 and hence dominates 𝑇 − 𝑣. Thus, 𝛾(𝑇 − 𝑣) ≤ |𝑆𝑢 | + 1 = 𝛾(𝑇) − 1 + 1 = 𝛾(𝑇). Let 𝑆 𝑣
be a 𝛾-set of 𝑇 − 𝑣. Note that 𝑢 ∈ 𝑆 𝑣 . If |𝑆 𝑣 \ {𝑢}| ≤ 𝛾(𝑇) − 2, then 𝑆 𝑣 \ {𝑢} ∪ {𝑣} is a dominating set of 𝑇 having fewer than 𝛾(𝑇) vertices, a contradiction. Hence, |𝑆 𝑣 \ {𝑢}| ≥ 𝛾(𝑇) − 1, that is, 𝛾(𝑇 − 𝑣) = |𝑆 𝑣 | ≥ 𝛾(𝑇). Consequently, 𝛾(𝑇 − 𝑣) = 𝛾(𝑇), as desired. Brigham et al. [118] showed that a forbidden subgraph characterization of vertexcritical graphs is not possible. They determined a sufficient condition to exclude a graph from being vertex-critical. Theorem 13.18 ([118]) If a graph 𝐺 has a vertex 𝑣 of degree at least 1 and the induced subgraph 𝐺 [N(𝑣)] is complete, then 𝐺 is not vertex-critical. Brigham et al. [118] concluded their study by posing the following four questions.

394

Chapter 13. Domination Critical and Stable Graphs

graph of order 𝑛. Question 13.19 Let 𝐺
be a vertex-critical
(a) Is 𝑛 ≥ 𝛿(𝐺) + 1
𝛾(𝐺) − 1
+ 1? (b) If 𝑛 = Δ(𝐺) + 1 𝛾(𝐺) − 1 + 1, is 𝐺 regular? (c) Is 𝑖(𝐺) = 𝛾(𝐺)?
(d) Is diam(𝐺) ≤ 2 𝛾(𝐺) − 1 ? These questions were subsequently answered by Fulman et al. [317] in 1995. They noted that the circulant graph 𝐺 = 𝐶17 ⟨1, 3, 5, 7⟩ is an 8-regular vertex-critical graph of order 𝑛 = 17 satisfying 𝑖(𝐺) =
5 and 𝛾(𝐺) = 3. This circulant 𝐺 has
order 𝑛 < 19 = 𝛿(𝐺) + 1 𝛾(𝐺) − 1 + 1 and 𝑖(𝐺) > 𝛾(𝐺), which resolves Questions 13.19(a) and (c). They settled Question 13.19(b) in the affirmative as follows. Theorem 13.20
([317]) If 𝐺 is a vertex-critical graph of order 𝑛 = Δ(𝐺) +
1 𝛾(𝐺) − 1 + 1, then 𝐺 is regular.

Proof Let 𝐺 be a vertex-critical graph of order 𝑛 = Δ(𝐺) + 1 𝛾(𝐺) − 1 + 1 and let 𝑆𝑢 denote a
𝛾-set of 𝐺 −
𝑢 for a vertex 𝑢. Since |𝑆𝑢 | = 𝛾(𝐺) − 1, in order that all of the Δ(𝐺) + 1 𝛾(𝐺) − 1 vertices of 𝐺 − 𝑢 are dominated, each element of 𝑆𝑢 must dominate exactly Δ(𝐺) + 1 vertices, and therefore have degree Δ(𝐺). This implies that 𝑆𝑢 is an independent set and no two vertices in 𝑆𝑢 have a common neighbor. To prove that 𝐺 is regular, it therefore suffices to show that an arbitrary vertex 𝑥 belongs to 𝑆 𝑣 for some vertex 𝑣. Let 𝑣 ∈ 𝑆 𝑥 . We show that 𝑥 ∈ 𝑆 𝑣 . Suppose, to the contrary, that 𝑥 ∉ 𝑆 𝑣 . Since 𝐺 is vertex-critical, 𝑆 𝑣 ∩ N[𝑣] = ∅. Each vertex of 𝑆 𝑥 \ {𝑣} dominates a unique vertex of 𝑆 𝑣 . But the remaining vertex in 𝑆 𝑣 must also be dominated by 𝑆 𝑥 , and so must be dominated by 𝑣, a contradiction. Hence, 𝑥 ∈ 𝑆 𝑣 .

A family of graphs 𝐺 𝑘,𝑡 of order 𝑛 = Δ(𝐺) + 1 𝛾(𝐺) − 1 + 1 illustrating Theorem 13.20 was described in [118]. For 𝑘 even and 𝑘, 𝑡 ≥ 2, let 𝑉 (𝐺 𝑘,𝑡 ) = {𝑣 0 , 𝑣 1 ,  . . . , 𝑣 (𝑡 −1) (𝑘+1) } and 𝐸 (𝐺 𝑘,𝑡 ) = 𝑣 𝑖 𝑣 𝑗 : 1 ≤ (𝑖− 𝑗) (mod ((𝑡 −1) (𝑘 +1) +1)) ≤ 𝑘/2 . Then 𝐺 𝑘,𝑡 is 𝑘-regular with 𝛾(𝐺 𝑘,𝑡 ) = 𝑡. For example, the graph 𝐺 2,2 is the cycle 𝐶4 and Figure 13.8 shows the graph 𝐺 4,3 . 𝑣0

𝑣 10

𝑣1

𝑣9

𝑣2

𝑣8

𝑣3 𝑣7

𝑣4 𝑣6

𝑣5

Figure 13.8 The graph 𝐺 4,3 The next theorem answers Question 13.19(d). Theorem 13.21 ([317]) If 𝐺 is a 𝑘-vertex-critical graph for 𝑘 ≥ 2, then diam(𝐺) ≤ 2(𝑘 − 1).

Section 13.3. Domination Vertex-Critical Graphs (CVR)

395

In 1995 Fulman et al. [317] gave a general construction for graphs that achieve the bound of Theorem 13.21. We describe a special case of this construction. Let 𝐺 be the graph formed by replacing each edge 𝑢𝑣 of a path on 𝑘 vertices by a four cycle 𝑢 𝑢 ′ 𝑣 𝑣 ′ 𝑢 such that 𝑢 ′ and 𝑣 ′ are new vertices. The resulting graph 𝐺 is 𝑘-vertex-critical and diam(𝐺) = 2(𝑘 − 1).

13.3.2

3-Vertex-Critical Graphs

The family of 3-vertex-critical graphs is the most studied of the vertex-critical graphs. In this section, we present some properties of 3-vertex-critical graphs. Recall that a matching 𝑀 ⊆ 𝐸 is perfect if every vertex 𝑣 ∈ 𝑉 is incident to exactly one edge in 𝑀 and is near perfect if every vertex except one in 𝑉 is incident to exactly one edge in 𝑀. In 2005 Ananchuen and Plummer [25] noted that 3-vertex-critical graphs of even order in general need not have a perfect matching. On the other hand, they established a sufficient condition for such a graph to have a perfect matching. Theorem 13.22 ([25]) If 𝐺 is a 𝐾1,5 -free 3-vertex-critical graph of even order, then 𝐺 has a perfect matching. A similar result for graphs of odd order was given by Ananchuen and Plummer [28] in 2007. Theorem 13.23 ([28]) If 𝐺 is a 𝐾1,5 -free 3-vertex-critical graph of odd order 𝑛 ≥ 11 with 𝛿(𝐺) ≥ 1, then 𝐺 has a near-perfect matching. The graph in Figure 13.9 is a 𝐾1,5 -free 3-vertex-critical graph of order 9 that does not have a near-perfect matching, showing that the assumption that the order be at least 11 in Theorem 13.23 is necessary.

Figure 13.9 A 𝐾1,5 -free 3-vertex-critical graph of order 9 having no near-perfect matching

In 2010 Wang and Yu [745] extended the results of Theorems 13.22 and 13.23 with the following two theorems. Theorem 13.24 ([745]) If 𝐺 is a 𝐾1,6 -free 3-vertex-critical graph with even order 𝑛 ≠ 12, then 𝐺 has a perfect matching. Let A be the set of three graphs in Figures 13.9, 13.10, and 13.11. Theorem 13.25 ([745]) If 𝐺 ∉ A is a 𝐾1,7 -free 3-vertex-critical graph of odd order 𝑛 ≠ 13, then 𝐺 has a near-perfect matching.

Chapter 13. Domination Critical and Stable Graphs

396

Figure 13.10 A graph in A

Figure 13.11 A graph in A

A graph 𝐺 with the property that 𝐺 − 𝑣 has a perfect matching, for every choice of 𝑣 ∈ 𝑉 (𝐺), is called factor-critical. In 2007 Ananchuen and Plummer [28] proved the following result concerning factor-critical graphs. Theorem 13.26 ([28]) If 𝐺 is a 𝐾1,4 -free 3-vertex-critical graph of odd order, then 𝐺 is factor-critical. The authors in [28] conjectured that any 2-connected 𝐾1,5 -free 3-vertex-critical graph 𝐺 of odd order with 𝛿(𝐺) ≥ 3 is factor-critical. In 2009 Wang and Yu [744] subsequently verified this conjecture except for two cases. Theorem 13.27 ([744]) If 𝐺 is a 2-connected 𝐾1,5 -free 3-vertex-critical graph of odd order with 𝛿(𝐺) ≥ 3, and 𝐺 is not one of the two graphs 𝐺 1 and 𝐺 2 shown in Figure 13.12, then 𝐺 is factor-critical.

(a) 𝐺 1

(b) 𝐺 2

Figure 13.12 The graphs 𝐺 1 and 𝐺 2 In 2004 Ananchuen and Plummer [24] also proved the following result concerning factor-critical graphs. Theorem 13.28 ([24]) If 𝐺 is a 2-connected 3-vertex-critical graph of odd order, then 𝐺 is factor-critical.

Section 13.4. Domination Edge-Critical Graphs (CEA)

397

In 2006 Ananchuen and Plummer [26] established the following property of 3-vertex-critical graphs. Theorem 13.29 ([26]) If 𝐺 is a connected 3-vertex-critical claw-free graph, then the following hold: (a) 𝐺 is 2-connected. (b) If 𝐺 is of even order or if 𝛿(𝐺) ≥ 3, then 𝐺 is 3-connected. (c) If 𝛿(𝐺) ≥ 5, then 𝐺 is 4-connected. A 2006 survey of results on 3-vertex-critical graphs is given by Plummer [642]. For more on domination vertex-critical graphs see [22, 365, 417, 598].

13.4 Domination Edge-Critical Graphs (CEA) Just as deleting an edge can increase the domination number by at most one, adding an edge can decrease it by at most one. Hence, a graph for which the domination number changes when an arbitrary edge is added has the property that 𝛾(𝐺 + 𝑒) = 𝛾(𝐺) − 1 for all 𝑒 ∈ 𝐸 ( 𝐺). This class of graphs was initially introduced and investigated in 1983 by Sumner and Blitch [694], who called them domination edge-critical graphs. Formally, a graph 𝐺 is domination 𝑘-edge-critical, or just 𝑘-edge-critical, if 𝛾(𝐺) = 𝑘 and the addition of any edge decreases the domination number. Since adding an edge cannot increase the domination number, these graphs are precisely the graphs in CEA. We remark that this collection of graphs is the most studied of the six families defined in Section 13.2. It is straightforward to characterize the 𝑘-edge-critical graphs for 𝑘 ≤ 2. Vacuously, a graph is 1-edge-critical if and only if it is the complete graph. The 2-edge-critical graphs are characterized in [694] as follows. Theorem 13.30 ([694]) A graph 𝐺 is 2-edge-critical if and only if its complement 𝐺 is the union of nontrivial stars. Although the 𝑘-edge-critical graphs for 𝑘 ≤ 2 are easy to characterize, the level of difficulty increases significantly for 𝑘 ≥ 3 and to date these graphs have not been characterized. However, many interesting properties of these graphs have been found. We present some of these properties of 𝑘-edge-critical graphs for general 𝑘 in Section 13.4.1 and focus on 3-edge-critical graphs in Section 13.4.2. For more details, see Chapter 5 of [417] and Chapter 16 of [416].

13.4.1

Properties of 𝒌-Edge-Critical Graphs

In 1990 Sumner [693] showed that the domination number of a 𝑘-edge-critical graph cannot be increased by deleting a vertex. Theorem 13.31 ([693]) If 𝐺 is a 𝑘-edge-critical graph with 𝑘 ≥ 2, then 𝑉 + = ∅. In 1999 Paris et al. [628] determined the structure of connected 𝑘-edge-critical graphs with a cut-vertex.

398

Chapter 13. Domination Critical and Stable Graphs

Theorem 13.32 ([628]) If 𝐺 is a 𝑘-edge-critical graph with a cut-vertex 𝑣, then 𝐺 − 𝑣 has exactly two components. In 1994 Favaron et al. [285] showed that the  diameter  of a connected 4-edgecritical graph is at most 5 and conjectured that 3𝑘/2 − 1 is an upper bound on the diameter for all 𝑘-edge-critical graphs with 𝑘 ≥ 4. Further, they showed that if this conjectured bound holds, then it is also tight. They proved the following bound. Theorem 13.33 ([285]) If 𝐺 is a 𝑘-edge-critical graph for 𝑘 ≥ 5, then diam(𝐺) ≤ 2𝑘 − 2. In 1990 Wojcicka [753] conjectured that every (𝑘 − 1)-connected 𝑘-edge-critical graph is Hamiltonian and proved the conjecture for 𝑘 = 3 (see Section 13.4.2). In 2005 Yang et al. [759] provided a counterexample for 𝑘 = 4 by constructing a family of 3-connected 4-edge-critical graphs that are not Hamiltonian. Wojcicka’s Conjecture remains open for 𝑘 ≥ 5. On the other hand, in 2017 Kaemawichanurat and Caccetta [514] proved that the conjecture holds for 3-connected 4-edge-critical claw-free graphs. Theorem 13.34 ([514]) If 𝐺 is a 3-connected 4-edge-critical claw-free graph, then 𝐺 is Hamiltonian.

13.4.2

3-Edge-Critical Graphs

Among the 𝑘-edge-critical graphs, 3-edge-critical graphs are the most studied. The corona 𝐾3 ◦ 𝐾1 shown in Figure 13.13 is the smallest (in terms of order) connected 3-edge-critical graph [694].

Figure 13.13 The smallest connected 3-edge-critical graph

Other examples of 3-edge-critical graphs are given in Figure 13.14, where the three graphs in Figure 13.14(b), (c), and (d) were given by Sumner and Blitch [694]. In 1990 Sumner [693] characterized the disconnected 3-edge-critical graphs. Theorem 13.35 ([693]) A disconnected graph 𝐺 is 3-edge-critical if and only if 𝐺 = 𝐴 ∪ 𝐵, where either 𝐴 is a trivial graph 𝐾1 and 𝐵 is a 2-edge-critical graph or 𝐴 is a complete graph and 𝐵 is a complete graph minus a 1-factor. In 2019 Furuya and Matsumoto [321] showed that the family of 3-edge-critical planar graphs is small.

Section 13.4. Domination Edge-Critical Graphs (CEA)

(a)

(b)

(c)

399

(d)

Figure 13.14 Examples of 3-edge-critical graphs

Theorem 13.36 ([321]) 𝑛 ≤ 23.

If 𝐺 is a 3-edge-critical planar graph of order 𝑛, then

In 1990 Wojcicka [753] considered longest paths in 3-edge-critical graphs. Theorem 13.37 ([753]) If 𝐺 is a connected 3-edge-critical graph of order 𝑛 ≥ 7, then 𝐺 has a Hamiltonian path. We remark that in 2002 Zhang and Tian [786] gave a short proof of Theorem 13.37. Further, they established the following upper bound on the independence number of a 3-edge-critical graph in terms of its minimum degree. Theorem 13.38 ([786]) If 𝐺 is a 𝑘-connected 3-edge-critical graph, then 𝛼(𝐺) ≤ 𝛿(𝐺) + 2, and if 𝑘 ≤ 𝛿(𝐺) − 1, then 𝛼(𝐺) ≤ 𝑘 + 1. Wojcicka [753] conjectured that every connected 3-edge-critical graph with minimum degree at least 2 has a Hamiltonian cycle. In 1997 Favaron et al. [286] verified Wojcicka’s Conjecture for graphs having independence number at most 𝛿(𝐺) + 1. Theorem 13.39 ([286]) If 𝐺 is a connected 3-edge-critical graph with 𝛿(𝐺) ≥ 2 and 𝛼(𝐺) ≤ 𝛿(𝐺) + 1, then 𝐺 is Hamiltonian. Finally, in 1999 Tian et al. [711] resolved the case 𝛼(𝐺) = 𝛿(𝐺) + 2 which combined with Theorem 13.39 proved Wojcicka’s Conjecture as follows. Theorem 13.40 ([711]) If 𝐺 is a connected 3-edge-critical graph with 𝛿(𝐺) ≥ 2, then 𝐺 is Hamiltonian. For related results, see also [23, 27, 49, 158, 159, 378, 757]. A graph 𝐺 is Hamilton-connected if every two distinct vertices of 𝐺 are connected by a Hamiltonian path. In 2009 Cheng et al. [162] showed that every 3-edge-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining their result with results from the four papers [157, 160, 161, 286], they solved a conjecture posed in 2003 by Chen et al. [160]. Their result is stated in terms of the toughness of a graph, which is a measure of the connectivity of a graph.

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Formally, let 𝑆 be a cutset of 𝐺 and 𝑐 𝑆 (𝐺) be the number of components in 𝐺 − 𝑆. The toughness 𝑡 (𝐺) is defined as   |𝑆| 𝑡 (𝐺) = min : 𝑆 ⊆ 𝑉 (𝐺) is a cutset of 𝐺 . 𝑐 𝑆 (𝐺) Theorem 13.41 ([162]) If 𝐺 is a connected 3-edge-critical graph, then 𝐺 is Hamilton-connected if and only if 𝑡 (𝐺) > 1. In 1983 Sumner and Blitch [694] conjectured that for any 𝑘-edge-critical graph 𝐺, 𝛾(𝐺) = 𝑖(𝐺) = 𝑘. In 1994 Ao [30] disproved this conjecture for 𝑘 = 4 and in 1996 Ao et al. [31] disproved it for all 𝑘 ≥ 4. However, for 𝑘 = 3 the conjecture remained open for several more years. It was listed in [417, 603] as a major outstanding conjecture and attracted attention of researchers worldwide. There was much support for its validity. For instance, it was shown in [694] that 𝛾(𝐺) = 𝑖(𝐺) = 3 for a 3-edge-critical graph 𝐺 with any of the following properties: 𝐺 has a leaf, 𝐺 has a cut-vertex, or 𝐺 has diameter 3. Moreover, Sumner and Wojcicka stated in [696] that an extensive computer search failed to find a counterexample. Further, Favaron et al. [286] showed that if a 3-edge-critical graph 𝐺 has minimum degree 𝛿(𝐺) ≥ 2 and independence number 𝛿(𝐺) + 2, then 𝛾(𝐺) = 𝑖(𝐺) = 3. Despite the support for the conjecture, in 1999 the conjecture for the case of 𝑘 = 3 was settled in the negative by Van der Merwe et al. [725]. The counterexample given in [725] to disprove the conjecture shows that not only is 𝛾(𝐺) not necessarily equal to 𝑖(𝐺) for 3-edge-critical graphs 𝐺, it constructs for each 𝑝 ≥ 3, a 3-edge-critical graph 𝐺 with 𝑖(𝐺) = 𝑝. We remark that for 𝑝 = 3, the constructed graph 𝐺 has 𝛾(𝐺) = 𝑖(𝐺) = 3 and so, in order to have a counterexample, we must have 𝑝 ≥ 4. Hence, the smallest counterexample to the Sumner-Blitch Conjecture for 𝑘 = 3 provided by this construction is the graph for 𝑝 = 4, which has 56 vertices. It is still unknown whether there exists a smaller counterexample. We conclude this section by giving the construction of the counterexample first presented in [721] and later in [725]. For 𝑝 ≥ 3, let 𝐺 𝑝 be the graph constructed from a factorization of the complete graph 𝐾2 𝑝 with vertex set 𝑣 1 , 𝑣 2 , . . . , 𝑣 2 𝑝 into the 1-factors 𝐹1 , 𝐹2 , . . . , 𝐹2 𝑝−1 as follows. Let the vertices of each 𝐹𝑖 be labeled 𝑣 𝑖,1 , 𝑣 𝑖,2 , . . . , 𝑣 𝑖,2 𝑝 such that 𝑣 𝑖, 𝑗 𝑣 𝑖,𝑞 ∈ 𝐸 (𝐹𝑖 ) if and only if 𝑣 𝑗 𝑣 𝑞 is an edge of the 1-factor 𝐹𝑖 . Then add the edges 𝑣 𝑖, 𝑗 𝑣 ℎ,𝑞 for all 𝑖 ≠ ℎ and 𝑗 ≠ 𝑞, that is, add edges such that vertex 𝑣 𝑖, 𝑗 is adjacent to every vertex in every other factor 𝐹ℎ , where ℎ ≠ 𝑖, except the vertex 𝑣 ℎ, 𝑗 for 𝑖 ∈ [2𝑝 − 1] and 𝑗 ∈ [2𝑝]. Let 𝑣 1,1 𝑣 1,2 be an edge in 𝐹1 . Proposition 13.42 ([721, 725]) For 𝑝 ≥ 3, the graph 𝐺 𝑝 is a 3-edge-critical graph satisfying 𝛾(𝐺 𝑝 ) = 3 and 𝑖(𝐺 𝑝 ) = 𝑝. Proof Consider the graph 𝐺 𝑝 , where 𝑝 ≥ 3. No two vertices in 𝐺 𝑝 dominate the graph and so 𝛾(𝐺 𝑝 ) ≥ 3. The set {𝑣 1,1 , 𝑣 2,1 , 𝑣 2,2 } is a dominating set of 𝐺 𝑝 and so 𝛾(𝐺 𝑝 ) ≤ 3. Consequently, 𝛾(𝐺 𝑝 ) = 3. There are only two possibilities to check to show that 𝐺 𝑝 is 3-edge-critical: adding an edge 𝑣 𝑖,𝑎 𝑣 𝑖,𝑏 in 𝐹𝑖 or adding an edge 𝑣 𝑖,𝑎 𝑣 ℎ,𝑎 for 𝑖 ≠ ℎ. If 𝑣 𝑖,𝑎 𝑣 𝑖,𝑏 is not an edge in 𝐺 𝑝 , then 𝑣 𝑗,𝑎 𝑣 𝑗,𝑏 is in some 𝐹 𝑗 ,

Section 13.5. Total Domination Edge-Critical Graphs

401

where 𝑗 ≠ 𝑖. Hence, {𝑣 𝑖,𝑎 , 𝑣 𝑗,𝑏 } is a dominating set of 𝐺 𝑝 + 𝑣 𝑖,𝑎 𝑣 𝑖,𝑏 . Further, if 𝑣 𝑖,𝑎 𝑣 𝑖,𝑏 ∈ 𝐸 (𝐹𝑖 ), then {𝑣 𝑖,𝑎 , 𝑣 ℎ,𝑏 } is a dominating set of 𝐺 𝑝 + 𝑣 𝑖,𝑎 𝑣 ℎ,𝑎 for ℎ ≠ 𝑖. In both cases, it follows that 𝐺 𝑝 is 3-edge-critical. To see that 𝑖(𝐺 𝑝 ) = 𝑝, observe that since 𝑣 𝑖, 𝑗 is adjacent to every vertex except 𝑣 ℎ, 𝑗 in 𝐹ℎ for every ℎ ≠ 𝑖, it follows that an ID-set containing vertices from different 1-factors must have at least 2𝑝 − 1 vertices (one from each 𝐹𝑖 ). On the other hand, an ID-set contained in a single 𝐹𝑖 needs only 𝑝 vertices (one from each adjacent pair). Moreover, no fewer than 𝑝 vertices form an ID-set of 𝐺 𝑝 and so 𝑖(𝐺) = 𝑝. Hence, the graph 𝐺 𝑝 is 3-edge-critical and has 𝑖(𝐺 𝑝 ) = 𝑝.

13.5

Total Domination Edge-Critical Graphs

Criticality and stability have been studied for several domination parameters. In this section, we consider total domination edge-critical graphs. For more details on edge-criticality and also for information on vertex-criticality of total domination, we refer the reader to Chapter 11 of [490]. A graph 𝐺 is total domination edge-critical if 𝛾t (𝐺 + 𝑒) < 𝛾t (𝐺) for every edge 𝑒 ∈ 𝐸 (𝐺) ≠ ∅. Further, if 𝛾t (𝐺) = 𝑘, then we say that 𝐺 is a 𝑘 t -edge-critical graph. Therefore, a 𝑘 t -edge-critical graph has total domination number 𝑘 and the addition of any edge decreases the total domination number. For examples of 𝑘 t -edge-critical graphs for small 𝑘 ∈ {3, 4, 5}, the cycle 𝐶5 is 3t -edge-critical, the graph 𝐺 4 in Figure 13.15(a) is 4t -edge-critical, and the graph 𝐺 5 in Figure 13.15(b) is 5t -edge-critical.

𝑢

𝑦

𝑣 𝑥

(a) 𝐺 4

(b) 𝐺 5

Figure 13.15 Examples of 𝑘 t -edge-critical graphs for 𝑘 ∈ {4, 5}

A graph 𝐺 is 2t -edge-critical if and only if 𝐺 is a complete graph 𝐾𝑛 , for 𝑛 ≥ 2. Complete graphs are vacuously 2t -edge-critical, and if 𝐺 has an edge, then 𝑘 = 3 is the smallest value of 𝑘 for which a graph can be 𝑘 t -edge-critical. The smallest 𝑘 for a non-complete graph to be domination 𝑘-edge-critical is 𝑘 = 2 and, as we have seen in Section 13.3, the 2-edge-critical graphs are easy to characterize. However, this is not the case for total domination. In fact, to date the problem of characterizing the 𝑘 t -edge-critical graphs remains open for every 𝑘 ≥ 3. Thus, the research on 𝑘 t -edgecritical graphs has focused on determining properties of these graphs. Properties of general 𝑘 t -edge-critical graphs are presented in Section 13.5.1 and 3t -edge-critical graphs are presented in Section 13.5.2.

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402

13.5.1

𝒌 t -Edge-Critical Graphs

In 1998 Van der Merwe et al. [724] showed that the addition of an edge to a graph can decrease the total domination number by at most two. Observation 13.43 ([724]) For any isolate-free graph 𝐺 and any edge 𝑒 ∈ 𝐸 ( 𝐺), 𝛾t (𝐺) − 2 ≤ 𝛾t (𝐺 + 𝑒) ≤ 𝛾t (𝐺). For the graph 𝐺 4 shown in Figure 13.15(a), adding an edge between two nonadjacent vertices in the same diamond decreases the total domination number by one, while adding edge 𝑢𝑣, for example, decreases it by two. Graphs for which the total domination number decreases by two no matter which edge is added, that is, graphs 𝐺 with the property that 𝛾t (𝐺) = 𝑘 and 𝛾t (𝐺 + 𝑒) = 𝑘 − 2 for every edge 𝑒 ∈ 𝐸 (𝐺) are called 𝑘 t -edge-supercritical graphs. In 1998 Van der Merwe et al. [721, 722] characterized the 4t -edge-supercritical graphs. Theorem 13.44 ([721, 722]) A graph 𝐺 is 4t -edge-supercritical if and only if 𝐺 is the disjoint union of two nontrivial complete graphs. This result was generalized by Henning and Yeo [490]. Theorem 13.45 ([490]) A graph 𝐺 is 𝑘 t -edge-supercritical for 𝑘 ≥ 4 if and only if 𝐺 is the disjoint union of 𝑘/2 nontrivial complete graphs. Tight upper and lower bounds on the diameter of 𝑘 t -edge-critical graphs have been determined for 𝑘 ∈ {3, 4}. Theorem 13.46 ([724]) If 𝐺 is a 3t -edge-critical graph, then 2 ≤ diam(𝐺) ≤ 3. Theorem 13.47 ([723]) If 𝐺 is a 4t -edge-critical graph, then 2 ≤ diam(𝐺) ≤ 4. Tight upper bounds on the diameter of 𝑘 t -edge-critical graphs for small 𝑘 were also given in 2012 by Henning and Van der Merwe in [479]. Theorem 13.48  ([479])  If 𝐺 is a 𝑘 t -edge-critical graph for 𝑘 ∈ {3, 4, 5, 6}, then diam(𝐺) ≤ 3(𝑘 − 1)/2 . A straightforward upper bound on the diameter of 𝑘 t -edge-critical graphs for general 𝑘 is also given in [479]. Theorem 13.49 ([479]) If 𝐺 is a 𝑘 t -edge-critical graph, then diam(𝐺) ≤ 2𝑘 − 3. Although Theorem 13.49 gives an upper bound, the problem of determining the maximum diameter of a 𝑘 t -edge-critical graphs remains open. However, the following was shown by Henning and Van der Merwe [479]. Theorem 13.50 ([479]) For every 𝑘 ≥ 3, there exists 𝑘 t -edge-critical graphs 𝐺  with diam(𝐺) ≥ 3(𝑘 − 1)/2 . For examples achieving the lower bound in Theorem 13.50, the graph in Figure 13.16 is a 5t -edge-critical graph of diameter 6 and the graph in Figure 13.17 is a 7t -edge-critical graph of diameter 9.

Section 13.5. Total Domination Edge-Critical Graphs

403

Figure 13.16 A 5t -edge-critical graph of diameter 6

Figure 13.17 A 7t -edge-critical graph of diameter 9

13.5.2

3t -Edge-Critical Graphs

As previously mentioned, the cycle 𝐶5 is a 3t -edge-critical graph. The graph shown in Figure 13.18(a) is another example of a 3t -edge-critical graph of diameter 2, while the graphs shown in Figure 13.18(b) and (c) are examples of 3t -edge-critical graphs of diameter 3.

(a)

(b)

(c)

Figure 13.18 3t -edge-critical graphs

Since 𝛾t (𝐺) ≥ 2 for all isolate-free graphs, the smallest value of 𝑘 for the total domination number to change when an edge is added is 3. The problem of characterizing 𝑘 t -edge-critical graphs, even in the simplest case when 𝑘 = 3, appears to be difficult and has attracted a lot of interest. Much research has been done on determining properties of 3t -edge-critical graphs. For example, Van der Merwe et al. [724] showed that any 3t -edge-critical graph with minimum degree at least two is 2-connected. Theorem 13.51 ([724]) If 𝐺 is a 3t -edge-critical graph with 𝛿(𝐺) ≥ 2, then 𝐺 is 2-connected.

404

Chapter 13. Domination Critical and Stable Graphs

Matching properties in 3t -edge-critical graphs were established in 2011 by Henning and Yeo [486] as follows. Recall that a graph 𝐺 is called factor-critical if 𝐺 − 𝑣 has a perfect matching for every vertex 𝑣 of 𝐺. Theorem 13.52 ([486]) Every 3t -edge-critical graph of even order has a perfect matching, while every 3t -edge-critical graph of odd order is factor-critical. Another reason that 3t -edge-critical graphs have received so much attention is their relationship to a long-standing conjecture involving diameter-2-critical graphs. This conjecture proposes an upper bound on the number of edges in a graph of diameter 2 having the property that the removal of any edge increases its diameter. Such graphs are called diameter-2-critical graphs. Clearly, removing an edge cannot decrease the diameter, so the diameter-critical graphs are graphs for which the diameter increases upon the removal of an arbitrary edge. Murty and Simon (see [128, 638]) independently made the following conjecture. If 𝐺 is a diameter-2-critical graph Conjecture 13.53 (Murty-Simon  Conjecture)  of order 𝑛 and size 𝑚, then 𝑚 ≤ 𝑛2 /4 , with equality if and only if 𝐺 is the complete bipartite graph 𝐾 ⌊ 𝑛2 ⌋, ⌈ 𝑛2 ⌉ . This conjecture was credited to Murty and Simon in the 1970s. However, according to Füredi [318], Erdős attributed it to the work of Ore in the 1960s. The Murty-Simon Conjecture has been proven for several families of graphs. Mantel’s [581] result in 1907 (a special case of a classic result of Turán [717]) proves the Murty-Simon Conjecture for triangle-free graphs. In 1979 Caccetta and Häg-
gkvist [128] proved that the conjecture holds for graphs of order 𝑛 which have O 𝑛3− 𝜀 triangles with a certain property. On the other hand, in 1986 Plesník [639] constructed families of diameter-2-critical graphs with the property that each edge belongs to at least one triangle of the graph. And in 1999 Madden [579] constructed families of diameter-2-critical graphs with both lower and upper bounds of approximately 𝑛3 triangles. See also [340–343]. In 1975 Plesník [638] proved that 𝑚 ≤ 3𝑛(𝑛 − 1)/8 for any diameter-2-critical graph with 𝑛 vertices and 𝑚 edges. In 1979 Caccetta and Häggkvist [128] proved that 𝑚 < 0.27𝑛2 for such graphs. In 1987 Fan [266] proved the bound of the conjecture for 𝑛 ≤ 24 and for 𝑛 = 26. For diameter-2-critical graphs of order 𝑛 ≥ 25, he showed
that 𝑚 < 𝑛2 /4 + 𝑛2 − 16.2𝑛 + 56 /320 < 0.2532𝑛2 . Perhaps Füredi’s [318] asymptotic result from 1992 is the most noteworthy contribution to date on the conjecture. He proved that the conjecture is true2 for large 𝑛, 2 that is, for 𝑛 > 𝑛0 where 𝑛0 is a tower of 2’s (nested powers of the form 22 ) of height 14 about 10 . However, even this striking result does not put the conjecture to rest. As remarked by Madden [579], “𝑛0 is an inconceivably (and inconveniently) large number: it is a tower of 2’s of height approximately 1014 . Since, for practical purposes, we are usually interested in graphs which are smaller than this, further investigation is warranted.”

Section 13.5. Total Domination Edge-Critical Graphs

405

We note that in 2017 Cao and Yan [131] generalized this result of Füredi. The Murty-Simon Conjecture has been studied by several other authors, see [86, 547, 602] for example. Although many impressive partial results have been obtained, the conjecture remains open for general 𝑛. Conventionally, the diameter of a disconnected graph is considered to be either undefined or defined as infinity. We adopt the former and hence require that a diameter-2-critical graph have minimum degree of two (since removing an edge incident to a vertex of degree one results in a disconnected graph). However, if we chose the latter and defined the diameter of disconnected graphs to be infinity, then the only additional diameter-2-critical graphs are stars with order at least 3. In 2003 Hanson and Wang [379] observed a relationship that essentially equates the Murty-Simon Conjecture with a conjecture involving total domination. This link gives a new way to look at the Murty-Simon Conjecture from the perspective of total domination in graphs. Note that this relationship is contingent on total domination being defined in the complement 𝐺 of the diameter-2-critical graph 𝐺, that is, 𝐺 is an isolate-free graph. Hence, our elimination of stars as diameter-2-critical graphs is necessary for this association. In the remainder of this section, we will discuss the link and progress made using it. The proof of this surprising result is short and simple. We present a slightly modified version of the proof in [379] here. First we make an observation. Since any pair of vertices at distance 3 or more apart in 𝐺 form a TD-set of 𝐺, it follows that if diam(𝐺) ≥ 3, then 𝛾t (𝐺) = 2. Moreover, if 𝑆 = {𝑢, 𝑣} is a TD-set of a graph 𝐺, then in 𝐺, the vertices 𝑢 and 𝑣 are not adjacent and have no common neighbors, that is, diam(𝐺) ≥ 3. Observation 13.54 A graph 𝐺 has 𝛾t (𝐺) = 2 if and only if diam(𝐺) ≥ 3. Theorem 13.55 ([379]) A graph is diameter-2-critical if and only if its complement is 3t -edge-critical or 4t -edge-supercritical. Proof Let 𝐺 be a diameter-2-critical graph. Observation 13.54 implies that 𝛾t (𝐺) ≥ 3. Moreover, diam(𝐺 − 𝑢𝑣) ≥ 3 for any edge 𝑢𝑣 ∈ 𝐸 (𝐺). Again, by Observation 13.54, we have that 𝛾t (𝐺 + 𝑢𝑣) = 2. It follows from Observation 13.43 that 𝐺 is 3t -edge-critical or 4t -edge-supercritical. Assume that 𝐺 is 3t -edge-critical or 4t -edge-supercritical. Thus, 𝛾t (𝐺) ∈ {3, 4} and 𝛾t (𝐺+𝑢𝑣) = 2 for any 𝑢𝑣 ∈ 𝐸 (𝐺). Observation 13.54 implies that diam(𝐺−𝑢𝑣) ≥ 3 for any edge 𝑢𝑣 ∈ 𝐸 (𝐺). If diam(𝐺) ≥ 3, then 𝛾t (𝐺) = 2, a contradiction. Hence, diam(𝐺) ≤ 2. If diam(𝐺) = 1, then 𝐺 is complete and 𝐸 (𝐺) is empty, contradicting the fact that 𝛾t (𝐺) ∈ {3, 4}. Thus, diam(𝐺) = 2 and so 𝐺 is a diameter-2-critical graph. For example, the self-complementary cycle 𝐶5 is both diameter-2-critical and 3t edge-critical. Figure 13.19 shows a diameter-2-critical graph 𝐺 and its complement 𝐺, a 3t -edge-critical graph. By Theorem 13.44, a graph 𝐺 is 4t -edge-supercritical if and only if 𝐺 is the disjoint union of two nontrivial complete graphs. Hence, the complement of a 4t edge-supercritical graph is a complete bipartite graph and the number of edges in a

Chapter 13. Domination Critical and Stable Graphs

406

𝑎

𝑎 𝑓

𝑏

𝑐

𝑒 𝑑

𝑔 𝑒

𝑐

𝑓

𝑏

𝑑

𝑔

(a) 𝐺

(b) 𝐺

Figure 13.19 A diameter-2-critical graph 𝐺 and its complement 𝐺

complete bipartite graph is maximized when the partite sets differ in cardinality by at most one. Thus, the Murty-Simon Conjecture 13.53 holds for diameter-2-critical graphs whose complements are 4t -edge-supercritical, and a subset of these graphs are the extremal  graphs of the conjecture. Noting also  that for a graph 𝐺 of order 𝑛, |𝐸 (𝐺)| < 𝑛2 /4 if and only if |𝐸 (𝐺)| > 𝑛(𝑛−2)/4 , it follows from Theorem 13.44 and Theorem 13.55 that proving the Murty-Simon Conjecture is equivalent to proving the following conjecture. Conjecture 13.56 If 𝐺 is a 3t -edge-critical graph of order 𝑛 and size 𝑚, then   𝑚 > 𝑛(𝑛−2) . 4 By Theorem 13.46, every 3t -edge-critical graph has diameter 2 or 3. In 2003 Hanson and Wang [379] proved the following result. Theorem 13.57 ([379]) If 𝐺 is a 3t -edge-critical graph of order 𝑛 and size 𝑚 with   diam(𝐺) = 3, then 𝑚 ≥ 𝑛(𝑛−2) . 4 Note that in order to prove that Conjecture 13.56 holds for 3t -edge-critical graphs of diameter 3, strict inequality is needed in the statement of Theorem 13.57. A lengthy, detailed argument, given in [429], proves strict inequality for such graphs with even order. A slight adaptation of the proof in [429] takes care of the graphs having odd order as shown in [431, 743]. In 2019 Dailly et al. [210] gave a simpler proof that this improved bound is not tight for 3t -edge-critical graphs of diameter 3. Thus, the Murty-Simon Conjecture is proven for the graphs whose complements have diameter 3. Hence, the problem is reduced to determining the minimum number of edges in 3t -edge-critical graphs of diameter 2. In 2014 Haynes et al. [430] showed the following for 3t -edge-critical graphs with sufficiently small minimum degree. Theorem 13.58 ([430]) If 𝐺 is a 3t -edge-critical graph of order 𝑛 and size 𝑚 with 𝛿(𝐺) ≥ 1, then the following hold:   (a) If 𝛿(𝐺) ≤ 0.3𝑛, then 𝑚 > 𝑛(𝑛 − 2)/4 .   (b) If 𝑛 ≥ 2000 and 𝛿(𝐺) ≤ 0.321𝑛, then 𝑚 > 𝑛(𝑛 − 2)/4 .

Section 13.5. Total Domination Edge-Critical Graphs

407

Since 𝛿(𝐺) = 𝑛 − 1 − Δ( 𝐺), Theorem 13.58 and the link between diameter-2critical graphs and 3t -edge-critical graphs proves the Murty-Simon Conjecture for diameter-2-critical graphs with sufficiently large maximum degree Δ(𝐺). Theorem 13.59 ([430]) If 𝐺 is a diameter-2-critical graph of order 𝑛 and size 𝑚, then the following hold:   (a) If Δ(𝐺) ≥ 0.7𝑛, then 𝑚 < 𝑛2 /4 .   (b) If 𝑛 ≥ 2000 and Δ(𝐺) ≥ 0.6787𝑛, then 𝑚 < 𝑛2 /4 . A summary of the graph families for which Conjecture 13.56 holds is given in Table 13.2. Thus, the Murty-Simon Conjecture has been verified for a number of infinite families of graphs, namely, the complements of the graphs listed in Table 13.2, where by a “connectivity-𝑘 graph” we mean a graph with vertex-connectivity 𝑘. More details on these results can be found in the respective references and in the 2015 survey by Haynes et al. [431].

Graph family

Reference

Diameter-3 graphs Connectivity-1 graphs Connectivity-2 graphs Connectivity-3 graphs Claw-free graphs 𝐶4 -free graphs Bull-free graphs Diamond-free graphs House-free graphs

[429, 431, 743] [429, 721] [431, 433, 743] [431, 433, 743] [432] [425] [50] [424] [426]

Table 13.2 Graph families for which Conjecture 13.56 holds

As we have seen, Conjecture 13.56 is true for the 3t -edge-critical graphs of diameter 3. Hence, all that remains is to prove the conjecture for 3t -edge-critical graphs of diameter 2. In 2015 Balbuena et al. [50] conjectured that Conjecture 13.56 can in fact be strengthened for that family of graphs. Conjecture 13.60 ([50]) If 𝐺 is a 3t -edge-critical graph of order 𝑛 and size 𝑚 with diam(𝐺) = 2, then 𝑚 ≥ (𝑛2 − 4)/4 . We remark that proving Conjecture 13.60 would prove both Conjecture 13.56 and the Murty-Simon Conjecture. It was proved in [50] that Conjecture 13.60 holds for a family H of graphs defined as follows. A pair of nonadjacent vertices 𝑢 and 𝑣 is called a dominating pair of 𝐺 if {𝑢, 𝑣} is a dominating set of 𝐺. Let H be the subset of 3t -edge-critical graphs of diameter 2 for which every pair of nonadjacent vertices is a dominating pair. It turns out that family H is precisely the family of bull-free, 3t -edge-critical graphs.

Chapter 13. Domination Critical and Stable Graphs

408

In order to characterize the extremal graphs in H , Balbuena et al. [50] defined a subfamily of H as follows. Let 𝐶5 (𝑛1 , 𝑛2 , 𝑛3 , 𝑛4 , 𝑛5 ) denote the graph that can be obtained from a 5-cycle 𝑥1 𝑥2 𝑥3 𝑥 4 𝑥5 𝑥1 by replacing each vertex 𝑥𝑖 with a nonempty clique 𝑋𝑖 , where |𝑋𝑖 | = 𝑛𝑖 ≥ 1 and 𝑖 ∈ [5], and adding all edges between 𝑋𝑖 and 𝑋𝑖+1 , where addition is taken modulo 5. For 𝑛 ≥ 5, let H𝑛 be the subfamily   of H defined by H𝑛 = 𝐶5 (𝑛1 , 𝑛2 , 𝑛3 , 𝑛4 , 𝑛5 ) : 𝑛1 = 𝑛3 = 1, 𝑛2 = (𝑛 − 3)/2 or 𝑛2 =   (𝑛 − 3)/2 and 𝑛 = 2 + 𝑛2 + 𝑛4 + 𝑛5 . As illustrated in [50], Figure 13.20 gives a schematic of graphs in the family H𝑛 , where in this diagram each 𝑋𝑖 represents the clique replacing 𝑥𝑖 , all edges exist between the vertices of 𝑋4 and 𝑋5 , the vertex 𝑥1 dominates 𝑋2 ∪ 𝑋5 , and the vertex 𝑥 3 dominates 𝑋2 ∪ 𝑋4 .

𝑋4

···

𝑋5 ···

···

𝑥3

𝑥1

·

··

·

·· 𝑋2

Figure 13.20 A graph in the family H𝑛 The following result shows that Conjecture 13.60 holds for the graphs in H and that the graphs H𝑛 are, in fact, the extremal ones in this family.   Theorem 13.61 ([50]) If 𝐺 ∈ H has order 𝑛 and size 𝑚, then 𝑚 ≥ (𝑛2 − 4)/4 , with equality if and only if 𝐺 ∈ H𝑛 . As an immediate consequence of Theorem 13.61, Conjecture 13.60, and hence Conjecture 13.56 holds for the graphs in family H . In other words, the Murty-Simon Conjecture is true for the diameter-2-critical graphs whose complements are in H . Recognizing that it was not necessary for every pair of nonadjacent vertices to be a dominating pair (as required in family H ) to prove Conjecture 13.60, Balbuena et al. [50] relaxed the condition to only requiring “many dominating pairs” as follows. Theorem 13.62 ([50]) Let 𝐺 be a 3t -edge-critical graph  2
 of order 𝑛 > 5 and size 𝑚 with diam(𝐺) = 2. If 𝐺 has fewer than 𝑛 / 4(𝑛 − 5) non-dominating pairs, then   𝑚 > 𝑛(𝑛 − 2)/4 . The second conjecture posed in [50] predicts that the 3t -edge-critical graphs of sufficiently large order that achieve equality in the lower bound of Conjecture 13.60 belong to the family H𝑛 .

Section 13.5. Total Domination Edge-Critical Graphs

409

Conjecture 13.63 ([50]) If 𝐺 is a 3t -edge-critical graph   of size 𝑚 and sufficiently large order 𝑛 with diam(𝐺) = 2, then 𝑚 ≥ (𝑛2 − 4)/4 , with equality if and only if 𝐺 ∈ H𝑛 . Dailly et al. [210] gave evidence suggesting that the bound of Conjecture 13.56 can be improved for 3t -edge-critical graphs of diameter 3 as well. In fact, they speculated that with one exception, the bound of Conjecture 13.60 can be extended to these graphs. Let 𝐹6 be the 3t -edge-critical graph of diameter 3 shown in Figure 13.18(b). Conjecture 13.64 ([210]) If 𝐺 ≠ 𝐹6 is a 3t -edge-critical graph of order 𝑛 and size 𝑚, then 𝑚 ≥ (𝑛2 − 4)/4 . They further conjectured that equality in the bound of Conjecture 13.64 holds if and only if 𝐺 ∈ H𝑛 or 𝐺 is one of thirteen special small graphs given in [210]. If this is true, then Conjecture 13.63 follows.

Chapter 14

Upper Domination Parameters 14.1

Introduction

In this chapter, we study the upper core domination parameters, namely the upper domination number, the upper total domination number, and the independence number. The independence number is very well-studied in the literature. Therefore, our focus is mainly on the upper domination and upper total domination numbers, although we do present several important results on the independence number. We first review some definitions from Chapter 2. A minimal dominating set in a graph 𝐺 is a dominating set that contains no dominating set of 𝐺 as a proper subset, and a minimal TD-set in 𝐺 is a TD-set that contains no TD-set of 𝐺 as a proper subset. A graph property is hereditary if the property is preserved when taking subsets, that is, if a set 𝑆 in a graph 𝐺 has a hereditary property P, then every subset of 𝑆 also has the property P. For example, the property that a set is independent is a hereditary property. However, the property that a set is a dominating set is not hereditary, since the removal of a vertex from a dominating set in a graph 𝐺 does not necessarily result in a dominating set of 𝐺. A graph property P is superhereditary if this property is preserved when taking supersets, that is, if a set 𝑆 in a graph 𝐺 has a superhereditary property P, then every superset of 𝑆 has the property P. As observed in Chapter 2, the property of being a dominating set is superhereditary. Thus, every superset of a dominating set is itself a dominating set of the graph. This means that a dominating set 𝐷 is minimal if and only if for every vertex 𝑣 ∈ 𝐷, the set 𝐷 \ {𝑣} is not a dominating set. Thus, it suffices, in order to show minimality of a dominating set 𝐷, to consider each vertex 𝑣 ∈ 𝐷 rather than all possible nonempty subsets of 𝐷. Similarly, the property of being a TD-set is superhereditary, implying that a TD-set 𝐷 is minimal if and only if for every vertex 𝑣 ∈ 𝐷, the set 𝐷 \ {𝑣} is not a TD-set. The upper domination number Γ(𝐺) of a graph 𝐺 is the maximum cardinality of a minimal dominating set in 𝐺. Similarly, the upper total domination number Γt (𝐺) of a graph 𝐺 is the maximum cardinality of a minimal TD-set in 𝐺. The independence © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_14

411

Chapter 14. Upper Domination Parameters

412

number 𝛼(𝐺) of 𝐺 is the maximum cardinality of an independent set of vertices in 𝐺. We call a minimal dominating set of cardinality Γ(𝐺) a Γ-set of 𝐺, and a minimal TD-set of cardinality Γt (𝐺) a Γt -set of 𝐺. An independent set of cardinality 𝛼(𝐺) is called an 𝛼-set of 𝐺. The following lemmas, which are also given in Chapters 4 and 6, characterize minimal dominating sets and minimal TD-sets, respectively, in a graph. Lemma 14.1 ([622]) A dominating set 𝑆 in a graph 𝐺 is a minimal dominating set of 𝐺 if and only if ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅ for every vertex 𝑣 ∈ 𝑆. Lemma 14.2 ([622]) A TD-set 𝑆 in a graph 𝐺 is a minimal TD-set in 𝐺 if and only if ipn(𝑣, 𝑆) ≠ ∅ or epn(𝑣, 𝑆) ≠ ∅ for every vertex 𝑣 ∈ 𝑆. If a graph 𝐺 contains a vertex 𝑣 that is not isolated, then 𝑉 \ {𝑣} is a dominating set in 𝐺, implying that the set 𝑉 is not a minimal dominating set. Hence, we have the following observation. Observation 14.3 If 𝐺 is a graph of order 𝑛 with Δ(𝐺) ≥ 1, then Γ(𝐺) ≤ 𝑛 − 1. That the trivial upper bound in Observation 14.3 is tight may be seen by taking 𝐺 to be a star 𝐾1,𝑛−1 , where 𝑛 ≥ 3. The set of 𝑛 − 1 leaves in the star forms a minimal dominating set in 𝐺 and so Γ(𝐺) ≥ 𝑛 − 1. Hence, by Observation 14.3, Γ(𝐺) = 𝑛 − 1. A similar observation may be made for the upper total domination number in an isolate-free graph with maximum degree at least 2. If 𝐺 is such a graph, then 𝐺 necessarily contains a vertex 𝑣 such that no neighbor of 𝑣 has degree 1. Further, 𝑉 \ {𝑣} is a TD-set in 𝐺, implying that the set 𝑉 is not a minimal TD-set. Hence, we have the following observation. Observation 14.4 If 𝐺 is an isolate-free graph of order 𝑛 with Δ(𝐺) ≥ 2, then Γt (𝐺) ≤ 𝑛 − 1. Again, the trivial upper bound in Observation 14.4 is tight, as may be seen by taking 𝐺 to be the graph obtained by subdividing every edge in a star 𝐾1,𝑘 , where 𝑘 ≥ 2, exactly once. The resulting graph 𝐺 has order 𝑛 = 2𝑘 + 1 ≥ 5. Since the set of 𝑘 leaves and 𝑘 support vertices forms a minimal TD-set in 𝐺, Γt (𝐺) ≥ 2𝑘 = 𝑛 − 1. Hence, by Observation 14.4, Γt (𝐺) = 𝑛 − 1.

14.2

Upper Bounds

In this section, we discuss upper bounds on the upper domination and upper total domination numbers. The examples that achieve tightness in Observations 14.3 and 14.4 have minimum degree equal to 1. Further, they are highly non-regular, in the sense that their minimum and maximum degrees differ significantly. They also have certain structural properties, such as having a claw, that is, an induced copy of 𝐾1,3 . Hence, it is a natural question to ask if the trivial upper bounds in Observations 14.3 and 14.4 can be improved if we impose certain conditions on a graph such as being regular, having a given minimum degree, or being claw-free. We consider some of these questions in this section.

Section 14.2. Upper Bounds

413

14.2.1 Upper Bounds in Terms of Minimum Degree In this section, we show that even if the minimum degree of a graph is large, both the upper domination and upper total domination numbers can be arbitrarily large. We first consider the upper domination number. The following general upper bound on the (independence number and) upper domination number of a general graph was proved in 1988 by Favaron [274]. We remark that a more general result was proven, showing that the upper bound in Theorem 14.5 also holds for the upper irredundance number IR(𝐺), that is, the maximum cardinality of an irredundant set in 𝐺. Theorem 14.5 ([274]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 1, then Γ(𝐺) ≤ 𝑛 − 𝛿(𝐺), and this bound is tight. Proof Let 𝐷 be an arbitrary set of vertices of 𝐺 such that |𝐷| = 𝑛 − 𝛿(𝐺). Let 𝐷 be the complement of the set 𝐷, and so 𝐷 = 𝑉 \ 𝐷 and |𝐷| = 𝑛 − |𝐷 | = 𝛿(𝐺). Every vertex in 𝐷 has at most 𝛿(𝐺) − 1 neighbors that belong to the set 𝐷, and therefore at least one neighbor that belongs to the set 𝐷, implying that the set 𝐷 is a dominating set of 𝐺. This implies that no minimal dominating set of 𝐺 has cardinality greater than 𝑛 − 𝛿(𝐺). Therefore, Γ(𝐺) ≤ 𝑛 − 𝛿(𝐺). That this bound is tight may be seen by taking 𝐺 as the complete graph 𝐾𝑛 , where 𝑛 ≥ 2. In this case, Γ(𝐺) = 1 = 𝑛 − 𝛿(𝐺). Moreover, for any fixed 𝛿 ≥ 1 and for 𝑥 ≥ 2𝛿 an arbitrary integer, let 𝐺 be the complete bipartite graph 𝐾 𝑥, 𝛿 with partite sets 𝑋 and 𝑌 , where |𝑋 | = 𝑥. The graph 𝐺 has order 𝑛 = 𝑥 + 𝛿 with minimum degree 𝛿(𝐺) = 𝛿. Since the set 𝑋 is a minimal dominating set of 𝐺, Γ(𝐺) ≥ |𝑋 | = 𝑛 − 𝛿. Consequently, Γ(𝐺) = 𝑛 − 𝛿(𝐺). Hence, there exist infinitely many connected graphs 𝐺 of order 𝑛 with minimum degree 𝛿 and Γ(𝐺) = 𝑛 − 𝛿. We consider next the upper total domination number. Theorem 14.6 If 𝐺 is a connected graph of order 𝑛 and 𝛿(𝐺) ≥ 2, then Γt (𝐺) ≤ 𝑛 − 𝛿(𝐺) + 1, and this bound is tight. Proof Let 𝐷 be an arbitrary set of vertices of 𝐺 such that |𝐷| = 𝑛 − 𝛿(𝐺) + 1, and let 𝐷 = 𝑉 \ 𝐷 and |𝐷| = 𝑛 − |𝐷| = 𝛿(𝐺) − 1. Every vertex in 𝐺 has at most 𝛿(𝐺) − 1 neighbors that belong to the set 𝐷, and therefore at least one neighbor that belongs to the set 𝐷, implying that the set 𝐷 is a TD-set of 𝐺. This implies that no minimal TD-set of 𝐺 has cardinality greater than 𝑛 − 𝛿(𝐺) + 1. Therefore, Γt (𝐺) ≤ 𝑛 − 𝛿(𝐺) + 1. That this bound is tight may be seen by taking 𝐺 as the complete graph 𝐾𝑛 , where 𝑛 ≥ 3. In this case, Γt (𝐺) = 2 = 𝑛 − 𝛿(𝐺) + 1. Moreover, for any fixed 𝛿 ≥ 1 and for 𝑥 ≥ 2𝛿 an arbitrary integer, let 𝐺 𝑥 be obtained from a complete bipartite graph 𝐾2𝑥, 𝛿−1 with partite sets 𝑋 and 𝑌 , where |𝑋 | = 2𝑥, by adding a perfect matching between the vertices of 𝑋. We note that 𝐺 𝑥 [𝑋] = 𝑥𝐾2 . The resulting graph 𝐺 𝑥 is a connected graph of order 𝑛 = 2𝑥 + 𝛿 − 1 with minimum degree 𝛿(𝐺) = 𝛿. Since the set 𝑋 is a minimal TD-set of 𝐺, Γt (𝐺) ≥ |𝑋 | = 2𝑥 = 𝑛 − 𝛿(𝐺) + 1. Consequently, Γt (𝐺) = 𝑛 − 𝛿(𝐺) + 1. Hence, there exist infinitely many connected graphs 𝐺 of order 𝑛, minimum degree 𝛿, and Γt (𝐺) = 𝑛 − 𝛿 + 1.

414

14.2.2

Chapter 14. Upper Domination Parameters

Upper Bounds in Regular Graphs

Recall from Theorem 6.89 in Chapter 6 that the independence number of a regular graph is at most one-half its order, that is, if 𝐺 is an isolate-free regular graph of order 𝑛, then 𝛼(𝐺) ≤ 21 𝑛. This bound is tight, as may be seen by considering regular graphs in which every component is a balanced complete bipartite graph. In this section, we show that if we impose a regularity condition on the graph, then the upper bounds of Theorems 14.5 and 14.6 can be significantly improved. Lemma 14.7 For 𝑘 and 𝑛 positive integers, there exists a (𝑘 − 1)-regular graph of order 𝑛 if and only if 𝑛 ≥ 𝑘 where 𝑛 is even whenever 𝑘 is even. Proof Since every graph has an even number of vertices of odd degree, every (𝑘 − 1)-regular graph on 𝑛 vertices satisfies 𝑛 ≥ 𝑘 with 𝑛 even whenever 𝑘 is even. To prove the converse, we note firstly that given any positive integer 𝑛, the empty graph on 𝑛 vertices is 0-regular, and given any positive even integer 𝑛, the graph comprising 𝑛/2 copies of 𝐾2 is 1-regular. Hence, to prove the converse, it suffices to show that given any two positive integers 𝑘 and 𝑛, where 𝑛 ≥ 𝑘 ≥ 3 and 𝑛 is even whenever 𝑘 is even, there always exist (𝑘 − 1)-regular graphs on 𝑛 vertices. For 𝑘 ≥ 4 even and 𝑛 ≥ 𝑘 with 𝑛 even, let 𝐿 𝑘,𝑛 = 1, 2, . . . , 12 𝑘 − 1, 12 𝑛 ; while for  𝑘 ≥ 3 odd and 𝑛 ≥ 𝑘, let 𝐿 𝑘,𝑛 = 1, 2, . . . , 12 (𝑘 − 1) . In both cases, the circulant graph 𝐶𝑛 ⟨𝐿 𝑘,𝑛 ⟩ is a (𝑘 − 1)-regular graph of order 𝑛, implying that for every two positive integers 𝑘 and 𝑛, where 𝑛 ≥ 𝑘 and 𝑛 is even whenever 𝑘 is even, there always exist (𝑘 − 1)-regular graphs on 𝑛 vertices. For example, the circulant graph 𝐶10 ⟨𝐿 6,10 ⟩ = 𝐶10 ⟨1, 2, 5⟩, shown in Figure 14.1, is a 5-regular graph of order 𝑛 = 10.

Figure 14.1 The circulant graph 𝐶10 ⟨1, 2, 5⟩ Let Breg be the family of connected bipartite regular graphs and consider a graph 𝐺 ∈ Breg . Let 𝐺 have partite sets 𝑉1 and 𝑉2 . We note that |𝑉1 | = |𝑉2 |. The set 𝑉1 is a minimal dominating set in 𝐺, implying that Γ(𝐺) ≥ |𝑉1 | = 12 |𝑉 (𝐺)|. A family of connected regular graphs Freg was constructed in 2013 by Southey and Henning [687] as follows. Let 𝑘 ≥ 1 and ℓ ≥ 𝑘 be arbitrary fixed integers, where ℓ is even whenever 𝑘 is even. By Lemma 14.7, there exist (𝑘 − 1)-regular graphs on ℓ vertices. Let 𝐹1 and 𝐹2 be disjoint (𝑘 − 1)-regular graphs (not necessarily connected) on ℓ vertices with 𝑉 (𝐹1 ) = {𝑢 1 , 𝑢 2 , . . . , 𝑢 ℓ } and 𝑉 (𝐹2 ) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 ℓ }. Let 𝐹 be the graph obtained from the disjoint union 𝐹1 ∪ 𝐹2 by joining 𝑢 𝑖 to 𝑣 𝑖 for each 𝑖 ∈ [ℓ]. The set 𝑉 (𝐹1 ) is a dominating set of 𝐹. Since the set 𝑉 (𝐹1 ) \ {𝑢 𝑖 } does not

Section 14.2. Upper Bounds

415

dominate the vertex 𝑣 𝑖 for every 𝑖 ∈ [ℓ], the set 𝑉 (𝐹1 ) is a minimal dominating set of 𝐹, implying that Γ(𝐹) ≥ |𝑉 (𝐹1 )| = 21 |𝑉 (𝐹)|. Let Freg be the family of all such graphs 𝐹 so constructed, with the additional property that 𝐹 is connected. We state the above observations formally as follows. Proposition 14.8 If 𝐺 ∈ Breg ∪ Freg has order 𝑛, then 𝐺 is a connected regular graph with Γ(𝐺) ≥ 12 𝑛. Southey and Henning [687] defined the concept of edge weighting functions on a dominating set as follows. Given a dominating set 𝑆 in a graph 𝐺 = (𝑉, 𝐸), an edge weight function of 𝑆 is the function 𝜓 𝑆 : 𝐸 → [0, 1] that assigns to each edge in 𝐺 [𝑆] and each edge in 𝐺 [𝑉 \ 𝑆] a weight of 0 and that assigns to each edge in [𝑆, 𝑉 \ 𝑆] a weight in (0, 1] in such a way that for each vertex 𝑣 ∈ 𝑉 \ 𝑆, the weight 1 is shared among the edges joining 𝑣 to 𝑆. Thus, ( 1 if 𝑒 ∈ [𝑆, 𝑉 \ 𝑆] joins a vertex 𝑣 ∈ 𝑉 \ 𝑆 to a vertex in 𝑆 𝜓 𝑆 (𝑒) = deg𝑆 (𝑣) 0 if 𝑒 ∈ [𝑆, 𝑆] ∪ [𝑉 \ 𝑆, 𝑉 \ 𝑆]. Further, a vertex weight function 𝜙 𝑆 was defined in [687] as a function that assigns to each vertex 𝑣 ∈ 𝑉 the sum of the weights of the edges incident with 𝑣. Recall that every edge in 𝐺 [𝑉 \ 𝑆] has weight 0 and for every vertex 𝑣 ∈ 𝑉 \ 𝑆 the weight 1 is shared among the edges joining 𝑣 to 𝑆. Thus, since 𝑆 is a dominating set in 𝐺, for every vertex 𝑣 ∈ 𝑉 \ 𝑆 the sum of the weights of the edges incident with 𝑣 is 1, that is, 𝜙 𝑆 (𝑣) = 1 for every vertex 𝑣 ∈ 𝑉 \ 𝑆. We define the vertex weight sum 𝜉 (𝑆) to be the sum over all vertices in 𝑆 of the weights assigned by 𝜙 𝑆 . Since every edge in 𝐺 [𝑆] and every edge in 𝐺 [𝑉 \ 𝑆] has weight 0, the sum of the weights of the vertices in 𝑆 is precisely the sum of the weights of all edges in 𝐺, that is, ∑︁ ∑︁ ∑︁ 𝜉 (𝑆) = 𝜙 𝑆 (𝑣) = 𝜓 𝑆 (𝑒) = 𝜙 𝑆 (𝑣) = |𝑉 \ 𝑆| = 𝑛 − |𝑆|. (14.1) 𝑣 ∈𝑆

𝑣 ∈𝑉\𝑆

𝑒∈𝐸

We are now in a position to establish the following bound on the upper domination number of a regular graph. Theorem 14.9 ([687]) If 𝐺 is an 𝑟-regular graph, for 𝑟 ≥ 1, of order 𝑛, then Γ(𝐺) ≤ 12 𝑛, with equality if and only if every component of 𝐺 belongs to the family Breg ∪ Freg . Proof Let 𝐺 be a 𝑟-regular graph of order 𝑛, where 𝑟 ≥ 1, and let 𝑆 be a Γ-set of 𝐺. We use the edge weight function 𝜓 𝑆 and vertex weight function 𝜙 𝑆 to count the number of vertices in 𝑆 in terms of the order 𝑛. Recall that if 𝑒 ∈ [𝑆, 𝑉 \ 𝑆] joins a vertex 𝑣 ∈ 𝑉 \ 𝑆 to a vertex in 𝑆, then 𝜓 𝑆 (𝑒) = 1/deg𝑆 (𝑣). Since 𝑆 is a dominating set and 𝐺 is an 𝑟-regular graph, deg𝑆 (𝑣) ∈ [𝑟]. Therefore, 1 𝑟

≤ 𝜓 𝑆 (𝑒) ≤ 1.

(14.2)

We show next that 𝜙 𝑆 (𝑣) ≥ 1 for every vertex 𝑣 ∈ 𝑆. Let 𝐴 be the set of isolated vertices in 𝐺 [𝑆] and let 𝐵 = 𝑆 \ 𝐴. Each vertex 𝑣 ∈ 𝐴 is incident with 𝑟 edges, each

Chapter 14. Upper Domination Parameters

416

of which joins 𝑣 to vertices in 𝑉 \ 𝑆. Hence, if 𝐸 𝑣 denotes the set of 𝑟 edges incident with 𝑣, then by Inequality (14.2), ∑︁ ∑︁ 1 1 𝜓 𝑆 (𝑒) ≥ 𝜙 𝑆 (𝑣) = (14.3) 𝑟 = 𝑟 × 𝑟 = 1. 𝑒∈𝐸𝑣

𝑒∈𝐸𝑣

We next consider vertices that belong to the set 𝐵 ⊆ 𝑆. Since the graph 𝐺 [𝐵] is isolate-free, ipn[𝑣, 𝑆] ≠ {𝑣} for every vertex 𝑣 ∈ 𝐵. Therefore, since ipn[𝑣, 𝑆] ∈  ∅, {𝑣} for every vertex 𝑣 ∈ 𝑆, we must have ipn[𝑣, 𝑆] = ∅. Hence, by Lemma 14.1, epn[𝑣, 𝑆] ≠ ∅ for every vertex 𝑣 ∈ 𝐵. We note that every edge that joins a vertex 𝑣 ∈ 𝐵 to a vertex in epn[𝑣, 𝑆] is assigned weight 1 under the function 𝜓 𝑆 , implying that 𝜙 𝑆 (𝑣) ≥ |epn[𝑣, 𝑆] | ≥ 1 (14.4) for every vertex 𝑣 ∈ 𝐵. By Inequalities (14.3) and (14.4), we have 𝜙 𝑆 (𝑣) ≥ 1 for each 𝑣 ∈ 𝑆, implying that ∑︁ 𝜉 (𝑆) = 𝜙 𝑆 (𝑣) ≥ |𝑆|. (14.5) 𝑣 ∈𝑆

Hence, by Equation (14.1) and Inequality (14.5), we have 𝑛 − |𝑆| = 𝜉 (𝑆) ≥ |𝑆| and so Γ(𝐺) = |𝑆| ≤ 12 𝑛. This establishes the desired upper bound in the statement of the theorem. Conversely, suppose that 𝐺 is a connected 𝑟-regular graph on 𝑛 vertices, where 𝑟 ≥ 1, with Γ(𝐺) = 12 𝑛. We show that 𝐺 ∈ Breg ∪ Freg . If 𝑟 = 1, then 𝐺 = 𝐾2 ∈ Breg . Hence, we may assume that 𝑟 ≥ 2. Let 𝑆 be a Γ-set of 𝐺 and let 𝑆 = 𝑉 \ 𝑆. By supposition, |𝑆| = Γ(𝐺) = 12 𝑛. Thus, Inequality (14.5) is an equality, which in turn implies there is equality throughout the Inequalities (14.3) and (14.4). In particular, 𝜙 𝑆 (𝑣) = 1 for every vertex 𝑣 ∈ 𝑆. Let 𝑆1 ⊆ 𝑆 be the set of vertices 𝑣 ∈ 𝑆 that are isolated in 𝐺 [𝑆] and such that every neighbor of 𝑣 is isolated in 𝐺 [𝑆]. Let 𝑆2 ⊆ 𝑆 be the set of vertices 𝑣 ∈ 𝑆 that have precisely one neighbor 𝑣 ′ in 𝑆 and this neighbor is its 𝑆-external private neighbor, that is, deg𝑆 (𝑣) = 1 and epn[𝑣, 𝑆] = {𝑣 ′ }. Since 𝑟 ≥ 2, 𝑆1 ∩ 𝑆2 = ∅. We proceed further with the following three claims. Claim 14.9.1 Every isolated vertex in 𝐺 [𝑆] belongs to the set 𝑆1 , while every non-isolated vertex in 𝐺 [𝑆] belongs to the set 𝑆2 . Proof Let 𝑣 ∈ 𝑆 and let N𝐺 (𝑣) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑟 }. On the one hand, suppose that 𝑣 is isolated in 𝐺 [𝑆] and so N𝐺 (𝑣) ⊆ 𝑆. In this case, since 𝜓 𝑆 (𝑒) ≥ 𝑟1 for every edge 𝑒 ∈ [𝑆, 𝑆], 𝑟 𝑟 ∑︁ ∑︁ 1 (14.6) 1 = 𝜙 𝑆 (𝑣) = 𝜓 𝑆 (𝑣𝑣 𝑖 ) ≥ 𝑟 = 1. 𝑖=1

𝑖=1

Hence, Inequality (14.6) must be an equality, implying that 𝜓 𝑆 (𝑣𝑣 𝑖 ) = 𝑟1 for all 𝑖 ∈ [𝑟]. This in turn implies that all 𝑟 neighbors of 𝑣 𝑖 belong to the set 𝑆, that is, N𝐺 (𝑣 𝑖 ) ⊆ 𝑆, and so 𝑣 𝑖 is isolated in 𝐺 [𝑆] for all 𝑖 ∈ [𝑟]. Therefore, 𝑣 ∈ 𝑆1 . Suppose, on the other hand, that 𝑣 is not isolated in 𝐺 [𝑆]. Thus, ipn[𝑣, 𝑆] ≠ {𝑣} and consequently, ipn[𝑣, 𝑆] = ∅. Hence, by Lemma 14.1, we have epn[𝑣, 𝑆] ≠ ∅.

Section 14.2. Upper Bounds

417

Renaming neighbors of 𝑣 if necessary, we may assume 𝑣 𝑟 ∈ epn[𝑣, 𝑆], and so 𝜓 𝑆 (𝑣𝑣 𝑟 ) = 1. Thus, since 𝜓 𝑆 (𝑒) ≥ 𝑟1 for every edge 𝑒 ∈ [𝑆, 𝑆], 1 = 𝜙 𝑆 (𝑣) =

𝑟 ∑︁

𝜓 𝑆 (𝑣𝑣 𝑖 ) ≥ 1 +

𝑖=1

𝑟 −1 ∑︁

𝜓 𝑆 (𝑣𝑣 𝑖 ) ≥ 1.

(14.7)

𝑖=1

Hence, Inequalities (14.6) and (14.7) must be equalities, implying that 𝜓 𝑆 (𝑣𝑣 𝑖 ) = 0 for all 𝑖 ∈ [𝑟 − 1]. This in turn implies that 𝑣 𝑖 ∈ 𝑆 for all 𝑖 ∈ [𝑟 − 1] and therefore that 𝑣 ∈ 𝑆2 . Claim 14.9.2

𝑆 = 𝑆1 ∪ 𝑆2 .

Proof By Claim 14.9.1, every vertex 𝑣 ∈ 𝑆 belongs to either the set 𝑆1 or the set 𝑆2 , implying that 𝑆 ⊆ 𝑆1 ∪ 𝑆2 . By definition of the sets 𝑆1 and 𝑆2 , 𝑆1 ∪ 𝑆2 ⊆ 𝑆. Consequently, 𝑆 = 𝑆1 ∪ 𝑆2 . Claim 14.9.3 Either 𝑆 = 𝑆1 or 𝑆 = 𝑆2 . Proof By Claim 14.9.2, 𝑆 = 𝑆1 ∪ 𝑆2 . As observed earlier, 𝑆1 ∩ 𝑆2 = ∅. Suppose, to the contrary, that 𝑆1 ≠ ∅ and 𝑆2 ≠ ∅. Since 𝐺 is a connected graph, every vertex in 𝑆1 is connected to a vertex in 𝑆2 by some path. Among all vertices in 𝑆1 and 𝑆2 , let 𝑤 ∈ 𝑆1 and 𝑥 ∈ 𝑆2 be chosen so that 𝑑𝐺 (𝑤, 𝑥) is a minimum. Let 𝑃 : 𝑤 1 𝑤 2 . . . 𝑤 𝑡 be a shortest path from 𝑤 to 𝑥 in 𝐺, where 𝑤 = 𝑤 1 and 𝑥 = 𝑤 𝑡 . By the definition of the set 𝑆1 , every neighbor of 𝑤 1 is an isolated vertex in 𝐺 [𝑆]. In particular, 𝑤 2 is an isolated vertex in 𝐺 [𝑆] and so 𝑤 3 ∈ 𝑆. If 𝑤 3 ∈ 𝑆2 , then by the definition of the set 𝑆2 , the vertex 𝑤 3 has precisely one neighbor in 𝑆 and this neighbor is its 𝑆-external private neighbor. Since 𝑤 2 is a neighbor of 𝑤 3 in 𝑆, this unique neighbor of 𝑤 3 in 𝑆 must be 𝑤 2 . However, 𝑤 2 is a common neighbor of at least two vertices in 𝑆, namely 𝑤 1 and 𝑤 3 , and therefore is not an 𝑆-external private neighbor, a contradiction. Hence, 𝑤 3 ∈ 𝑆1 . But then 𝑑𝐺 (𝑤 3 , 𝑤 𝑡 ) = 𝑡 − 2 < 𝑡 = 𝑑𝐺 (𝑤 1 , 𝑥), contradicting our choice of 𝑤 and 𝑥. Therefore, either 𝑆 = 𝑆1 or 𝑆 = 𝑆2 . By Claim 14.9.3, either 𝑆 = 𝑆1 or 𝑆 = 𝑆2 . If 𝑆 = 𝑆1 , then by the definition of 𝑆1 , every vertex in 𝑆 is isolated in 𝐺 [𝑆] and every vertex in 𝑆 is isolated in 𝐺 [𝑆]. Therefore, 𝐺 is a connected 𝑟-regular bipartite graph with partite sets 𝑆 and 𝑆, and so 𝐺 ∈ Breg . Hence, we may assume that 𝑆 = 𝑆2 , for otherwise the desired result follows. Let 𝐹1 = 𝐺 [𝑆] and let 𝐹2 = 𝐺 [𝑆]. Let 𝑆 = {𝑢 1 , 𝑢 2 , . . . , 𝑢 ℓ }, where ℓ = |𝑆| = 12 𝑛. By the definition of 𝑆2 , the vertex 𝑢 𝑖 has exactly one neighbor 𝑣 𝑖 in 𝑆 and this neighbor is an 𝑆-external private neighbor. Thus, 𝑉 (𝐹2 ) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 ℓ }. Furthermore, 𝐺 [𝑆] and 𝐺 [𝑆] are disjoint (𝑘 − 1)-regular graphs and 𝐺 is the connected graph obtained from the disjoint union 𝐺 [𝑆] ∪ 𝐺 [𝑆] by joining 𝑢 𝑖 to 𝑣 𝑖 for each 𝑖 ∈ [ℓ]. But this is precisely the definition of a graph in the family Freg . We conclude that 𝐺 ∈ Freg , as desired. As an immediate consequence of Proposition 14.8, Theorem 14.9, and the proof of Theorem 14.9, we have the following result.

Chapter 14. Upper Domination Parameters

418

Corollary 14.10 If 𝐺 is an 𝑟-regular graph, for 𝑟 ≥ 1, of order 𝑛, then 𝛼(𝐺) ≤ 21 𝑛, with equality if and only if every component of 𝐺 belongs to the family Breg . We consider next an upper bound on the upper total domination number of a regular graph. Theorem 14.11 ([687]) If 𝐺 is an 𝑟-regular graph, for 𝑟 ≥ 1, of order 𝑛, then Γt (𝐺) ≤

𝑛 . 2−

1 𝑟

Proof Let 𝐺 be an 𝑟-regular graph of order 𝑛, where 𝑟 ≥ 1, and let 𝑆 be a Γt -set of 𝐺. We use the edge weight function 𝜓 𝑆 and vertex weight function 𝜙 𝑆 to count the number of vertices in 𝑆 in terms of the order 𝑛. We show that, on average, 𝜙 𝑆 (𝑣) ≥ 1 − 𝑟1 for every vertex 𝑣 ∈ 𝑆. Let 𝐶 be an arbitrary component of 𝐺 [𝑆]. Since 𝑆 is a TD-set in 𝐺, every vertex in 𝑆 has at least one neighbor in 𝑆, and so |𝑉 (𝐶)| ≥ 2. Suppose that |𝑉 (𝐶)| = 2. Since 𝐺 is 𝑟-regular, each vertex in 𝐶 has 𝑟 − 1 edges to 𝑉 \ 𝑆, each of weight at least 1/𝑟, implying that 𝜙 𝑆 (𝑣) ≥ (𝑟 − 1)/𝑟 for each vertex 𝑣 ∈ 𝑉 (𝐶). Hence, we may assume that |𝑉 (𝐶)| ≥ 3. Let 𝐴 be the set of vertices of degree 1 in 𝐶 and let 𝐵 = N𝐶 ( 𝐴). Since |𝑉 (𝐶)| ≥ 3, the set 𝐴 is independent. Every vertex in 𝐵 has at least one neighbor in 𝐴, and no two vertices in 𝐵 have a common neighbor in 𝐴, implying that | 𝐴| ≥ |𝐵|. Since every vertex in 𝐵 has degree at least 2 in 𝐶, ipn(𝑣, 𝑆) = ∅ for every vertex 𝑣 ∈ 𝐴. Hence, by Lemma 14.2, |epn(𝑣, 𝑆)| ≥ 1 for every 𝑣 ∈ 𝐴. Every edge that joins a vertex 𝑣 ∈ 𝐴 to one of its 𝑆-external private neighbors has weight 1. Thus, since a vertex 𝑣 ∈ 𝐴 has 𝑟 − 1 neighbors in 𝑉 \ 𝑆, and since each edge joining 𝑣 to one of these neighbors in 𝑉 \ 𝑆 that is not an 𝑆-external private neighbor has weight at least 1/𝑟, we have 𝜙 𝑆 (𝑣) ≥ |epn(𝑣, 𝑆)| +

1 𝑟


2(𝑟 − 1) 𝑟 − 1 − |epn(𝑣, 𝑆)| ≥ . 𝑟

(14.8)

Furthermore, for each vertex 𝑣 ∈ 𝑉 (𝐶) \ ( 𝐴 ∪ 𝐵), we have ipn(𝑣, 𝑆) = ∅, which implies that epn(𝑣, 𝑆) ≠ ∅ and therefore that some edge incident with 𝑣 has weight 1. Hence, since 2| 𝐴| ≥ | 𝐴| + |𝐵|, ∑︁ ∑︁ ∑︁ 𝜙 𝑆 (𝑣) ≥ 𝜙 𝑆 (𝑣) + 𝜙 𝑆 (𝑣) 𝑣 ∈𝑉 (𝐶 )

𝑣∈ 𝐴

𝑣 ∈𝑉 (𝐶 )\( 𝐴∪𝐵)



 2(𝑟 − 1) ≥ | 𝐴| + |𝑉 (𝐶) \ ( 𝐴 ∪ 𝐵)| 𝑟    
𝑟 −1 𝑟 −1 ≥ | 𝐴| + |𝐵| + |𝑉 (𝐶) \ ( 𝐴 ∪ 𝐵)| 𝑟 𝑟   𝑟 −1 = |𝑉 (𝐶)|. 𝑟 Since 𝐶 is an arbitrary component of 𝐺 [𝑆], we have therefore shown that on average, 𝜙 𝑆 (𝑣) ≥ 1 − 𝑟1 for every vertex 𝑣 ∈ 𝑆. Hence, by Equation (14.1),

Section 14.2. Upper Bounds

419 

 𝑟 −1 𝑛 − |𝑆| = |𝑉 \ 𝑆| = 𝜙 𝑆 (𝑣) ≥ |𝑆|. 𝑟 𝑣 ∈𝑆

∑︁



1 Therefore, 𝑛 ≥ 1 + 𝑟 −1 𝑟 |𝑆|, and so Γt (𝐺) = |𝑆| ≤ 𝑛/ 2 − 𝑟 . This establishes the desired upper bound. The extremal graphs that achieve equality in the upper bound of Theorem 14.11 are characterized in [687]. This characterization is also given in [490].

14.2.3

Upper Bounds in Claw-free Graphs

For 𝑘 ≥ 2 a graph is 𝐾1,𝑘+1 -free if it does not contain the star 𝐾1,𝑘+1 as an induced subgraph. Recall that in Theorem 14.5, we showed that if 𝐺 is a connected graph of order 𝑛, then 𝛼(𝐺) ≤ Γ(𝐺) ≤ 𝑛 − 𝛿(𝐺). In 1992 Faudree et al. [272] showed that this upper bound on the independence number can be improved if the graph is 𝐾1,𝑘+1 -free for 𝑘 ≥ 2. Theorem 14.12 ([272]) For 𝑘 ≥ 2 an integer, if 𝐺 is a 𝐾1,𝑘+1 -free graph of order 𝑛 with minimum degree 𝛿, then 𝑘𝑛 𝛼(𝐺) ≤ . 𝛿+𝑘 Proof Let 𝐼 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑝 } be an 𝛼-set of 𝐺. Since 𝐺 is 𝐾1,𝑘+1 -free, every vertex outside the set 𝐼 has at most 𝑘 neighbors in the set 𝐼. Thus, counting edges in the set [𝐼, 𝑉 \ 𝐼] between vertices in 𝐼 and 𝑉 \ 𝐼, 𝑝×𝛿 ≤

𝑝 ∑︁

deg𝐺 (𝑣 𝑖 ) = | [𝐼, 𝑉 \ 𝐼] | ≤ 𝑘 × |𝑉 \ 𝐼 | = 𝑘 (𝑛 − 𝑝),

𝑖=1

implying that 𝛼(𝐺) = |𝐼 | = 𝑝 ≤ 𝑘𝑛/(𝛿 + 𝑘). In the special case when 𝐺 is claw-free, that is, when 𝑘 = 2, we have the 1990 result due to Li and Virlouvet [560]. Theorem 14.13 ([560]) If 𝐺 is a claw-free graph of order 𝑛 with minimum degree 𝛿, then 2𝑛 𝛼(𝐺) ≤ . 𝛿+2 In 1995 Ryjáček and Schiermeyer [660] established the following upper bound on the independence number of a 𝐾1,𝑘+1 -free graph for 𝑘 ≥ 2. Theorem 14.14 ([660]) For 𝑘 ≥ 2, if 𝐺 is a 𝐾1,𝑘+1 -free graph of order 𝑛 ≥ 2 and size 𝑚, then either 𝐺 = 𝐾𝑛 and 𝛼(𝐺) = 1 or 𝐺 ≠ 𝐾𝑛 and   √︁ 𝛼(𝐺) ≤ 12 2𝑛 + 2𝑘 − 1 − 8𝑚 + (2𝑘 − 1) 2 , and this bound is tight.

Chapter 14. Upper Domination Parameters

420

Proof Let 𝐺 be a connected 𝐾1,𝑘+1 -free graph of order 𝑛 ≥ 2 and size 𝑚. If 𝐺 = 𝐾𝑛 , then 𝛼(𝐺) = 1. Hence, we may assume that 𝐺 ≠ 𝐾𝑛 . Let 𝛼 = 𝛼(𝐺). Let 𝐼 be an 𝛼-set of 𝐺. Let 𝐼 = 𝑉 \ 𝐼, and so |𝐼 | = 𝑛 − |𝐼 | = 𝑛 − 𝛼. Since 𝐺 is 𝐾1,𝑘+1 -free, every vertex outside the set 𝐼 has at most 𝑘 neighbors in the set 𝐼, and so | [𝐼, 𝐼] | ≤ 𝑘 |𝐼 | = 𝑘 (𝑛 − 𝛼). Counting edges in 𝐺,

𝑚 ≤ 𝑚 𝐺 [𝐼] + | [𝐼, 𝐼] | + 𝑚 𝐺 [𝐼]
≤ 0 + 𝑘 (𝑛 − 𝛼) + 𝑛−2 𝛼 = 12 (𝑛 − 𝛼) (𝑛 − 𝛼 + 2𝑘 − 1) or equivalently, 𝛼2 − (2𝑛 + 2𝑘 − 1)𝛼 + 𝑛2 + (2𝑘 − 1)𝑛 − 2𝑚 ≥ 0. Solving this quadratic equation,   √︁ 𝛼 ≤ 12 2𝑛 + 2𝑘 − 1 − 8𝑚 + (2𝑘 − 1) 2 .

(14.9)

This establishes the desired upper bound. That the upper bound is tight may be seen as follows. Let 𝑘, ℓ, 𝑛 be arbitrary integers such that 2 ≤ 𝑘 ≤ ℓ < 𝑛. Let 𝐺 𝑘,ℓ,𝑛 be obtained from the disjoint union of an empty graph 𝐾 ℓ with vertex set 𝐴 and a complete graph 𝐾𝑛−ℓ with vertex set 𝐵 by selecting an arbitrary vertex 𝑣 ∈ 𝐴 and joining it to every vertex in 𝐵, and by joining every vertex in 𝐵 to exactly 𝑘 − 1 vertices in 𝐴 \ {𝑣}. The resulting graph 𝐺 = 𝐺 𝑘,ℓ,𝑛 is a 𝐾1,𝑘+1 -free graph of order 𝑛 and size 𝑚 such that   𝑛−ℓ+1 𝑚 = (𝑘 − 1) (𝑛 − ℓ) + = 12 (𝑛 − ℓ) (𝑛 − ℓ + 2𝑘 − 1). 2 By Inequality (14.9),   √︁ 𝛼(𝐺) ≤ 12 2𝑛 + 2𝑘 − 1 − 4(𝑛 − ℓ) (𝑛 − ℓ + 2𝑘 − 1) + (2𝑘 − 1) 2   √︁ = 12 2𝑛 + 2𝑘 − 1 − (2𝑛 + 2𝑘 − 1 − 2ℓ) 2 = ℓ. Since the set 𝐴 is an independent set of 𝐺, we have 𝛼(𝐺) ≥ | 𝐴| = ℓ. Consequently, the graph 𝐺 achieves equality in the upper bound in the statement of the theorem. As a consequence of Theorem 14.14, in the special case when 𝐺 is clawfree, the following result holds, which was established independently in 1999 by Gasharov [327]. Theorem 14.15 ([327, 660]) If 𝐺 is a connected claw-free graph of order 𝑛, then 𝛼(𝐺) ≤ 12 (𝑛 + 1), with equality if and only if 𝐺 = 𝐾1 or 𝐺 = 𝑃3 .

Section 14.2. Upper Bounds

421

Proof Let 𝐺 be a connected claw-free graph of order 𝑛 and size 𝑚, and so 𝑚 ≥ 𝑛 − 1. If 𝐺 = 𝐾𝑛 , then 𝛼(𝐺) = 1 ≤ 21 (𝑛 + 1), with strict inequality if 𝑛 ≥ 2. Hence, we may assume that 𝐺 ≠ 𝐾𝑛 , and so 𝑛 ≥ 3. By Theorem 14.14 with 𝑘 = 2, and noting that 𝑚 ≥ 𝑛 − 1 and 𝑛 ≥ 3, √
𝛼(𝐺) ≤ 12 2𝑛 + 3 − 8𝑛 + 1 ≤ 12 (𝑛 + 1), with strict inequality if 𝑛 ≥ 4. We remark that when 𝛿 ≥ 2, the bound in Theorem 14.13 improves on the bound in Theorem 14.15. The upper bound in Theorem 14.13 does not hold for the upper domination number. For example, if 𝐺 is the graph obtained from two disjoint cliques 𝐾 𝑘 , for 𝑘 ≥ 3, by adding a perfect matching between these cliques, then the resulting graph 𝐺 has order 𝑛 and minimum degree 𝛿 = 𝑘 and satisfies Γ(𝐺) = 𝛿 = 12 𝑛 > 2𝑛/(𝛿 + 2). However, Favaron [276] showed that the upper bound in Theorem 14.15 holds for the upper domination number (and the upper irredundance number). Theorem 14.16 ([276]) If 𝐺 is a connected claw-free graph of order 𝑛, then Γ(𝐺) ≤ 12 (𝑛 + 1). Further, if Γ(𝐺) = 12 (𝑛 + 1), then every Γ-set of 𝐺 is an independent set. In particular, if Γ(𝐺) = 12 (𝑛 + 1), then Γ(𝐺) = 𝛼(𝐺). Proof Among all Γ-sets of 𝐺, let 𝑆 be chosen so that 𝐺 [𝑆] has maximum size. Let 𝐴 be the set of isolated vertices in 𝐺 [𝑆]. We consider first the case when 𝑆 is an independent set. Claim 14.16.1 If 𝐴 = 𝑆, then Γ(𝐺) ≤ 12 (𝑛 + 1). Further, if Γ(𝐺) = 12 (𝑛 + 1), then every Γ-set of 𝐺 is an independent set. Proof Suppose that 𝐴 = 𝑆. Thus, 𝑆 is an independent set. By the Domination Chain and Theorem 14.15, we have 𝛼(𝐺) ≤ Γ(𝐺) = |𝑆| ≤ 𝛼(𝐺) ≤ 12 (𝑛 + 1). If Γ(𝐺) = 12 (𝑛 + 1), then by our choice of the set 𝑆, every Γ-set of 𝐺 is an independent set. In particular, this implies that Γ(𝐺) = 𝛼(𝐺). By Claim 14.16.1, we may assume that the set 𝑆 is not an independent set. Thus, 𝐴 ⊂ 𝑆. Let 𝐵 = 𝑆 \ 𝐴 and so 𝐵 consists of the vertices of 𝑆 that are not isolated in 𝐺 [𝑆]. By our earlier assumption, 𝐵 ≠ ∅. Since 𝑆 is a minimal dominating set, by Lemma 14.1, ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅ for every vertex 𝑣 ∈ 𝑆. If 𝑣 is an arbitrary vertex in 𝐵, then ipn[𝑣, 𝑆] = ∅, implying that epn[𝑣, 𝑆] ≠ ∅. Let Ø Ø 𝐶𝐴 = epn[𝑣, 𝑆] and 𝐶 𝐵 = epn[𝑣, 𝑆]. 𝑣∈𝐵

𝑣∈ 𝐴



Further, let 𝐶 = 𝐶 𝐴 ∪ 𝐶 𝐵 and so 𝐶 = 𝑣 ∈ 𝑉 \ 𝑆 : deg𝑆 (𝑣) = 1 . Let the sets 𝐷, 𝐷 𝐴, and 𝐷 𝐵 be defined as follows:  𝐷 = 𝑣 ∈ 𝑉 \ 𝑆 : deg𝑆 (𝑣) ≥ 2 ,  𝐷 𝐴 = 𝑣 ∈ 𝐷 : deg 𝐴 (𝑣) ≥ 2 , and 𝐷 𝐵 = 𝐷 \ 𝐷 𝐴.

422

Chapter 14. Upper Domination Parameters

Since 𝑆 is a dominating set, 𝑉 = 𝐴 ∪ 𝐵 ∪ 𝐶 ∪ 𝐷. Since 𝐺 is a claw-free graph, for every 𝑣 ∈ 𝐵, the component of 𝐺 [𝑆] that contains the vertex 𝑣 is a clique. Further, since 𝐺 is claw-free, the set epn[𝑣, 𝑆] induces a clique for every 𝑣 ∈ 𝐵. Moreover, for every vertex 𝑣 ∈ 𝐷 𝐴, deg 𝐴 (𝑣) = 2 and deg 𝐵 (𝑣) = 0, while for every vertex 𝑣 ∈ 𝐷 𝐵 , deg 𝐵 (𝑣) ≥ 1, and deg 𝐴 (𝑣) + deg 𝐵 (𝑣) = deg𝑆 (𝑣) ≥ 2. We define next the sets 𝑉1 , 𝑉2 , and 𝑉3 as follows: 𝑉1 = 𝐴 ∪ 𝐷 𝐴, 𝑉2 = 𝐵 ∪ 𝐶 𝐵 , and 𝑉3 = 𝐶 𝐴 ∪ 𝐷 𝐵 . Thus, 𝑉 = 𝑉1 ∪ 𝑉2 ∪ 𝑉3 . By our earlier observations, |epn[𝑣, 𝑆] | ≥ 1 for every vertex 𝑣 ∈ 𝐵 and so ∑︁ (14.10) |𝑉2 | − |𝐵| = |𝐶 𝐵 | = |epn[𝑣, 𝑆] | ≥ |𝐵|, 𝑣∈𝐵

or equivalently, |𝐵| ≤ 21 |𝑉2 |,

(14.11)

with equality if and only if |epn[𝑣, 𝑆] | = 1 for every vertex 𝑣 ∈ 𝐵. If 𝐴 = ∅, then Γ(𝐺) = |𝐵| ≤ 12 |𝑉2 | ≤ 12 |𝑉 | = 12 𝑛. Hence, we may assume that 𝐴 ≠ ∅. Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑡 be the components of 𝐺 [𝑉1 ]. Let 𝑋𝑖 = 𝑉 (𝐺 𝑖 ) and let 𝐺 𝑖 have order 𝑛𝑖 = |𝑋𝑖 | for 𝑖 ∈ [𝑡]. Each component of 𝐺 [𝑉1 ] is a claw-free graph, and so by Theorem 14.15, 𝛼(𝐺 𝑖 ) ≤ 12 (𝑛𝑖 + 1)

for each 𝑖 ∈ [𝑡].

(14.12)

Let 𝐴𝑖 = 𝐴 ∩ 𝑋𝑖 for 𝑖 ∈ [𝑡]. Since 𝐴𝑖 is an independent set in 𝐺 𝑖 , we have | 𝐴𝑖 | ≤ 𝛼(𝐺 𝑖 ) for 𝑖 ∈ [𝑡]. Therefore, by Inequality (14.12), | 𝐴𝑖 | ≤ 12 (𝑛𝑖 + 1)

for each 𝑖 ∈ [𝑡].

(14.13)

We now define an injective map 𝑓 : 𝑋𝑖 → 𝑉3 for each 𝑖 ∈ [𝑡] as follows: (i) If there exists a vertex 𝑣 𝑖 ∈ 𝐴𝑖 such that epn[𝑣 𝑖 , 𝑆] ≠ ∅, then we select a vertex 𝑣 𝑖′ ∈ epn[𝑣 𝑖 , 𝑆]. In this case, we note that 𝑣 𝑖′ ∈ 𝐶 𝐴 ⊆ 𝑉3 and we define 𝑓 (𝑋𝑖 ) = {𝑣 𝑖′ }. (ii) If epn[𝑣, 𝑆] = ∅ for all 𝑣 ∈ 𝐴𝑖 , then by the connectedness of 𝐺, there is an edge 𝑢 𝑖 𝑤 𝑖 , where 𝑢 𝑖 ∈ 𝑋𝑖 and 𝑤 𝑖 ∉ 𝑋𝑖 . We note that either 𝑢 𝑖 ∈ 𝐴𝑖 or 𝑢 𝑖 ∈ 𝐷 𝐴. By our earlier observations, if 𝑢 𝑖 ∈ 𝐴𝑖 , then 𝑤 𝑖 ∈ 𝐷 𝐵 . If 𝑢 𝑖 ∈ 𝐷 𝐴, then 𝑢 𝑖 has two neighbors in 𝐴 and no neighbor in 𝐵 ∪ 𝐶, implying that 𝑤 𝑖 ∈ 𝐷 𝐵 . In both cases, we define 𝑓 (𝑋𝑖 ) = {𝑤 𝑖 }. We show next that 𝑓 is injective. Let 𝑖, 𝑗 ∈ [𝑡], where 𝑖 ≠ 𝑗. Adopting our earlier notation, if 𝑓 (𝑋𝑖 ) = 𝑣 𝑖′ ∈ epn[𝑣 𝑖 , 𝑆] and 𝑓 (𝑋 𝑗 ) = 𝑣 ′𝑗 ∈ epn[𝑣 𝑗 , 𝑆], then by the definition of 𝑆-external private neighbor, 𝑓 (𝑋𝑖 ) ≠ 𝑓 (𝑋 𝑗 ). If 𝑓 (𝑋𝑖 ) = 𝑣 𝑖′ ∈ 𝐶 𝐴 and 𝑓 (𝑋 𝑗 ) = 𝑤 𝑗 ∈ 𝐷 𝐵 , then 𝑓 (𝑋𝑖 ) ≠ 𝑓 (𝑋 𝑗 ) since 𝐶 𝐴 and 𝐷 𝐵 are disjoint. Hence, it remains to consider the case when 𝑓 (𝑋𝑖 ) = 𝑤 𝑖 ∈ 𝐷 𝐵 and 𝑓 (𝑋 𝑗 ) = 𝑤 𝑗 ∈ 𝐷 𝐵 . Suppose, to the contrary, that 𝑤 𝑖 = 𝑤 𝑗 . Let 𝑤 be a neighbor of 𝑤 𝑖 that belongs to the set 𝐵. Since 𝑢 𝑖 and 𝑢 𝑗 belong to different components of 𝐺 [𝑉1 ], they are not adjacent.

Section 14.2. Upper Bounds

423

If 𝑢 𝑖 ∈ 𝐴𝑖 , then 𝑢 𝑖 is isolated in 𝐺 [𝑆] and therefore is not adjacent to 𝑤; while if 𝑢 𝑖 ∈ 𝐷 𝐴, then by our earlier observations, deg 𝐵 (𝑢 𝑖 ) = 0, and so once again 𝑢 𝑖 is not adjacent to 𝑤. Similarly, 𝑢 𝑗 is not adjacent to 𝑤. But then the subgraph of 𝐺 induced by the set {𝑢 𝑖 , 𝑢 𝑗 , 𝑤 𝑖 , 𝑤} is a claw centered at 𝑤 𝑖 = 𝑤 𝑗 , a contradiction. Therefore, 𝑤 𝑖 ≠ 𝑤 𝑗 , implying that 𝑓 is injective. Hence, the 𝑡 sets 𝑋𝑖 ∪ { 𝑓 (𝑋𝑖 )} are vertex-disjoint for 𝑖 ∈ [𝑡] and 𝑡 Ø
𝑋𝑖 ∪ { 𝑓 (𝑋𝑖 )} ⊆ 𝑉1 ∪ 𝑉3 , 𝑖=1

and so 𝑡 ∑︁


|𝑋𝑖 | + | 𝑓 (𝑋𝑖 )| ≤ |𝑉1 | + |𝑉3 |.

(14.14)

𝑖=1

By Inequalities (14.13) and (14.14), 𝑡 𝑡 ∑︁ ∑︁ | 𝐴𝑖 | | 𝐴 ∩ 𝑋𝑖 | = | 𝐴| = 𝑖=1

𝑖=1

≤

= ≤

𝑡 ∑︁ 𝑖=1 𝑡 ∑︁

1 2 (𝑛𝑖

1 2

+ 1)

|𝑋𝑖 | + | 𝑓 (𝑋𝑖 )|

𝑖=1 1 2 |𝑉1 |

(14.15)



+ |𝑉3 | .

Hence, by Inequalities (14.11) and (14.15), Γ(𝐺) = |𝑆| = | 𝐴| + |𝐵| ≤

1 2


|𝑉1 | + |𝑉3 | + 12 |𝑉2 | = 12 |𝑉 | = 12 𝑛.

Thus, we have shown that when the Γ-set 𝑆 is not an independent set, Γ(𝐺) ≤ 12 𝑛. This completes the proof of Theorem 14.16. A simplicial vertex is a vertex 𝑣 whose open neighborhood N(𝑣) induces a complete graph. The line graph 𝐿(𝐺) of a graph 𝐺 has vertex set 𝑉 (𝐿(𝐺)) = 𝐸 (𝐺) and two vertices of 𝐿 (𝐺) are adjacent if and only if the corresponding edges are adjacent in 𝐺. The clique graph 𝐾 (𝐺) of a graph 𝐺 has as its vertices the set of maximal cliques in 𝐺 and two vertices of 𝐾 (𝐺) are adjacent if and only if they intersect as cliques of 𝐺. Favaron [276] characterized the (infinite) family of graphs achieving equality in the upper bound in Theorem 14.16 that Γ(𝐺) = 12 (𝑛 + 1). Let Fclaw-free be the family of claw-free graphs constructed as follows. Let 𝑇1 , 𝑇2 , . . . , 𝑇𝑞 be 𝑞 nontrivial trees. Let 𝐿 𝑖 = 𝐿(𝑇𝑖 ◦ 𝐾1 ) be the line graph of the corona 𝑇𝑖 ◦ 𝐾1 of the tree 𝑇𝑖 for 𝑖 ∈ [𝑞]. We note that each graph 𝐿 𝑖 contains |𝑉 (𝑇𝑖 )| simplicial vertices (namely, the vertices of 𝐿 𝑖 corresponding to the pendant edges in the corona 𝑇𝑖 ◦ 𝐾1 ). Let 𝐺 be the graph constructed from the vertex-disjoint union of the graphs 𝐿 1 , 𝐿 2 , . . . , 𝐿 𝑞 as follows. We choose 𝑞 − 1 pairs {𝑣 𝑖 𝑗 , 𝑣 𝑗𝑖 }, where 𝑣 𝑖 𝑗 and 𝑣 𝑗𝑖 are simplicial vertices

Chapter 14. Upper Domination Parameters

424

in 𝐿 𝑖 and 𝐿 𝑗 , respectively, and contract each pair of vertices into one new vertex 𝑐 𝑖 𝑗 (that is adjacent to all neighbors of 𝑣 𝑖 𝑗 in 𝐿 𝑖 and all neighbors of 𝑣 𝑗𝑖 in 𝐿 𝑗 ), in such a way that the following hold: (i) The 2(𝑞 − 1) chosen vertices from the pairs {𝑣 𝑖 𝑗 , 𝑣 𝑗𝑖 } are all distinct. (ii) The clique graph of 𝐺 is a tree. Equivalently, the graph 𝐺 is connected and no cycle of 𝐺 contains two distinct contracted vertices 𝑐 𝑖 𝑗 and 𝑐 𝑖 ′ 𝑗 ′ . Let Fclaw-free be the family of all such graphs 𝐺. To illustrate the above construction when 𝑞 = 3, let 𝑇1 , 𝑇2 , and 𝑇3 be the trees shown in Figure 14.2(a), (b), and (c), respectively. The coronas 𝑇1 ◦ 𝐾1 , 𝑇2 ◦ 𝐾1 , and 𝑇3 ◦ 𝐾1 are shown in Figure 14.2(d), (e), and (f), respectively. The line graphs 𝐿 1 = 𝐿 (𝑇1 ◦ 𝐾1 ), 𝐿 2 = 𝐿(𝑇2 ◦ 𝐾1 ), and 𝐿 3 = 𝐿(𝑇3 ◦ 𝐾1 ) are shown in Figure 14.2(g), (h), and (i), respectively. The graph 𝐺 constructed from the pairs {𝑣 12 , 𝑣 21 } and {𝑣 23 , 𝑣 32 } is illustrated in Figure 14.2(j).

(a) 𝑇1

(b) 𝑇2

(c) 𝑇3

(d) 𝑇1 ◦ 𝐾1

(e) 𝑇2 ◦ 𝐾1

(f) 𝑇3 ◦ 𝐾1

𝑣 12

𝑣 23

(g) 𝐿 1

(h) 𝐿 2

𝑐 12

𝑣 32 (i) 𝐿 3

𝑐 23

(j) The graph 𝐺

Figure 14.2 A graph 𝐺 in the family Fclaw-free

We are now in a position to state the characterization of Favaron [276], albeit without proof. Theorem 14.17 ([276]) If 𝐺 is a connected claw-free graph of order 𝑛, then Γ(𝐺) ≤ 12 (𝑛 + 1), with equality if and only if 𝐺 ∈ Fclaw-free . We note that if 𝐺 is an arbitrary graph of order 𝑛 in the family Fclaw-free , then 𝑛 ≥ 3 is odd and the vertex set of 𝐺 can be partitioned into two sets 𝐴 and 𝐵 such that the following hold: (i) | 𝐴| = 12 (𝑛 − 1) and |𝐵| = 12 (𝑛 + 1). (ii) The set 𝐵 is an independent set.

Section 14.2. Upper Bounds

425

(iii) Each vertex in 𝐴 has exactly two neighbors in 𝐵. We refer to the partition ( 𝐴, 𝐵) as the partition associated with 𝐺. For the graph 𝐺 ∈ Fclaw-free illustrated in Figure 14.2(j), the set 𝐴 consists of the red highlighted vertices and the set 𝐵 consists of the blue highlighted vertices. The following property of graphs in the family Fclaw-free is given in Henning and Rall [476]. Lemma 14.18 ([476]) If 𝐺 ∈ Fclaw-free and ( 𝐴, 𝐵) is the partition associated with 𝐺, then the set 𝐵 is the unique Γ-set of 𝐺 (and the unique IR-set of 𝐺). Favaron [276] also presented the following result on connected claw-free graphs of even order 𝑛 with maximum possible upper domination number. Theorem 14.19 ([276]) Let 𝐺 be a connected claw-free graph of even order 𝑛 ≥ 8 with minimum degree 𝛿. If Γ(𝐺) = 21 𝑛, then 𝛿 = 12 𝑛 or 1 ≤ 𝛿 ≤ 14 𝑛. Moreover, for every integer 𝑘, where 1 ≤ 𝑘 ≤ 14 𝑛 or 𝑘 = 12 𝑛, there exists a connected claw-free graph of even order 𝑛 ≥ 8 with minimum degree 𝛿 = 𝑘 and Γ(𝐺) = 12 𝑛. Next we consider upper bounds on the upper total domination numbers of clawfree graphs. In 2003 Favaron and Henning [278] showed that the upper bound in Theorem 14.6 can be significantly improved for the class of claw-free graphs. For this purpose, we first define two families Gclaw-free and Hclaw-free of connected claw-free graphs. Let Gclaw-free be the family of connected claw-free graphs 𝐺 = (𝑉, 𝐸) that admit a vertex partition 𝑉 = 𝑋 ∪ 𝐶 such that 𝐺 [𝑋] = 𝑞𝐾2 and each vertex of 𝐶 is adjacent to vertices of 𝑋 from exactly two 𝐾2 components of 𝐺 [𝑋] and possibly to other vertices of 𝐶. The following properties of graphs in the family Gclaw-free are shown in [278]. Lemma 14.20 ([278]) If 𝐺 ∈ Gclaw-free has minimum degree 𝛿, then the following hold: (a) |𝑋 | ≤ 2|𝐶 | + 2. | (b) If 𝛿 ≥ 3, then |𝑋 | ≤ 4|𝐶 𝛿−1 . For 𝛿 ≥ 1, let G 𝛿,claw-free be the subfamily of graphs in Gclaw-free defined as follows: ( 𝐺 ∈ Gclaw-free : 𝛿(𝐺) ≥ 𝛿 and |𝑋 | = 2|𝐶 | + 2 for 𝛿 ∈ [2], G 𝛿,claw-free =  4|𝐶 | 𝐺 ∈ Gclaw-free : 𝛿(𝐺) ≥ 𝛿 and |𝑋 | = 𝛿−1 for 𝛿 ≥ 3. For 𝑘 ≥ 1, the path 𝑃3𝑘+2 : 𝑣 1 𝑣 2 . . . 𝑣 3𝑘+2 of order 3𝑘 of a  + 2 is an example graph in the family G1,claw-free . In this example, letting 𝐶 = 𝑣 3𝑖 : 𝑖 = [𝑘] and letting 𝑋 = 𝑉 (𝑃3𝑘+2 ) \ 𝐶, we note that 𝐺 [𝑋] = (𝑘 + 1)𝐾2 and each vertex of 𝐶 is adjacent to vertices of 𝑋 from exactly two 𝐾2 components of 𝐺 [𝑋]. Hence, 𝑃3𝑘+2 ∈ Gclaw-free . Since |𝑋 | = 2|𝐶 | + 2, we have 𝑃3𝑘+2 ∈ G1,claw-free . Examples of graphs in the families G 𝛿,claw-free with 𝛿 ∈ {2, 3, 4, 5} are shown in Figure 14.3(a), (b), (c), and (d), respectively, where the vertices highlighted in red form the set 𝑋 and the vertices highlighted in blue the set 𝐶. We remark that each of the families G𝑖,claw-free with 𝑖 ∈ [5] contains graphs of arbitrarily large order. For 𝛿 ≥ 5, let 𝐻 be a connected graph with minimum degree at least 𝛿 whose vertex set can be partitioned into two sets 𝑉1 and 𝑉2 , where the edges between 𝑉1

Chapter 14. Upper Domination Parameters

426

(a) 𝐺 ∈ G2,claw-free

(b) 𝐺 ∈ G3,claw-free

(c) 𝐺 ∈ G4,claw-free

(d) 𝐺 ∈ G5,claw-free

Figure 14.3 Graphs in the families G 𝛿,claw-free for 𝛿 ∈ {2, 3, 4, 5}

and 𝑉2 induce a perfect matching and where the subgraph induced by each of 𝑉1 and 𝑉2 is a clique or a disjoint union of cliques. Let Hclaw-free denote the family of all such graphs 𝐻. We are now in a position to state the result from [278] which provides tight upper bounds on the upper total domination number of a connected claw-free graph in terms of its order and minimum degree. Theorem 14.21 ([278]) If 𝐺 is a connected claw-free graph of order 𝑛 with minimum degree 𝛿, then 2   (𝑛 + 1) if 𝛿 ∈ {1, 2}    3 4𝑛 Γt (𝐺) ≤ 𝛿+3 if 𝛿 ∈ {3, 4, 5}    1  𝑛 if 𝛿 ≥ 6. 2 Furthermore, the following hold: (a) Γt (𝐺) = 23 (𝑛 + 1) if and only if 𝐺 ∈ G1,claw-free ∪ G2,claw-free . 4𝑛 (b) If 𝛿 ∈ {3, 4, 5}, then Γt (𝐺) = 𝛿+3 if and only if 𝐺 ∈ G 𝛿,claw-free or 𝛿 = 5 and 𝐺 ∈ Hclaw-free . (c) If 𝛿 ≥ 6, then Γ(𝐺) = 12 𝑛 if and only if 𝐺 ∈ Hclaw-free . We remark that the upper bounds in Theorem 14.21 are tight even for connected claw-free graphs of arbitrarily large order.

14.3

Upper Domination Number

As we saw in Chapter 2, the complement of a minimal dominating set is a maximal enclaveless set and vice versa. Hence, we have the following result, first noted in 1977 by Slater [679]. Let 𝜓(𝐺) equal the minimum cardinality of a maximal enclaveless set. Theorem 14.22 ([679]) For any graph 𝐺 of order 𝑛, Γ(𝐺) + 𝜓(𝐺) = 𝑛. We recall some definitions from Chapter 2. Let 𝑆 be a set of vertices in a graph 𝐺. A vertex 𝑣 ∈ 𝑉 is called 𝑆-perfect if |N[𝑣] ∩ 𝑆| = 1. A vertex in 𝑉 is called almost 𝑆-perfect if it is not 𝑆-perfect but is adjacent to an 𝑆-perfect vertex. A set 𝑆 ⊆ 𝑉 is called a perfect neighborhood set if every vertex 𝑣 ∈ 𝑉 is either 𝑆-perfect or almost 𝑆-perfect. The perfect neighborhood number 𝜃 (𝐺) is the minimum cardinality of a

Section 14.3. Upper Domination Number

427

perfect neighborhood set in 𝐺, while the upper perfect neighborhood number Θ(𝐺) is the maximum cardinality of a perfect neighborhood set in 𝐺. In 1999 Fricke et al. [310] showed that the upper perfect neighborhood number of a graph is precisely its upper domination number. They first proved the following lemma. Lemma 14.23 ([310]) For every minimal dominating set 𝐷 of a graph 𝐺, there exists a perfect neighborhood set of 𝐺 of cardinality |𝐷|. Proof Let 𝐷 be an arbitrary minimal dominating set of 𝐺. By Lemma 14.1, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. Let 𝐴 be the set of isolated vertices in 𝐺 [𝐷] and let 𝐵 = 𝐷 \ 𝐴. For each vertex 𝑣 ∈ 𝐵, we have ipn[𝑣, 𝐷] = ∅, implying that epn[𝑣, 𝐷] ≠ ∅. For each vertex 𝑣 ∈ 𝐵, we select a vertex 𝑣 ′ ∈ epn[𝑣, 𝐷] and we let 𝐶 = {𝑣 ′ : 𝑣 ∈ 𝐵}. We now consider the set 𝑆 = 𝐴 ∪ 𝐶. We note that every vertex 𝑣 ∈ 𝐴 is isolated in the graph 𝐺 [𝑆] and so N[𝑣] ∩ 𝑆 = {𝑣}. For every vertex 𝑣 ∈ 𝐵, we have N[𝑣] ∩ 𝑆 = {𝑣 ′ }. Hence, |N[𝑣] ∩ 𝑆| = 1 for every vertex 𝑣 ∈ 𝐷 and so every vertex in 𝐷 is an 𝑆-perfect vertex. Since 𝐷 is a dominating set, every vertex in 𝑉 \ 𝐷 is adjacent to a vertex of 𝐷, and is therefore adjacent to an 𝑆-perfect vertex. Hence, every vertex 𝑣 ∈ 𝑉 is either 𝑆-perfect or almost 𝑆-perfect. Thus, 𝑆 is a perfect neighborhood set of cardinality |𝐷 |. As an immediate consequence of Lemma 14.23, we have the following inequality chain. Corollary 14.24 ([310]) For every graph 𝐺, 𝜃 (𝐺) ≤ 𝛾(𝐺) ≤ Γ(𝐺) ≤ Θ(𝐺). We are now in a position to present the main result in [310]. The boundary 𝜕 (𝑆) of a set 𝑆 in a graph 𝐺 is the set of vertices that do not belong to 𝑆 but have a neighbor in 𝑆, that is, 𝜕 (𝑆) = N(𝑆) \ 𝑆. We note that N[𝑆] = 𝑆 ∪ 𝜕 (𝑆). Theorem 14.25 ([310]) For every graph 𝐺, Γ(𝐺) = Θ(𝐺). Proof Let 𝑆 be a perfect neighborhood set of maximum cardinality and so |𝑆| = Θ(𝐺). Let 𝐴 be the set of all vertices in 𝑆 that are adjacent to an 𝑆-perfect vertex that belongs to 𝑉 \𝑆 and let 𝐵 = 𝑆 \ 𝐴. If 𝑣 ∈ 𝐵 has a neighbor in 𝑆, then |N[𝑣] ∩𝑆| ≥ 2 and so 𝑣 is not an 𝑆-perfect vertex and therefore is adjacent to an 𝑆-perfect vertex which necessarily belongs to 𝑉 \𝑆. However, this would imply that 𝑣 ∈ 𝐴, a contradiction. Hence, every vertex in 𝐵 is an isolated vertex in 𝐺 [𝑆] and is therefore an 𝑆-perfect vertex. We now consider the boundary 𝐶 = 𝜕 (𝑆) of the set 𝑆. Let 𝐶1 be the set of 𝑆-perfect vertices in 𝜕 (𝑆). Thus, if 𝑣 ∈ 𝐶1 , then 𝑣 ∈ 𝜕 (𝑆) and |N[𝑣] ∩ 𝑆| = 1. We note that each vertex in 𝐶1 is adjacent to a unique vertex in 𝐴 and to no vertex in 𝐵. Every vertex in 𝜕 (𝑆) \ 𝐶1 has at least two neighbors in 𝑆 and is therefore not an 𝑆-perfect vertex. Let 𝐶2 be the set of vertices in 𝜕 (𝑆) \ 𝐶1 that have a neighbor in 𝐵 ∪ 𝐶1 and let 𝐶3 = 𝜕 (𝑆) \ (𝐶1 ∪ 𝐶2 ). We note that each vertex in 𝐶3 has at least two neighbors in 𝐴 and no neighbor in 𝐵. Let 𝐷 = 𝑉 \ N[𝑆], and so 𝑉 = 𝐴 ∪ 𝐵 ∪ 𝐶 ∪ 𝐷. Since N[𝑣] ∩ 𝑆 = ∅ for every vertex 𝑣 ∈ 𝐷, no vertex in the set 𝐷 is 𝑆-perfect. By our earlier observations, no vertex in 𝐶2 ∪ 𝐶3 is 𝑆-perfect. Further, no vertex in 𝐶3 has a neighbor in 𝐵 ∪ 𝐶1 . Each vertex in 𝐶3 must therefore have a neighbor in 𝐴 that

428

Chapter 14. Upper Domination Parameters

is 𝑆-perfect. Among all subsets of 𝑆-perfect vertices in 𝐴 that dominate the set 𝐶3 , let 𝐴1 be chosen to have minimum cardinality. By the minimality of the set 𝐴1 , each vertex in 𝐴1 uniquely dominates at least one vertex in 𝐶3 ; that is, for each 𝑣 ∈ 𝐴1 , there exists a vertex 𝑣 ′ ∈ 𝐶3 such that N(𝑣 ′ ) ∩ 𝐴1 = {𝑣}. Let 𝐴2 = 𝐴 \ 𝐴1 . We now consider the set 𝑋 = 𝐴1 ∪ 𝐵 ∪ 𝐶1 . By definition, each vertex of 𝐴 has a neighbor in 𝐶1 . In particular, the set 𝐶1 dominates the set 𝐴2 . By definition, each vertex of 𝐶2 has a neighbor in 𝐵 ∪ 𝐶1 and 𝐵 ∪ 𝐶1 dominates the set 𝐶2 . By definition, the set 𝐴1 dominates the set 𝐶3 . As observed earlier, no vertex in the set 𝐷 is 𝑆-perfect and no vertex in 𝐶2 ∪ 𝐶3 is 𝑆-perfect. Hence, the only 𝑆-perfect neighbors of vertices in 𝐷 must belong to the set 𝐶1 and so the set 𝐶1 dominates the set 𝐷. Therefore, the set 𝑋 is a dominating set of 𝐺. This implies that there must exist a subset 𝑋 ★ of 𝑋 that is a minimal dominating set of 𝐺, where possibly 𝑋 = 𝑋 ★. We show next that |𝑋 ★ | ≥ |𝑆|. Since every vertex in 𝐵 is an isolated vertex in 𝐺 [𝑋], we note that 𝐵 ⊆ 𝑋 ★. By our choice of the set 𝐴1 , every vertex in 𝐴1 uniquely dominates a vertex in 𝐶3 . Since no vertex in 𝐶3 has a neighbor in 𝐵 ∪ 𝐶1 , this implies that 𝐴1 ⊆ 𝑋 ★. Since each vertex 𝑣 in 𝐴1 is 𝑆-perfect, 𝑣 is an isolated vertex in 𝐺 [𝑆], implying that no vertex of 𝐴2 has a neighbor in 𝐴1 ∪ 𝐵. Moreover, every vertex in 𝐴2 is adjacent to an 𝑆-perfect vertex in 𝐶1 . Since each vertex in 𝐶1 is adjacent to at most one vertex in 𝐴2 , this implies that |𝑋 ★ ∩ 𝐶1 | ≥ | 𝐴2 |. Hence, 𝑋 ★ is a minimal dominating set of 𝐺 satisfying |𝑋 ★ | ≥ | 𝐴1 | + | 𝐴2 | + |𝐵| = |𝑆| = Θ(𝐺). Therefore, Γ(𝐺) ≥ |𝑋 ★ | ≥ Θ(𝐺). By Corollary 14.24, we have Γ(𝐺) ≤ Θ(𝐺). Consequently, Γ(𝐺) = Θ(𝐺).

14.4

Independence Number

As mentioned in the introductory comments in this chapter, the independence number is one of the most fundamental and well-studied graph parameters. In this section, we present a small sample of selected lower bounds on the independence number, including a fundamental 1941 result due to Turán [717] and classical results due to Fajtlowicz [264, 265] in 1978 and Chvátal and McDiarmid [176] in 1992. We also present the important Caro-Wei Theorem due to Caro [132] and Wei [748]. In 2014 Henning et al. [469] gave the following lower bound on the independence number of a graph, which strengthens results due to Fajtlowicz [264, 265]. The main idea is to carefully choose a maximum independent set 𝑆 in the graph 𝐺 such that the number of edges from 𝑆 to vertices outside 𝑆 is minimized. With this choice of 𝑆, we establish a property on the graph 𝐺 by replacing a vertex in 𝑆 with a vertex not in 𝑆 in order to get a smaller number of edges between 𝑆 and vertices outside 𝑆 to reach a contradiction. Theorem 14.26 ([469]) If 𝐺 is a graph of order 𝑛 and 𝑝 is an integer such that Property 𝑃1 below holds, then 𝛼(𝐺) ≥ 2𝑛 𝑝. Property 𝑷1 . For every clique 𝑋 in 𝐺, there exists a vertex 𝑥 ∈ 𝑋 such that deg(𝑥) < 𝑝 − |𝑋 |. Proof Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 and let 𝑝 be an integer such that Property 𝑃1 holds. Among all 𝛼-sets of 𝐺, let 𝑆 be chosen so that the number of edges

Section 14.4. Independence Number

429

in [𝑆, 𝑉 \ 𝑆] is minimized. Thus, 𝑆 is an independent set in 𝐺 such that |𝑆| = 𝛼(𝐺) and | [𝑆, 𝑉 \ 𝑆] | is minimized. Let 𝛼𝑖 (𝑆) denote the number of vertices in 𝑉 \ 𝑆 with exactly 𝑖 neighbors in 𝑆. Since 𝑆 is a maximum independent set, we note that 𝛼0 (𝑆) = 0. Therefore, 𝑛 − |𝑆| = 𝛼1 (𝑆) + 𝛼2 (𝑆) + · · · + 𝛼 |𝑆 | (𝑆).

(14.16)

Furthermore, counting the number of edges in [𝑆, 𝑉 \ 𝑆], ∑︁ | [𝑆, 𝑉 \ 𝑆] | = deg(𝑠) = 𝛼1 (𝑆) + 2𝛼2 (𝑆) + 3𝛼3 (𝑆) + · · · + |𝑆|𝛼 |𝑆 | (𝑆). (14.17) 𝑠∈𝑆

Multiplying Equation (14.16) by 2 and subtracting Equation (14.17), we obtain the following. ∑︁
deg(𝑠) = 𝛼1 (𝑆) − 𝛼3 (𝑆) − 2𝛼4 (𝑆) − · · · − |𝑆| − 2 𝛼 |𝑆 | (𝑆) 2𝑛 − 2|𝑆| − 𝑠∈𝑆

≤ 𝛼1 (𝑆).

(14.18)

For each vertex 𝑣 ∈ 𝑆, let 𝑌𝑣 be the set of all vertices in 𝑉 \ 𝑆 adjacent to 𝑣 but to no other vertex of 𝑆 and so every vertex in 𝑌𝑣 has no neighbor in 𝑆 \ {𝑣}. If 𝑌𝑣 does not induce
a clique, then let 𝑦 1 , 𝑦 2 ∈ 𝑌𝑠 be nonadjacent vertices and note that 𝑆 ∪ {𝑦 1 , 𝑦 2 } \ {𝑣} is an independent set in 𝐺 of cardinality greater than |𝑆|, a contradiction. Therefore, 𝑌𝑣 ∪ {𝑣} induces a clique in 𝐺 for every vertex 𝑣 ∈ 𝑆. Suppose that deg(𝑣) + |𝑌𝑣 | + 1 ≥ 𝑝 for some
vertex 𝑣 ∈ 𝑆. If there is a vertex 𝑦 ∈ 𝑌𝑣 such that deg(𝑦) < deg(𝑣), then 𝑆 ∪ {𝑦} \ {𝑣} contradicts the minimality of | [𝑆, 𝑉 \ 𝑆]|. Therefore, for all 𝑦 ∈ 𝑌𝑣 ∪ {𝑣}, we have deg(𝑦) + |𝑌𝑣 ∪ {𝑣}| ≥ deg(𝑣) + |𝑌𝑣 | + 1 ≥ 𝑝. Thus, there exists a vertex 𝑦 that belongs to a clique 𝑌𝑣 ∪ {𝑣} in 𝐺 such that deg(𝑦) ≥ 𝑝 − |𝑌𝑣 ∪ {𝑣}|, a contradiction to Property 𝑃1 . Hence, deg(𝑣) + |𝑌𝑣 | + 1 < 𝑝 for all vertices 𝑣 ∈ 𝑆. Since deg(𝑣), |𝑌𝑣 |, and 𝑝 are integers, this implies that deg(𝑣) + |𝑌𝑣 | + 1 ≤ 𝑝 − 1 for all vertices 𝑣 ∈ 𝑆. Therefore, by Inequality (14.18), ∑︁ ∑︁
∑︁ deg(𝑣) + 2|𝑆| = |𝑌𝑣 | + deg(𝑣) + 2 ≤ 𝑝 = |𝑆| × 𝑝, 2𝑛 ≤ 𝛼1 (𝑆) + implying that 𝛼(𝐺) = |𝑆| ≥

𝑣 ∈𝑆

𝑣 ∈𝑆

𝑣 ∈𝑆 2𝑛 𝑝

as desired.

As an immediate consequence of Theorem 14.26, we have the following result due to Fajtlowicz [264, 265] on the independence number of graph. Corollary 14.27 (Fajtlowicz’ Theorem [264, 265]) If 𝐺 is a graph of order 𝑛 and maximum degree Δ containing no clique of size 𝑞, then 𝛼(𝐺) ≥

2𝑛 . Δ+𝑞

Proof Let 𝐺 be a graph of order 𝑛 containing no clique of size 𝑞 and let 𝑝 = Δ(𝐺) +𝑞. For every clique 𝑋 in 𝐺 and for all vertices 𝑥 ∈ 𝑋, we have |𝑋 | ≤ 𝑞 − 1, implying that

Chapter 14. Upper Domination Parameters

430

deg(𝑥) < Δ(𝐺) + 1 ≤ Δ(𝐺) + 𝑞 − |𝑋 | = 𝑝 − |𝑋 |, and therefore Property 𝑃1 holds in Theorem 14.26. By Theorem 14.26, 𝛼(𝐺) ≥

2𝑛 2𝑛 = . 𝑝 Δ+𝑞

In 1984 Fajtlowicz [265] studied classes of graphs that achieve equality in the bound of Corollary 14.27. We next present a simple probabilistic proof by Alon and Spencer [19] of the well-known result, known as the Caro-Wei Theorem, established independently by Caro [132] in 1979 and Wei [748] in 1981 that presents a lower bound on the independence number of a graph in terms of its degree sequence. Theorem 14.28 (Caro-Wei Theorem [132, 748]) For every graph 𝐺, 𝛼(𝐺) ≥

∑︁

𝑣 ∈𝑉 (𝐺)

1 . deg(𝑣) + 1

Proof Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 and let 𝜋 be a random permutation, or ordering, of the vertices in 𝑉, where 𝜋(𝑣) denotes the position of 𝑣 with respect to the permutation 𝜋. A vertex 𝑣 ∈ 𝑉 is a 𝜋-least vertex if 𝜋(𝑣) < 𝜋(𝑤) for all neighbors 𝑤 of 𝑣 in 𝐺, that is, if 𝑣 appears before all of its neighbors in the ordering 𝜋. Let 𝐼 be the set of all 𝜋-least vertices. By construction, the set 𝐼 is an independent set. For each vertex 𝑣 ∈ 𝑉, let 𝑋𝑣 be the indicator random variable for the vertex 𝑣 and the event 𝐼 and so 𝑋𝑣 (𝐼) = 1 if 𝑣 ∈ 𝐼 and 𝑋𝑣 (𝐼) = 0 otherwise. Thus, ∑︁ |𝐼 | = 𝑋𝑣 (𝐼). 𝑣 ∈𝑉


For a given vertex 𝑣 ∈ 𝐼, there are deg(𝑣) + 1 ! possible orderings of 𝑣 and its neighbors and of these orderings, deg(𝑣)! have the vertex 𝑣 as the least element in the ordering. Thus, the probability that a given vertex 𝑣 is included in the set 𝐼 is Pr(𝑣 ∈ 𝐼) =

deg(𝑣)! 1
= . +1 deg(𝑣) deg(𝑣) + 1 !

By linearity of expectation, ∑︁
∑︁
∑︁ E |𝐼 | = E 𝑋𝑣 (𝐼) = Pr(𝑣 ∈ 𝐼) = 𝑣 ∈𝑉

𝑣 ∈𝑉

𝑣 ∈𝑉

1 . deg(𝑣) + 1

Since expectation is an average value, there exists a permutation 𝜋 of the vertices in 𝐺 and an associated independent set 𝐼 (namely, the set of all 𝜋-least vertices) such that ∑︁ 1 . |𝐼 | ≥ deg(𝑣) + 1 𝑣 ∈𝑉 The desired bound now follows since 𝛼(𝐺) ≥ |𝐼 |.

Section 14.4. Independence Number

431

A fundamental result in graph theory is Turán’s Theorem on the independence number of a graph. As a consequence of the Caro-Wei Theorem (Theorem 14.28), we can readily prove this 1941 result of Turán [717]. For this purpose, we recall the well-known Cauchy-Schwarz Inequality which states that 𝑛 ∑︁

! 𝑥𝑖2

𝑖=1

𝑛 ∑︁

! 𝑦 2𝑖

≥

𝑖=1

𝑛 ∑︁

!2 𝑥𝑖 𝑦 𝑖 ,

𝑖=1

where 𝑥𝑖 and 𝑦 𝑖 , 1 ≤ 𝑖 ≤ 𝑛, are real numbers. Theorem 14.29 (Turán’s Theorem [717]) If 𝐺 is a graph of order 𝑛 and size 𝑚, then 𝑛 𝛼(𝐺) ≥ , degav (𝐺) + 1 where degav (𝐺) = 2𝑚/𝑛 is the average degree in 𝐺. Proof Let 𝐺 have order 𝑛 and size 𝑚. Let 𝑘 = degav (𝐺) be the average degree in 𝐺, and so 𝑚 = 12 𝑛𝑘. Let 𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 denote the degree sequence of 𝐺. Applying the Cauchy-Schwarz Inequality with 𝑥𝑖 =

√︁

𝑑𝑖 + 1

and

𝑦𝑖 = √

1 𝑑𝑖 + 1

for 𝑖 ∈ [𝑛], (2𝑚 + 𝑛)

𝑛 ∑︁ 𝑖=1

! ! 𝑛 ! !2 𝑛 𝑛 ∑︁ 1 ∑︁ ∑︁ 1 = (𝑑𝑖 + 1) ≥ 1 = 𝑛2 . 𝑑𝑖 + 1 𝑑 + 1 𝑖 𝑖=1 𝑖=1 𝑖=1

Hence, by the Caro-Wei Theorem, 𝛼(𝐺) ≥

𝑛 ∑︁ 𝑖=1

1 𝑛2 𝑛 𝑛 . ≥ = = 𝑑𝑖 + 1 2𝑚 + 𝑛 𝑘 + 1 degav (𝐺) + 1

Recall that a transversal (also called a vertex cover) in 𝐺 is a set 𝑇 of vertices that covers all the edges of 𝐺. The transversal number 𝜏(𝐺) (also called the vertex cover number 𝛽(𝐺)) is the minimum cardinality of a transversal in 𝐺. A transversal of 𝐺 of cardinality 𝜏(𝐺) is called a 𝜏-transversal of 𝐺. The complement of a transversal in 𝐺 is an independent set of 𝐺 and vice versa. Hence, by definition, we have the following relationship between transversals and independent sets, first observed in 1959 by Gallai [324] and previously mentioned in Chapters 1 and 4. Theorem 14.30 If 𝐺 is a graph of order 𝑛, then 𝜏(𝐺) + 𝛼(𝐺) = 𝑛. In view of Theorem 14.30, a lower bound on the independence number yields an upper bound on the transversal number, and vice versa. For example, the upper bound on the transversal number given in Theorem 14.31 yields the lower bound on the independence number given in Theorem 14.32. (We will shortly show that Chvátal and McDiarmid [176] proved a much stronger result.)

432

Chapter 14. Upper Domination Parameters

Theorem 14.31 ([176]) If 𝐺 is a connected graph of order 𝑛 and size 𝑚, then 𝜏(𝐺) ≤ 31 (𝑛 + 𝑚). Proof We proceed by induction on the order 𝑛 ≥ 1. If 𝑛 = 1, then 𝑚 = 0 and 𝜏(𝐺) = 0 < 13 (𝑛+𝑚). If 𝑛 = 2, then 𝑚 = 1 and 𝜏(𝐺) = 1 = 13 (𝑛+𝑚). This establishes the base cases. Let 𝑛 ≥ 3 and assume that if 𝐺 ′ is a connected graph of order 𝑛′ and size 𝑚 ′ , where 𝑛′ < 𝑛, then 𝜏(𝐺 ′ ) ≤ 13 (𝑛′ + 𝑚 ′ ). Let 𝐺 be a connected graph of order 𝑛 and size 𝑚. Since 𝑛 ≥ 3, we note that Δ(𝐺) ≥ 2. Let 𝑣 be a vertex of maximum degree in 𝐺 and so deg𝐺 (𝑣) = Δ(𝐺). We now consider the graph 𝐺 ′ = 𝐺 − 𝑣. Let 𝐺 ′ have order 𝑛′ and size 𝑚 ′ and so 𝑛′ = 𝑛 − 1 and 𝑚 ′ = 𝑚 − deg𝐺 (𝑣) ≤ 𝑚 − 2. Let 𝑇 ′ be a 𝜏-transversal in 𝐺 ′ and so 𝑇 ′ covers all edges in 𝑇 ′ . Applying the induction hypothesis to each component of 𝐺 ′ , 𝜏(𝐺 ′ ) = |𝑇 ′ | ≤ 13 (𝑛′ + 𝑚 ′ ) ≤ 13 (𝑛 + 𝑚) − 1. Since 𝑇 ′ ∪ {𝑣} is a transversal of 𝐺, 𝜏(𝐺) ≤ |𝑇 ′ | + 1 ≤ 13 (𝑛 + 𝑚), establishing the desired upper bound. As an immediate consequence of Theorems 14.30 and 14.31, we have the following lower bound on the independence number. Theorem 14.32 ([176]) If 𝐺 is a connected graph of order 𝑛 and size 𝑚, then 𝛼(𝐺) ≥ 23 𝑛 − 13 𝑚. Let 𝐼2 be the set of all points (𝜉, 𝜆) of nonnegative integers 𝜉 and 𝜆 for which the inequality 𝛼(𝐺) ≥ 𝜉𝑛 − 𝜆𝑚 holds for all connected graphs 𝐺 of order 𝑛 and size 𝑚.
As shown in Theorem 14.32, the point 23 , 13 ∈ 𝐼2 . A natural problem is to determine the (infinite) set 𝐼2 . First we show that 𝐼2 is convex. Theorem 14.33 The set 𝐼2 is a convex set. Proof Let (𝜉1 , 𝜆1 ) and (𝜉2 , 𝜆2 ) be two arbitrary points in the set 𝐼2 . Let 𝐺 be a connected graph of order 𝑛 and size 𝑚. Thus, 𝛼(𝐺) ≥ 𝜉1 𝑛 − 𝜆1 𝑚 and 𝛼(𝐺) ≥ 𝜉2 𝑛 − 𝜆2 𝑚. We show that the line segment 𝐿 joining the points (𝜉1 , 𝜆1 ) and (𝜉2 , 𝜆2 ) lies completely within the set 𝐼2 . Let (𝜉, 𝜆) be an arbitrary point on the line segment 𝐿. Thus,
(𝜉, 𝜆) = 𝜀(𝜉1 , 𝜆1 ) + (1 − 𝜀) (𝜉2 , 𝜆2 ) = 𝜀𝜉1 + (1 − 𝜀)𝜉2 , 𝜀𝜆1 + (1 − 𝜀)𝜆 2 for some real number 𝜀, where 0 ≤ 𝜀 ≤ 1. The point (𝜉, 𝜆) satisfies

𝜉𝑛 − 𝜆𝑚 = 𝜀𝜉1 + (1 − 𝜀)𝜉2 𝑛 − 𝜀𝜆1 + (1 − 𝜀)𝜆2 𝑚 = 𝜀(𝜉1 𝑛 − 𝜆1 𝑚) + (1 − 𝜀) (𝜉2 𝑛 − 𝜆2 𝑚) ≤ 𝜀𝛼(𝐺) + (1 − 𝜀)𝛼(𝐺) = 𝛼(𝐺), implying that the point (𝜉, 𝜆) belongs to the set 𝐼2 . By Theorem 14.33, the set 𝐼2 is a convex set. Let 𝑆2 be the set of all points (𝜉, 𝜆) of nonnegative integers 𝜉 and 𝜆 for which the inequality 𝜏(𝐺) ≤ 𝜉𝑛 + 𝜆𝑚 holds for all connected graphs 𝐺 of order 𝑛 and size 𝑚. A proof similar to the proof of

Section 14.4. Independence Number

433

Theorem 14.33 shows that the set 𝑆2 is a convex set. By Theorem 14.30, the inequality 𝛼(𝐺) ≥ 𝜉𝑛 − 𝜆𝑚 holds if and only if the inequality 𝜏(𝐺) ≤ (1 − 𝜉)𝑛 + 𝜆𝑚 holds. Therefore, finding all extreme points for the convex set 𝐼2 is equivalent to finding all extreme points for the convex set 𝑆2 . We say that a point (𝜉, 𝜆) of nonnegative integers 𝜉 and 𝜆 is tight in the convex set 𝐼2 if the point (𝜉, 𝜆) belongs to the set 𝐼2 and 𝛼(𝐺) = 𝜉𝑛 − 𝜆𝑚 for some connected graph 𝐺 of order 𝑛 and size 𝑚. Further, if (𝜉, 𝜆) is tight in the convex set 𝐼2 and 𝐺 is a connected graph 𝐺 of order 𝑛 and size 𝑚 satisfying 𝛼(𝐺) = 𝜉𝑛 − 𝜆𝑚, then we call 𝐺 a realizable graph for (𝜉, 𝜆). We shall need the following lemma. Lemma 14.34 If (𝜉𝑖 , 𝜆𝑖 ) and (𝜉𝑖+1 , 𝜆𝑖+1 ) are both tight in the convex set 𝐼2 for some connected graph 𝐹 of order 𝑛 and size 𝑚, then all points 𝜀(𝜉𝑖 , 𝜆𝑖 ) + (1 − 𝜀) (𝜉𝑖+1 , 𝜆𝑖+1 ), where 0 ≤ 𝜀 ≤ 1 are also tight in 𝐼2 for 𝐹. Proof Since (𝜉𝑖 , 𝜆𝑖 ) is tight for the graph 𝐹, the point (𝜉𝑖 , 𝜆𝑖 ) belongs to the convex set 𝐼2 , and so 𝛼(𝐺) ≥ 𝜉𝑖 𝑛 + 𝜆𝑖 𝑚 for all connected graphs 𝐺 of order 𝑛 and size 𝑚. Analogously, 𝛼(𝐺) ≥ 𝜉𝑖+1 𝑛 + 𝜆𝑖+1 𝑚 for all connected graphs 𝐺 of order 𝑛 and size 𝑚. Thus, for all connected graphs 𝐺 of order 𝑛 and size 𝑚, 𝛼(𝐺) = 𝜀𝛼(𝐺) + (1 − 𝜀)𝛼(𝐺) ≥ 𝜀(𝜉𝑖 𝑛 + 𝜆𝑖 𝑚) + (1 − 𝜀) (𝜉𝑖+1 𝑛 + 𝜆𝑖+1 𝑚)

= 𝜀𝜉𝑖 + (1 − 𝜀)𝜉𝑖+1 𝑛 + 𝜀𝜆𝑖 + (1 − 𝜀)𝜆𝑖+1 𝑚. Therefore, the point 𝜀(𝜉𝑖 , 𝜆𝑖 ) + (1 − 𝜀) (𝜉𝑖+1 , 𝜆𝑖+1 ) lies in the convex set 𝐼2 . Furthermore, since 𝐹 is a realizable graph for both points (𝜉𝑖 , 𝜆𝑖 ) and (𝜉𝑖+1 , 𝜆𝑖+1 ), we have 𝜉𝑖 𝑛 + 𝜆𝑖 𝑚 = 𝛼(𝐹) = 𝜉𝑖+1 𝑛 + 𝜆𝑖+1 𝑚. Therefore, if we considered 𝐹 instead of 𝐺 above, we would have equality everywhere, implying that the point 𝜀(𝜉𝑖 , 𝜆𝑖 ) + (1 − 𝜀) (𝜉𝑖+1 , 𝜆𝑖+1 ) is also tight in 𝐼2 , and 𝐹 is a realizable graph for this point. In 1992 Chvátal and McDiarmid [176] established the following upper bounds on the transversal number of a graph. A proof of this result can be found in [492]. Theorem 14.35 ([176]) For every integer 𝑘 ≥ 1, if 𝐺 is a graph of order 𝑛 and size 𝑚, then     𝑘 −1 2 𝜏(𝐺) ≤ 𝑛+ 𝑚, 𝑘 +1 𝑘 (𝑘 + 1) with equality if and only if 𝐺 is a complete graph of order 𝑘 or 𝑘 + 1. As an immediate consequence of Theorems 14.30 and 14.35, we have the following lower bound on the independence number. Theorem 14.36 ([176]) For every integer 𝑘 ≥ 1, if 𝐺 is a graph of order 𝑛 and size 𝑚, then     2 2 𝛼(𝐺) ≥ 𝑛− 𝑚, 𝑘 +1 𝑘 (𝑘 + 1) with equality if and only if 𝐺 is a complete graph of order 𝑘 or 𝑘 + 1.

Chapter 14. Upper Domination Parameters

434

As a special case of Theorem 14.36 for small 𝑘 ∈ [5], we have the following lower bounds on the independence number of a connected graph 𝐺 of order 𝑛 and size 𝑚: 𝑘=1:

𝛼(𝐺) ≥ 𝑛 − 𝑚

𝑘=2:

𝛼(𝐺) ≥ 32 𝑛 − 13 𝑚

𝑘=3:

𝛼(𝐺) ≥ 12 𝑛 − 16 𝑚

𝑘=4:

𝛼(𝐺) ≥ 25 𝑛 −

𝑘=5:

𝛼(𝐺) ≥ 13 𝑛 −

1 10 𝑚 1 15 𝑚.

We are now in a position to determine all extreme points in the convex set 𝐼2 . Theorem 14.37 ([176]) The set     2 2 , : 𝑘 = 1, 2, 3, . . . ∪ (1, 1) 𝑘 + 1 𝑘 (𝑘 + 1) is the set of extreme points of the convex set 𝐼2 . Proof We note that the point (1, 1) is an accumulation point (also called a limit point in the literature). For 𝑘 ≥ 1, let   2 2 (𝜉𝑖 , 𝜆𝑖 ) = , . 𝑘 + 1 𝑘 (𝑘 + 1) By Theorem 14.36, the points (𝜉𝑖 , 𝜆𝑖 ) and (𝜉𝑖+1 , 𝜆𝑖+1 ) are both tight for the complete graph of order 𝑘 + 1. Hence, by Lemma 14.34, every point on the line segment joining the points (𝜉𝑖 , 𝜆𝑖 ) and (𝜉𝑖+1 , 𝜆𝑖+1 ) is tight in 𝐼2 (for the complete graph of order 𝑘 + 1). As remarked earlier, finding all extreme points for the convex set 𝐼2 is equivalent to finding all extreme points for the convex set 𝑆2 .

Chapter 15

Relating the Core Parameters 15.1

Introduction

The Domination Chain, whose development is detailed in Chapter 2, establishes the following basic relationships among domination parameters for all graphs 𝐺: ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). In their 1998 book on domination in graphs [417], Haynes et al. presented an 8-page discussion of the following questions: Q1. Given an integer sequence 1 ≤ 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑 ≤ 𝑒 ≤ 𝑓 , does there exist a graph 𝐺 whose six parameters in the Domination Chain take on these values? Q2. Under what conditions are any two of the six parameters in the Domination Chain equal? Q3. Are there variants of these parameters for which a similar chain of inequalities exists? Q4. Are there other inequalities involving other graph parameters and any of these six parameters? To this list of questions, we would add the following. Q5. Does there exist a natural parameter pair, say 𝑎(𝐺) and 𝐴(𝐺), for which 𝑎(𝐺) ≤ ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺) ≤ 𝐴(𝐺)? In this chapter, we consider the inequalities of the restricted Domination Chain for only the core parameters as follows. Theorem 15.1 ([196]) For every graph 𝐺, 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺). The bounds in Theorem 15.1 are tight, as may be seen by taking 𝐺 to be the corona 𝐻 ◦ 𝐾1 of any graph 𝐻, which has 𝛾(𝐺) = 𝑖(𝐺) = 𝛼(𝐺) = Γ(𝐺) = |𝑉 (𝐻)|. However, the difference between each of these parameters can be made arbitrarily large. For example, for 𝑘 ≥ 1, if 𝐺 is the double star 𝑆(𝑘, 𝑘), then 𝛾(𝐺) = 2, 𝑖(𝐺) = 𝑘 + 1, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_15

435

436

Chapter 15. Relating the Core Parameters

and 𝛼(𝐺) = Γ(𝐺) = 2𝑘. For 𝑘 ≥ 2, if 𝐺 is the Cartesian product 𝐾 𝑘 □ 𝐾2 , then 𝛼(𝐺) = 2 and Γ(𝐺) = 𝑘. In 1979 Bollobás and Cockayne [84] were the first to observe the following relationship between the domination and total domination numbers of a graph without isolated vertices. Theorem 15.2 ([84]) If 𝐺 is an isolate-free graph, then 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ 2𝛾(𝐺). Proof By definition, every TD-set in an isolate-free graph 𝐺 is a dominating set, implying that 𝛾(𝐺) ≤ 𝛾t (𝐺). Consider next a 𝛾-set 𝑆 in the graph 𝐺. For each vertex Ð 𝑣 ∈ 𝑆, let 𝑣 ′ be an arbitrary neighbor of 𝑣 and let 𝑆 ′ = 𝑣 ∈𝑆 {𝑣 ′ }. The set 𝑆 ∪ 𝑆 ′ is a TD-set of 𝐺, implying that 𝛾t (𝐺) ≤ |𝑆 ∪ 𝑆 ′ | ≤ |𝑆| + |𝑆 ′ | ≤ 2|𝑆| = 2𝛾(𝐺). We note that 𝑖(𝐺) and 𝛾t (𝐺) are, in general, incomparable. For instance, if 𝐺 is a nontrivial complete graph, then 𝑖(𝐺) = 1 < 2 = 𝛾t (𝐺), while if 𝐺 is a double star 𝑆(𝑟, 𝑠) for 2 ≤ 𝑟 ≤ 𝑠, then 𝛾t (𝐺) = 2 < 𝑟 + 1 = 𝑖(𝐺). Examples abound for tightness of the inequalities in Theorem 15.2. For example, if 𝐺 is a double star, then 𝛾(𝐺) = 𝛾t (𝐺) = 2. If 𝐺 = 𝐹 ◦ 𝑃2 has order 𝑛, then by Observation 4.28 in Chapter 4, we have 𝛾t (𝐺) = 23 𝑛. Moreover, 𝛾(𝐺) = 13 𝑛, implying that in this case 𝛾t (𝐺) = 2𝛾(𝐺). Theorems 15.1 and 15.2 along with the fact that 𝛾t (𝐺) ≤ Γt (𝐺) give fundamental relationships among the core domination parameters. In this chapter, we explore these relationships in more depth. For any two graph parameters 𝜆 and 𝜇, we define a graph 𝐺 to be a (𝜆, 𝜇)-graph if 𝜆(𝐺) = 𝜇(𝐺). For example, a (𝛾, 𝑖)-graph is a graph 𝐺 satisfying 𝛾(𝐺) = 𝑖(𝐺).

15.2

Well-covered and Well-dominated Graphs

The (𝑖, 𝛼)-graphs are called well-covered and the (𝛾, Γ)-graphs (respectively, (𝛾t , Γt )graphs) are called well-dominated (respectively, well-total-dominated) in the literature. We give a brief summary of these concepts in this section. We provide only a sampling of the many known results on well-covered and well-dominated graphs. We review some definitions before preceding. A vertex 𝑣 in a graph 𝐺 is called simplicial if every two neighbors of 𝑣 are adjacent, or equivalently, if 𝐺 [N(𝑣)] is a complete graph. A simplicial vertex is a vertex that appears in exactly one maximal clique. A maximal clique of a graph 𝐺 containing at least one simplicial vertex is called a simplex. A graph 𝐺 is called simplicial if every vertex in 𝐺 is either simplicial or is adjacent to a simplicial vertex. A block in a graph 𝐺 is a maximal connected subgraph having the property that it contains no cut vertex. A graph 𝐺 is a block graph if every block of 𝐺 is a complete graph. A graph 𝐺 is called a block-cactus graph if every block of 𝐺 is either a complete graph or a cycle. A 5-cycle subgraph of a graph 𝐺 is called basic if it does not contain two adjacent vertices of degree three or more in 𝐺. A 4-cycle subgraph of 𝐺 is called basic if it contains two adjacent vertices of degree two in 𝐺 and the remaining two vertices belong to either a simplex or a basic 5-cycle.

Section 15.2. Well-covered and Well-dominated Graphs

437

15.2.1 Well-covered Graphs A graph 𝐺 is called well-covered if 𝑖(𝐺) = 𝛼(𝐺). Equivalently, a graph is wellcovered if every maximal independent set is in fact a maximum independent set. Thus, a graph 𝐺 is well-covered if every greedy algorithm for finding a maximal independent set produces an 𝛼-set of 𝐺. The concept of well-covered graphs was introduced by Plummer [640] in 1970. This was partly motivated by the observation that, whereas the problem of computing the independence number 𝛼(𝐺) of an arbitrary graph 𝐺 is NP-complete, for wellcovered graphs one can simply use any greedy maximal independent set algorithm to compute the value of 𝛼(𝐺) in polynomial time. The problem of deciding if a graph 𝐺 is well-covered is co-NP-complete, or equivalently, the problem of deciding if a graph 𝐺 is not well-covered is an NP-complete problem. This was proved independently by Sankaranarayana and Stewart [667] in 1992 and by Chvátal and Slater [177] in 1993. Also, the well-covered decision problem is co-NP-complete for 𝐾1,4 -free graphs (see [136]). On the other hand, the recognition problem for well-covered graphs can be solved in polynomial time for several classes of graphs including the following: (a) 𝐾1,3 -free graphs: Tankus and Tarsi [703, 704] (b) graphs of girth 𝑔 ≥ 5: Finbow et al. [294] (c) graphs of bounded maximum degree: Caro et al. [133] (d) simplicial graphs: Currie and Nowakowski [208], Prisner et al. [643] (e) chordal graphs, circular-arc graphs: Prisner et al. [643] (f) bipartite graphs: Ravindra [654] (g) graphs having no 4-cycle or 5-cycle subgraphs: Finbow et al. [295] (h) cubic graphs: Campbell et al. [129] (i) 4-connected 4-regular claw-free graphs, 4-connected planar claw-free graphs: Hartnell and Plummer [392] (j) block graphs: Topp and Volkmann [713] (k) unicyclic graphs: Topp and Volkmann [715]. Although a good deal of effort has been expended in an attempt to obtain a characterization of the well-covered graphs, this problem remains unsolved. But characterizations have been obtained for some families of graphs. In 1977 Ravindra [654] gave the following characterization of well-covered bipartite graphs. Theorem 15.3 ([654]) If 𝐺 is an isolate-free bipartite graph, then 𝐺 is well-covered if and only if 𝐺 has a perfect matching 𝑀 and, for every edge 𝑢𝑣 ∈ 𝑀, the subgraph induced by the vertices in N({𝑢, 𝑣}) is a complete bipartite graph. In 1990 Topp and Volkmann [713] provided the following characterization of well-covered block graphs. Theorem 15.4 ([713]) If 𝐺 is a block graph, then 𝐺 is well-covered if and only if every vertex of 𝐺 belongs to exactly one simplex of 𝐺. A graph 𝐺 is in the family SQC if 𝑉 (𝐺) can be partitioned into three disjoint subsets 𝑆, 𝑄, and 𝐶, where: (i) the subset 𝑆 contains all vertices in the simplexes

438

Chapter 15. Relating the Core Parameters

of 𝐺, and the simplexes of 𝐺 are vertex-disjoint, (ii) the subset 𝐶 consists of the vertices of the basic 5-cycles and the basic 5-cycles form a partition of 𝐶, and (iii) the remaining set 𝑄 contains all vertices of degree 2 in the basic 4-cycles. In 1991 Topp and Volkmann [715] provided the following characterization of well-covered unicyclic graphs. Theorem 15.5 ([715]) If 𝐺 is a unicyclic graph, then 𝐺 is well-covered if and only if 𝐺 ∈ {𝐶4 , 𝐶7 } ∪ SQC. In 1993 and 1994 Finbow et al. [294, 295] studied well-covered graphs of girth 5 and (𝐶4 , 𝐶5 )-free well-covered graphs. A corollary of their results is that the (𝐶4 , 𝐶5 )-free well-covered graphs can be recognized in polynomial time. In 1994 Randerath and Volkmann [647] characterized well-covered block-cactus graphs. Theorem 15.6 ([647]) If 𝐺 is a connected block-cactus graph, then 𝐺 is wellcovered if and only if 𝐺 ∈ {𝐶4 , 𝐶7 } ∪ SQC. In 2006 Randerath and Vestergaard [646] showed that every isolate-free wellcovered graph 𝐺 has a perfect [1, 2]-factor, that is, a spanning subgraph each component of which is either 1-regular or 2-regular. In 2018 Hartnell et al. [397] provided a variety of results on well-covered Cartesian product graphs. We give the following representative result on Cartesian products. Theorem 15.7 ([397]) If 𝐺 and 𝐻 are nontrivial connected graphs, both of which have girth at least 5, then the Cartesian product 𝐺 □ 𝐻 is well-covered if and only if it is isomorphic to either 𝐾2 □ 𝐾2 or 𝐶5 □ 𝐾2 . One of the most challenging problems in this area, posed in the survey of Plummer [641], is to find a good characterization of well-covered graphs of girth 4. In closing this section, we give the following conjecture about well-covered graphs due to Barbosa and Ellingham [58]. Conjecture 15.8 (Barbosa-Ellingham [58]) For any fixed integer 𝑑 ≥ 3, there is a finite collection G = {𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 } of well-covered graphs having maximum degree at most 𝑑, and a list of rules for connecting the elements of G together with edges, such that with finitely many exceptions, every well-covered graph with maximum degree at most 𝑑 can be obtained by joining elements of G according to these rules.

15.2.2

Well-dominated Graphs

A graph 𝐺 is called well-dominated if 𝛾(𝐺) = Γ(𝐺). Equivalently, a graph is well-dominated if every minimal dominating set is, in fact, a minimum dominating set. Thus, a graph 𝐺 is well-dominated if every greedy algorithm for finding a minimal dominating set produces a 𝛾-set of 𝐺. Among the simplest examples of well-dominated graphs are the complete graphs 𝐾𝑛 , for any 𝑛 ≥ 1, the path 𝑃4 , and the cycles 𝐶4 and 𝐶5 .

Section 15.2. Well-covered and Well-dominated Graphs

439

The study of well-dominated graphs was initiated in 1988 by Finbow et al. [293], who characterized well-dominated graphs having girth 𝑔 ≥ 5 as well as the welldominated bipartite graphs. It follows from the Domination Chain that if 𝛾(𝐺) = Γ(𝐺), then necessarily 𝑖(𝐺) = 𝛼(𝐺). Therefore, if a graph 𝐺 is well-dominated then it must also be well-covered, and thus, well-dominated graphs form a subset of the well-covered graphs. However, there exist well-covered graphs that are not well-dominated. Simple examples are Cartesian products 𝐺 = 𝐾𝑛 □ 𝐾2 , for 𝑛 ≥ 3, for which 𝛾(𝐺) = 𝑖(𝐺) = 𝛼(𝐺) = 2 < Γ(𝐺) = 𝑛. In 1990 Topp and Volkmann [714] gave the following characterizations of well-dominated and well-covered block graphs and unicyclic graphs. Theorem 15.9 ([714]) For a block graph 𝐺, the following statements are equivalent: (a) 𝛾(𝐺) = Γ(𝐺) (b) 𝛾(𝐺) = 𝛼(𝐺) (c) 𝑖(𝐺) = 𝛼(𝐺) (d) 𝑖(𝐺) = Γ(𝐺). In the following theorem, KU denotes the class of coronas of the form 𝐺 = 𝑈 ◦𝐾1 , where 𝑈 is a unicyclic graph, and S𝑘 denotes the family of unicyclic graphs whose unique cycle has length 𝑘. Theorem 15.10 ([714]) For a unicyclic graph 𝐺, the following statements are equivalent: (a) 𝛾(𝐺) = Γ(𝐺) (b) 𝛾(𝐺) = 𝛼(𝐺) (c) 𝐺 ∈ {𝐶3 , 𝐶4 , 𝐶5 , 𝐶7 } ∪ KU ∪ S3 ∪ S5 . In 1994 Finbow et al. [295] showed that there is a polynomial algorithm for recognizing well-dominated graphs with girth 𝑔 ≥ 6. However, the complexity of recognizing well-dominated graphs is not known. It is not even known if this problem is in the class NP. In 2011 Gionet et al. [337] characterized the class of well-dominated graphs that are claw-free, 4-connected, and 4-regular. In 2017 Levit and Tankus [559] proved that a (𝐶4 , 𝐶5 )-free graph is well-dominated if and only if it is well-covered. Thus, the problem of recognizing well-dominated (𝐶4 , 𝐶5 )-free graphs can be solved in polynomial time. In 2017 Gözüpek et al. [363] characterized well-dominated lexicographic product graphs as follows. Theorem 15.11 ([363]) If 𝐺◦𝐻 is a nontrivial lexicographic product of a connected graph 𝐺 and a graph 𝐻, then 𝐺 ◦ 𝐻 is well-dominated if and only if one of the following two conditions holds: (a) 𝐺 is well-dominated and 𝐻 is a complete graph, or (b) 𝐺 is a complete graph and 𝐻 is well-dominated with 𝛾(𝐻) = 2. In 2021 Anderson et al. [29] showed that there are exactly 11 well-dominated graphs that are connected and triangle-free, and whose domination number is at

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440

most 3. They proved that if 𝐺 and 𝐻 are both nontrivial connected graphs with girth 𝑔 ≥ 4, then the Cartesian product 𝐺 □ 𝐻 is well-dominated if and only if 𝐺 = 𝐻 = 𝐾2 , and, in general, for 𝐺 and 𝐻 both connected, if 𝐺 □ 𝐻 is well-dominated, then either 𝐺 or 𝐻 is well-dominated. They also obtained results for other product graphs. We close this section by noting that in 2021 Alizadeh and Gözüpek [14] considered families of graphs 𝐺 in which 𝜇 𝑑 (𝐺) = Γ(𝐺) − 𝛾(𝐺) is a fixed constant. Thus, if 𝜇 𝑑 (𝐺) = 0, then 𝐺 is a well-dominated graph.

15.2.3

Well-total Dominated Graphs

A graph 𝐺 is called well-total-dominated if 𝛾t (𝐺) = Γt (𝐺). For simplicity of notation, we refer to these graphs as WTD graphs. Thus, a graph is a WTD graph if every minimal TD-set is, in fact, a minimum TD-set, and any greedy algorithm for finding a minimal TD-set always finds a 𝛾t -set in the graph. Among the simplest examples of WTD graphs are the complete graphs 𝐾𝑛 , for any 𝑛 ≥ 2, and the cycles 𝐶3 , 𝐶4 , and 𝐶5 . Well-total-dominated graphs were introduced in 1997 by Hartnell [391], who characterized WTD paths and cycles and gave several constructions of WTD trees. Theorem 15.12 ([391]) The only WTD paths are 𝑃2 , 𝑃3 , 𝑃4 , 𝑃6 , 𝑃7 , and 𝑃10 . Theorem 15.13 ([391]) The only WTD cycles are 𝐶3 , 𝐶4 , 𝐶5 , 𝐶6 , 𝐶7 , 𝐶8 , 𝐶10 , and 𝐶14 . The author also proved the following. Theorem 15.14 ([391]) If 𝐺 is a connected graph with 𝛿(𝐺) ≥ 2 and girth 𝑔(𝐺) ≥ 15, then 𝛾t (𝐺) < Γt (𝐺). Corollary 15.15 If 𝐺 is a connected WTD graph with 𝛿(𝐺) ≥ 2, then 𝑔(𝐺) ≤ 14. In 2009 Frendrup and Vestergaard [309] proved that any WTD tree can be reduced to, or constructed from, three small trees: 𝑃2 , 𝑃4 , and the corona 𝑃3 ◦ 𝐾1 , using a small set of operations on trees. In 2021 Bahadir et al. [47] showed that although the complexity of recognizing WTD graphs is unknown, WTD graphs with a bounded total domination number can be recognized in polynomial time. In particular, they showed the following. Theorem 15.16 ([47]) If 𝐺 is a planar WTD graph of order 𝑛 with 𝛾t (𝐺) = 2 and 𝛿(𝐺) ≥ 3, then 𝑛 ≤ 16. The authors also improved Theorem 15.14 as follows. Theorem 15.17 ([47]) If 𝐺 is a connected WTD graph with 𝛿(𝐺) ≥ 3, then girth 𝑔(𝐺) ≤ 12.

15.3

Domination Versus Independent Domination

In this section, we explore relationships between the domination and independent domination numbers.

Section 15.3. Domination Versus Independent Domination

15.3.1

441

(𝜸, 𝒊)-graphs

By Theorem 15.1, for every graph 𝐺, we have 𝛾(𝐺) ≤ 𝑖(𝐺). It remains an open problem to characterize the (𝛾, 𝑖)-graphs. A necessary and sufficient forbiddensubgraph list characterizing (𝛾, 𝑖)-graphs is impossible, since the addition of a new vertex adjacent to all vertices of a graph 𝐺 produces a graph 𝐺 ′ containing 𝐺 as an induced subgraph with 𝛾(𝐺 ′ ) = 𝑖(𝐺 ′ ) = 1. The first result involving forbidden subgraphs that implies equality of the parameters 𝛾 and 𝑖 was Theorem 10.67 presented in Chapter 10, which showed that every claw-free graph, and hence, every line graph, is a (𝛾, 𝑖)-graph. Subsequently, in 1991 Topp and Volkmann [716] found 16 graphs such that if 𝐺 is 𝐹-free, where 𝐹 is one of the 16 graphs, then 𝐺 is a (𝛾, 𝑖)-graph. Certain properties of a graph also imply that the graph is a (𝛾, 𝑖)-graph. One example is the property that the vertices of degree 3 or more form an independent set. The proof of Theorem 15.18 by Goddard and Henning [352] in 2013 is similar to that of Theorem 10.67. Theorem 15.18 ([352]) Every graph in which the vertices of degree 3 or more in the graph form an independent set is a (𝛾, 𝑖)-graph. Proof Let 𝐺 be a graph in which the vertices of degree 3 or more in the graph form an independent set. Among all 𝛾-sets of 𝐺, let 𝑆 be chosen so that the subgraph 𝐺 [𝑆] induced by 𝑆 contains the fewest edges. We show that 𝑆 is an independent set. Suppose, to the contrary, that there exist vertices 𝑢 and 𝑣 in 𝑆 that are adjacent. Since 𝑆 is a minimal dominating set, by Lemma 2.72, we have ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. Since 𝑢 and 𝑣 are adjacent vertices, ipn[𝑢, 𝑆] = ∅ and ipn[𝑣, 𝑆] = ∅, implying that |epn[𝑢, 𝑆] | ≥ 1 and |epn[𝑣, 𝑆] | ≥ 1. In particular, both 𝑢 and 𝑣 have degree at least 2 in 𝐺. By supposition, at most one of 𝑢 and 𝑣 has degree 3 or more in 𝐺. Renaming 𝑢 and 𝑣 if necessary, we may assume that deg(𝑣) = 2. Let 𝑣 ′ be the neighbor of 𝑣 different from 𝑢, and so epn[𝑣, 𝑆] = {𝑣 ′ }. Replacing the vertex 𝑣 in 𝑆 with the vertex 𝑣 ′ produces a new 𝛾-set of 𝐺 that induces a subgraph that contains fewer edges than does 𝐺 [𝑆], a contradiction. Therefore, the set 𝑆 is an independent set and hence, is an ID-set of 𝐺, and so 𝑖(𝐺) ≤ |𝑆| = 𝛾(𝐺). By definition, 𝛾(𝐺) ≤ 𝑖(𝐺). Consequently, 𝛾(𝐺) = 𝑖(𝐺). Recall that a set 𝑆 of vertices in a graph 𝐺 is a packing if the vertices in 𝑆 are pairwise at distance at least 3 in 𝐺. The packing number 𝜌(𝐺) is the maximum cardinality of a packing. Since 𝜌(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺), we note that every (𝜌, 𝑖)-graph is a (𝛾, 𝑖)-graph. As stated in Chapter 5, Meir and Moon [589] showed that every tree is a (𝜌, 𝛾)-tree and so the (𝜌, 𝑖)-trees are precisely the (𝛾, 𝑖)-trees. The (𝛾, 𝑖)-trees were first described by Harary and Livingston [385] in 1986, but this description is rather complex. In 2006 a constructive characterization of (𝜌, 𝑖)-graphs was given by Dorfling et al. [240]. This characterization uses a construction of labeled graphs (𝐺, 𝜎), where the labeling function 𝜎 assigns every vertex 𝑣 of 𝐺 a label or status, denoted sta(𝑣). Let H be the family of {𝐴, 𝐵, 𝐶, 𝐷}-labeled graphs that can be obtained from a disjoint union of isolated vertices labeled 𝐷, and a disjoint union of isolated edges,

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each with a vertex labeled 𝐴 and a vertex labeled 𝐶, using Operations G1 , G2 , and G3 listed below and illustrated in Figure 15.1. Operation G1 . Given sta(𝑦) ∈ { 𝐴, 𝐷}, add a vertex 𝑥 and the edge 𝑥𝑦; let sta(𝑥) = 𝐵. Operation G2 . Given sta(𝑦) = 𝐴 and sta(𝑧) = 𝐶, add a vertex 𝑥 and the edges 𝑥𝑦 and 𝑥𝑧; let sta(𝑥) = 𝐵. Operation G3 . Given sta(𝑥) ∈ {𝐴, 𝐵} and sta(𝑦) = 𝐵, add the edge 𝑥𝑦. We remark that if (𝐺, 𝜎) ∈ H , then the vertices labeled 𝐴 or 𝐷 form an 𝑖-set of 𝐺, the vertices labeled 𝐶 or 𝐷 form a maximum packing in 𝐺, and there are equal numbers of 𝐴’s and 𝐶’s. 𝐵 𝐴 or 𝐷

𝐵

𝐵

𝐴 𝐶

𝐴 or 𝐵 (a) G1

(b) G2

(c) G3

Figure 15.1 Operations G1 , G2 , and G3

Theorem 15.19 ([240]) The (𝜌, 𝑖)-graphs are precisely those graphs 𝐺 such that (𝐺, 𝜎) ∈ H for some labeling 𝜎.

15.3.2

Domination Perfect Graphs

Recall that a clique in a graph is a complete subgraph in the graph. A graph is perfect if the chromatic number of every induced subgraph equals the cardinality of the largest clique of that subgraph, that is, a graph 𝐺 is perfect if and only if for every subset 𝑆 ⊆ 𝑉, we have 𝜒(𝐺 [𝑆]) = 𝜔(𝐺 [𝑆]), where 𝜒 and 𝜔 denote the chromatic number and the clique number, respectively. Motivated by the concept of perfect graphs in the chromatic sense, Sumner and Moore [695] defined domination perfect graphs in 1979. Definition 15.20 A graph 𝐺 is domination perfect if 𝛾(𝐻) = 𝑖(𝐻) for every induced subgraph 𝐻 of 𝐺. One can quickly generalize this notion to any two graph parameters 𝜆 and 𝜇 for which 𝜆(𝐺) ≤ 𝜇(𝐺), and say that a graph 𝐺 is (𝜆, 𝜇)-perfect if for every induced subgraph 𝐻 of 𝐺, 𝜆(𝐻) ≤ 𝜇(𝐻). We should make a distinction here since there are papers in which the authors say that a family F of graphs is (𝜆, 𝜇)-perfect only if for every graph 𝐺 ∈ F , 𝜆(𝐺) = 𝜇(𝐺), there is no requirement that 𝜆(𝐻) = 𝜇(𝐻) for every induced subgraph 𝐻 of 𝐺. In this section, we will only consider the Sumner and Moore definition of domination perfect, that 𝛾(𝐻) = 𝑖(𝐻) for every induced subgraph 𝐻 of a graph 𝐺.

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Perhaps the first family of domination perfect graphs was discovered accidentally in 1977 by Mitchell and Hedetniemi [594], who observed that in any tree 𝑇, a smallest dominating set of edges can always be found in which no two edges are adjacent. Of course, a minimum dominating set of edges in a tree 𝑇 is equivalent to a minimum dominating set of vertices in the line graph 𝐿 (𝑇) of 𝑇. Theorem 15.21 ([594]) For every tree 𝑇, 𝛾(𝐿(𝑇)) = 𝑖(𝐿 (𝑇)). This was soon generalized in 1978 by Allan and Laskar [15], who realized that line graphs of trees are examples of claw-free (that is, 𝐾1,3 -free) graphs. We restate their result here, a proof of which is given in Theorem 10.67 in Chapter 10. Theorem 15.22 ([15]) If 𝐺 is a claw-free graph, then 𝛾(𝐺) = 𝑖(𝐺). Since every subgraph of a claw-free graph is claw-free, Theorem 15.22 yields the following result. Corollary 15.23 Every claw-free graph is domination perfect. In 1965 Van Rooij and Wilf [726] proved that every line graph is claw-free. Hence, as a consequence of Corollary 15.23, we have the following result. Corollary 15.24 ([726]) Every line graph is domination perfect. The subdivision graph 𝑆(𝐺) of a graph 𝐺 is the graph obtained by subdividing each edge of 𝐺 exactly once. The middle graph 𝑀 (𝐺) of a graph 𝐺 is the graph obtained from the subdivision graph 𝑆(𝐺) by adding an edge between any two subdivision vertices if and only if the edges they subdivide are adjacent in 𝐺. In 1976 Hamada and Yoshimura [376] showed that for any graph 𝐺, 𝑀 (𝐺) = 𝐿(𝐺 ◦ 𝐾1 ), that is, the middle graph 𝑀 (𝐺) of any graph 𝐺 is isomorphic to the line graph of the corona 𝐺 ◦ 𝐾1 . From this we get the following corollary. Corollary 15.25 ([376]) Every middle graph is domination perfect. In 1978 Cockayne et al. [196] showed the following about middle graphs, where once again, 𝛽′ (𝐺) denotes the edge covering number of 𝐺. Theorem 15.26 ([196]) For any connected graph 𝐺, ir(𝑀 (𝐺)) = 𝛾(𝑀 (𝐺)) = 𝑖(𝑀 (𝐺)) = 𝛽′ (𝐺). In 1986 Favaron [273] generalized the Allan and Laskar result given in Theorem 15.22 by adding a second forbidden subgraph for domination perfect graphs, called the A-L graph, which is the unicyclic graph of order 𝑛 = 7 consisting of a triangle, at two vertices of which are attached paths of length two. This graph is also the line graph of the spider 𝑆1,3,3 . Theorem 15.27 ([273]) If 𝐺 is a graph containing no claw or no A-L graph, then ir(𝐺) = 𝛾(𝐺) = 𝑖(𝐺).

444

Chapter 15. Relating the Core Parameters

A graph 𝐺 is called minimal domination imperfect if 𝛾(𝐺) < 𝑖(𝐺), yet for all proper induced subgraphs 𝐻 of 𝐺, 𝛾(𝐻) = 𝑖(𝐻). A simple check of all graphs 𝐺 of order 𝑛 ≤ 5 shows that all of them have a 𝛾-set that is independent and so for each of them 𝛾(𝐺) = 𝑖(𝐺). And since every induced subgraph of these graphs is also contained in this set, all graphs of order 𝑛 ≤ 5 are domination perfect. Of the 156 graphs of order 𝑛 = 6, there are only four for which 𝛾(𝐺) < 𝑖(𝐺). The smallest in terms of size is the double star 𝑆(2, 2), consisting of two adjacent vertices which we will label 𝑣 3 and 𝑣 4 . Two leaves 𝑣 1 and 𝑣 2 are adjacent to vertex 𝑣 3 and two leaves 𝑣 5 and 𝑣 6 are adjacent to vertex 𝑣 4 . Additional edges can be added to this double star to produce the four minimal domination imperfect graphs, 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 , of order 𝑛 = 6, which are listed below. (a) 𝐻1 = 𝑆(2, 2). (b) 𝐻2 is obtained from 𝐻1 by adding the edge 𝑣 1 𝑣 6 . (c) 𝐻3 is obtained from 𝐻2 by adding the edge 𝑣 1 𝑣 5 . (d) 𝐻4 is obtained from 𝐻3 by adding the edges 𝑣 2 𝑣 5 and 𝑣 2 𝑣 6 , that is, 𝐻4 = 𝐾3,3 . The four graphs 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 are illustrated in Figure 15.2(a), (b), (c), and (d), respectively. Of the 1044 graphs of order 𝑛 = 7, there are six additional minimal domination imperfect graphs. Two of these six graphs, which we name 𝐻5 and 𝐻6 , are shown in Figure 15.2(e) and (f), respectively. As the order 𝑛 grows, the number of minimal domination imperfect graphs of order 𝑛 grows exponentially. In 1979 Sumner and Moore [695] established that it is not necessary to check every induced subgraph of a graph in order to determine if it is domination perfect. They proved the following key theorem, which was instrumental in obtaining a full characterization of domination perfect graphs some 16 years later. Theorem 15.28 ([695]) A graph 𝐺 is domination perfect if and only if 𝛾(𝐻) = 𝑖(𝐻) for every induced subgraph 𝐻 of 𝐺 with 𝛾(𝐻) = 2. In the remainder of this section, we discuss results that have been obtained since the 1990 survey by Sumner [693] on domination perfect graphs. In 1990 Sumner [693] characterized planar graphs that are domination perfect. Theorem 15.29 ([693]) If 𝐺 is planar, then 𝐺 is domination perfect if and only if 𝐺 does not contain an induced subgraph 𝐻 of order at most 8 such that 𝛾(𝐻) = 2 and 𝑖(𝐻) > 2. In 1991 Zverovich and Zverovich [789] offered a forbidden induced subgraph characterization of domination perfect graphs that are triangle-free, involving the four forbidden subgraphs 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 , of order 6. Theorem 15.30 ([789]) If 𝐺 is a triangle-free graph, then 𝐺 is domination perfect if and only if 𝐺 does not contain any of the four graphs 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 as an induced subgraph. In 1991 Topp and Volkmann [716] identified thirteen graphs 𝐹 such that being 𝐹free implies that a graph is domination perfect. In 1991 Zverovich and Zverovich [789]

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offered a finite forbidden induced subgraph characterization of domination perfect graphs. However, in 1993 Fulman [315] showed that this characterization is not correct, and he presented a sufficient condition for a graph to be domination perfect in terms of eight forbidden induced subgraphs. Subsequently, in 1995 Zverovich and Zverovich [790] provided a characterization of domination perfect graphs in terms of seventeen forbidden induced subgraphs. Each of these seventeen graphs 𝐺 are minimal domination imperfect. Theorem 15.31 ([790]) A graph is domination perfect if and only if it does not contain any of the seventeen graphs 𝐻1 , 𝐻2 , . . . , 𝐻17 shown in Figure 15.2 as an induced subgraph. It is worth noting that the only tree among the 17 forbidden subgraphs 𝐻1 , 𝐻2 , . . . , 𝐻17 shown in Figure 15.2 is the double star 𝑆(2, 2). The following was pointed out in 2006 by Dorfling et al. [240], a proof of which can be constructed from observations by any of Fulman [315], Sumner [693], and Zverovich and Zverovich [790]. Theorem 15.32 ([240]) If 𝑇 is a tree, then 𝑇 is domination perfect if and only if it does not contain two adjacent vertices of degree 3 or more, or equivalently, a tree 𝑇 is domination perfect if and only if 𝑇 is 𝑆(2, 2)-free. Sumner [693] proved a more general result. Theorem 15.33 ([693]) If 𝐺 is a chordal graph, then 𝐺 is domination perfect if and only if 𝐺 is 𝑆(2, 2)-free. We remark that several other maximal subclasses of domination perfect graphs can be found at the website graphclasses.org. While it is known that proper interval graphs are domination perfect, several other maximal subclasses of domination perfect graphs are as well, including the following, where 𝐻1 and 𝐻2 are the graphs shown in Figure 15.2(a) and (b): (i) (𝐻1 , 𝐻2 , 𝐾3,3 , 𝑋45 , 𝐾3 )-free (proper subclass) (ii) (𝐶𝑛+4 , 𝐻)-free (proper subclass) (iii) chordal ∩ domination perfect (proper subclass) (iv) planar ∩ domination perfect (possibly equal) (v) triangle-free ∩ domination perfect (proper subclass) (vi) strong domination perfect (possibly equal; see Rautenbach and Zverovich [653]).

15.3.3

Regular Graphs

In 2012 Goddard et al. [358] initiated the study of the independent domination number versus the domination number in regular graphs. If 𝐺 is a connected 𝑘regular graph of order  𝑛, where 𝑘 = 2, then 𝐺 is a cycle 𝐶𝑛 of order 𝑛. In this case, 𝑖(𝐺) = 𝛾(𝐺) = 31 𝑛 and 𝑖(𝐺)/𝛾(𝐺) = 1. Hence, the ratio of the independent domination number to the domination number in 𝑘-regular graphs is only of interest when 𝑘 ≥ 3. For 𝑘 = 3, it is shown in [358] that if 𝐺 is a connected 𝑘-regular graph, then the ratio 𝑖(𝐺)/𝛾(𝐺) ≤ 3/2, with equality if and only if 𝐺 = 𝐾3,3 . This result was

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(a) 𝐻1

(b) 𝐻2

(c) 𝐻3

(d) 𝐻4

(e) 𝐻5

(f) 𝐻6

(g) 𝐻7

(h) 𝐻8

(i) 𝐻9

(j) 𝐻10

(k) 𝐻11

(l) 𝐻12

(m) 𝐻13

(n) 𝐻14

(o) 𝐻15

(p) 𝐻16

(q) 𝐻17

Figure 15.2 Minimal domination imperfect graphs 𝐻1 , 𝐻2 , . . . , 𝐻17 extended in 2020 by Babikir and Henning [43] to larger values of 𝑘, namely 𝑘 ∈ {4, 5, 6}. The authors in [43] posed the open question of whether this result is true for all integers 𝑘 ≥ 3. This question was answered in 2021 by Knor et al. [533], who proved the following result. Theorem 15.34 ([533]) For all integers 𝑘 ≥ 3, if 𝐺 is a connected 𝑘-regular graph, then

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447

𝑖(𝐺) 𝑘 ≤ , 𝛾(𝐺) 2 with equality if and only if 𝐺 = 𝐾 𝑘,𝑘 . Proof For 𝑘 ≥ 3, let 𝐺 be a connected 𝑘-regular graph. Let 𝑋 be a 𝛾-set of 𝐺 and let 𝑌 = 𝑉 (𝐺) \ 𝑋. Let ℓ be the number of edges in the subgraph 𝐺 [𝑋]. We proceed further with the following claim. Claim 15.34.1 If ℓ < 12 𝛾(𝐺), then

𝑖 (𝐺) 𝛾 (𝐺)

< 𝑘2 .

Proof Suppose that ℓ < 12 𝛾(𝐺). If ℓ = 0, then 𝑋 is an ID-set of 𝐺, implying that 𝑖(𝐺) = 𝛾(𝐺). Hence, we may assume that ℓ ≥ 1, for otherwise 𝑖(𝐺)/𝛾(𝐺) = 1 < 𝑘/2. Let 𝑋1 be a maximal independent set in 𝐺 [𝑋], and let 𝑋2 = 𝑋 \ 𝑋1 . By the maximality of the set 𝑋1 , every vertex in 𝑋2 has at least one neighbor in 𝑋1 , implying that ℓ ≥ |𝑋2 |. Thus, |𝑋 | = |𝑋1 | + |𝑋2 | ≤ |𝑋1 | + ℓ, and so |𝑋1 | ≥ |𝑋 | − ℓ. Thus, |𝑋1 | = |𝑋 | − ℓ + 𝑝 for some integer 𝑝 ≥ 0. Let 𝑋2 = {𝑥 1 , 𝑥2 , . . . , 𝑥ℓ − 𝑝 }, and let 𝑁𝑖 be the set of neighbors of 𝑥𝑖 in 𝑌 , that is, Ð −𝑝 𝑁𝑖 = N(𝑥𝑖 ) ∩ 𝑌 for 𝑖 ∈ [ℓ − 𝑝]. Let 𝑌1 = ℓ𝑖=1 𝑁𝑖 . Each vertex 𝑥𝑖 ∈ 𝑋2 has at least one neighbor in 𝑋1 and therefore at most 𝑘 − 1 neighbors in 𝑌 , and so |𝑁𝑖 | ≤ 𝑘 − 1 for 𝑖 ∈ [ℓ − 𝑝], implying that |𝑌1 | ≤ (𝑘 − 1) (ℓ − 𝑝). Let 𝑌2 be the set of vertices in 𝑌1 that have a neighbor in 𝑋1 and let 𝑌3 = 𝑌1 \ 𝑌2 . Further, let 𝑌4 be an ID-set in 𝐺 [𝑌3 ]. Since 𝑌4 ⊆ 𝑌3 ⊆ 𝑌1 , we have |𝑌4 | ≤ |𝑌1 | ≤ (𝑘 − 1) (ℓ − 𝑝). By construction and our earlier observations, the set 𝑋1 dominates the set 𝑋2 ∪ 𝑌2 ∪ (𝑌 \ 𝑌1 ), and the set 𝑌4 dominates the set 𝑌3 . Further, the set 𝑋1 ∪ 𝑌4 is an ID-set of 𝐺, implying that 𝑖(𝐺) ≤ |𝑋1 | + |𝑌4 |
≤ |𝑋 | − ℓ + 𝑝 + (𝑘 − 1) (ℓ − 𝑝) = |𝑋 | + (𝑘 − 2) (ℓ − 𝑝) ≤ |𝑋 | + (𝑘 − 2)ℓ < 𝛾(𝐺) + (𝑘 − 2) × 12 𝛾(𝐺) = 𝑘2 𝛾(𝐺), or equivalently, 𝑖(𝐺)/𝛾(𝐺) < 𝑘/2. By Claim 15.34.1, we may assume that ℓ ≥ 12 𝛾(𝐺), for otherwise the desired result follows (with strict inequality). Claim 15.34.2

𝛾(𝐺) ≥ 𝑛𝑘 .

Proof We count the number of edges | [𝑋, 𝑌 ] | between the vertices in 𝑋 and the vertices in 𝑌 . Since 𝑋 is a dominating set of 𝐺, every vertex of 𝑌 has at least one neighbor in 𝑋, and so | [𝑋, 𝑌 ] | ≥ |𝑌 |. Counting edges emanating from the set 𝑋, the 𝑘-regularity of the graph 𝐺 implies that | [𝑋, 𝑌 ] | = 𝑘 |𝑋 | − 2ℓ. Hence, since |𝑋 | = 𝛾(𝐺) and ℓ ≥ 12 𝛾(𝐺), |𝑌 | ≤ |[𝑋, 𝑌 ] | = 𝑘 |𝑋 | − 2ℓ ≤ 𝑘𝛾(𝐺) − 𝛾(𝐺) = (𝑘 − 1)𝛾(𝐺),

Chapter 15. Relating the Core Parameters

448

and so 𝑛 = |𝑋 | + |𝑌 | ≤ 𝛾(𝐺) + (𝑘 − 1)𝛾(𝐺) = 𝑘𝛾(𝐺), or equivalently, 𝛾(𝐺) ≥ 𝑛𝑘 . We now return to the proof of Theorem 15.34. By Theorem 6.90, we have that 𝑖(𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 = 𝐾 𝑘,𝑘 . By Claim 15.34.2, we have 𝛾(𝐺) ≥ 𝑘1 𝑛. Therefore, 1 𝑛 𝑘 𝑖(𝐺) ≤ 21 = . (15.1) 𝛾(𝐺) 2 𝑘𝑛 This establishes the desired upper bound in the statement of the theorem. Suppose, next, that 𝐺 is a connected 𝑘-regular graph satisfying 𝑖(𝐺)/𝛾(𝐺) = 𝑘/2. In this case, we must have equality in Inequality (15.1). In particular, 𝑖(𝐺) = 12 𝑛, implying by Theorem 6.90 that 𝐺 = 𝐾 𝑘,𝑘 . A natural problem is to determine, for each value of 𝑘 ≥ 3, an improved upper bound on the ratio 𝑖(𝐺)/𝛾(𝐺) given in Theorem 15.34 if we forbid the exceptional 𝑘 denote the family of all connected 𝑘-regular graph 𝐺 = 𝐾 𝑘,𝑘 . For 𝑘 ≥ 3, let R reg graphs different from 𝐾 𝑘,𝑘 , and define 𝑟 𝑘,dom = sup 𝑘 𝐺 ∈ R reg

𝑖(𝐺) . 𝛾(𝐺)

In 2013 Southey and Henning [686] showed that the 32 -ratio given in Theorem 15.34 can be strengthened to a 43 -ratio if we forbid 𝐾3,3 , that is, they showed that 𝑟 3,dom = 43 . The proof we present of this result in [686] is similar to the proof given previously of Theorem 15.34. Theorem 15.35 ([686]) For connected cubic graphs, 𝑟 3,dom = 43 . Proof Let 𝐺 be a connected cubic graph. If 𝐺 is the 5-prism 𝐶5 □ 𝐾2 , then 𝑖(𝐺) = 4 and 𝛾(𝐺) = 3, implying that 𝑟 3,dom ≥ 4/3. Hence, it suffices to show that 𝑟 3,dom ≤ 4/3. Let 𝐺 ≠ 𝐾3,3 be a connected cubic graph. Let 𝑋 be a 𝛾-set of 𝐺, let 𝑌 = 𝑉 (𝐺) \ 𝑋, and let ℓ be the number of edges in the subgraph 𝐺 [𝑋]. Claim 15.35.1 If ℓ < 13 𝛾(𝐺), then

𝑖 (𝐺) 𝛾 (𝐺)

< 43 .

Proof Suppose that ℓ < 13 𝛾(𝐺). If ℓ = 0, then 𝑋 is an ID-set of 𝐺, implying that 𝑖(𝐺) = 𝛾(𝐺). Hence, we may assume that ℓ ≥ 1, for otherwise 𝑖(𝐺)/𝛾(𝐺) = 1 < 4/3. Adopting the identical notation we used in the proof of Claim 15.34.1 and noting that 𝑘 = 3, we have 𝑖(𝐺) ≤ |𝑋 | + ℓ < 𝛾(𝐺) + 13 𝛾(𝐺) = 43 𝛾(𝐺) or equivalently, 𝑖(𝐺)/𝛾(𝐺) < 4/3. By Claim 15.35.1, we may assume that ℓ ≥ 13 𝛾(𝐺), for otherwise the desired result follows (with strict inequality). Claim 15.35.2

𝛾(𝐺) ≥

3 10 𝑛.

Proof Counting the number of edges | [𝑋, 𝑌 ] | and noting that ℓ ≥ 13 𝛾(𝐺), |𝑌 | ≤ |[𝑋, 𝑌 ] | = 3|𝑋 | − 2ℓ ≤ 3𝛾(𝐺) − 32 𝛾(𝐺) = 73 𝛾(𝐺),

Section 15.3. Domination Versus Independent Domination and so 𝑛 = |𝑋 | + |𝑌 | ≤ 𝛾(𝐺) + 37 𝛾(𝐺) =

10 3 𝛾(𝐺),

449

or equivalently, 𝛾(𝐺) ≥

3 10 𝑛.

We now return to the proof of Theorem 15.35. Since 𝐺 ≠ 𝐾3,3 is a connected cubic 3 graph, applying Theorem 6.93, we have 𝑖(𝐺) ≤ 25 𝑛. By Claim 15.35.2, 𝛾(𝐺) ≥ 10 𝑛. Therefore, 2 𝑛 𝑖(𝐺) 4 ≤ 53 = . (15.2) 𝛾(𝐺) 3 𝑛 10 This proves that 𝑟 3,dom ≤ 43 . As observed earlier, 𝑟 3,dom ≥ 43 . Consequently, 𝑟 3,dom = 4 3. If 𝐺 ≠ 𝐾3,3 is a connected cubic graph satisfying 𝑖(𝐺)/𝛾(𝐺) = 4/3, then there must be equality in Inequality (15.2). This implies that all the inequalities in the 3 proof of Theorem 15.35 must be equalities. In particular, 𝑖(𝐺) = 25 𝑛, 𝛾(𝐺) = 10 𝑛, 1 and ℓ = 3 𝛾(𝐺). Further, |𝑋 | = 3ℓ and |𝑌 | = 7ℓ. Thus, 𝑖(𝐺) = 4ℓ, 𝛾(𝐺) = 3ℓ, and 𝑛 = 10ℓ. Each vertex of 𝑌 has exactly one neighbor in 𝑋. Choosing the 𝛾-set 𝑋 of 𝐺 such that 𝐺 [𝑋] has minimum size among all 𝛾-sets, we can further deduce that 𝐺 [𝑋] = ℓ𝐾1 ∪ ℓ𝐾2 . Using these structural properties of the graph 𝐺, one can show that if ℓ ≥ 2, then 𝑖(𝐺) < 4ℓ, a contradiction. Hence, ℓ = 1 and one can readily deduce that 𝐺 = 𝐶5 □ 𝐾2 . The full details of the proof that if 𝐺 ≠ 𝐾3,3 is a connected cubic graph satisfying 𝑖(𝐺)/𝛾(𝐺) = 4/3, then 𝐺 = 𝐶5 □ 𝐾2 can be found in [686]. The authors in [686] conjectured that if 𝐺 is a connected cubic graph of sufficiently large order, then 𝑖(𝐺)/𝛾(𝐺) ≤ 6/5. This conjecture was disproved in 2016 by O and West [618], who constructed an infinite family of 2-connected cubic graphs 𝐻ℓ with the ratio of the independent domination number to the domination number equal to 54 . In order to describe their construction, let 𝐹 be obtained from the 14-cycle 𝑥 𝑎 1 𝑎 2 . . . 𝑎 6 𝑦 𝑏 6 𝑏 5 . . . 𝑏 1 𝑥 by adding the chords 𝑎 3 𝑏 4 , 𝑎 4 𝑏 3 , and 𝑎 𝑖 𝑏 𝑖 for 𝑖 ∈ {1, 2, 5, 6}, as illustrated in Figure 15.3. 𝑏1

𝑏2

𝑏3

𝑏4

𝑏5

𝑏6 𝑦

𝑥 𝑎1

𝑎2

𝑎3

𝑎4

𝑎5

𝑎6

Figure 15.3 A subcubic graph 𝐹 constructed by O and West Given ℓ ≥ 1 disjoint copies 𝐹1 , 𝐹2 , . . . , 𝐹ℓ of 𝐹, with 𝑥𝑖 and 𝑦 𝑖 being the names of the vertices 𝑥 and 𝑦 in 𝐹𝑖 , form 𝐻ℓ by adding the edges of the form 𝑦 𝑖−1 𝑥𝑖 (with indices taken modulo ℓ). As remarked by O and West [618], when ℓ = 1, the indices 𝑖 − 1 and 𝑖 are congruent modulo ℓ, and in this case the construction simply adds the edge 𝑦𝑥 to 𝐹. The resulting graph 𝐻1 is 3-connected, while for ℓ ≥ 2 the constructed graph 𝐻ℓ has connectivity 2. In all cases, 𝐻ℓ is a cubic graph of order 14ℓ. The graph 𝐻2 , for example, is illustrated in Figure 15.4. We are now in a position to state the O-West result.

Chapter 15. Relating the Core Parameters

450 𝑥1

𝑦 1 𝑥2

𝑦2

Figure 15.4 The cubic graph 𝐻2 constructed by O and West

Theorem 15.36 ([618]) For a graph 𝐻ℓ with ℓ ≥ 1, 𝑖(𝐻ℓ ) = 5ℓ and 𝛾(𝐻ℓ ) = 4ℓ. Proof First, we prove that 𝛾(𝐻ℓ ) = 4ℓ. The set {𝑎 1 , 𝑎 6 , 𝑏 3 , 𝑏 4 }, for example, is a dominating set in the graph 𝐹 shown in Figure 15.3, implying that 𝛾(𝐻ℓ ) ≤ 4ℓ. Hence, it suffices to show that 𝛾(𝐻ℓ ) ≥ 4ℓ. Suppose, to the contrary, that 𝛾(𝐻ℓ ) < 4ℓ. Let 𝑆 be a 𝛾-set of 𝐻ℓ and let 𝑆𝑖 = 𝑆 ∩ 𝑉 (𝐹𝑖 ) for 𝑖 ∈ [ℓ]. By the Pigeonhole Principle, |𝑆𝑖 | ≤ 3 for some 𝑖 ∈ [ℓ]. Since each vertex dominates four vertices (including itself), this implies that neither vertex 𝑥𝑖 nor 𝑦 𝑖 is dominated from within 𝐹𝑖 by the set 𝑆𝑖 . Therefore, the set 𝑆𝑖 does not contain the vertices labeled 𝑎 1 , 𝑏 1 , 𝑎 6 , 𝑏 6 in 𝐹𝑖 , implying that 𝑆𝑖 does contain the vertices labeled 𝑎 2 , 𝑏 2 , 𝑎 5 , 𝑏 5 in 𝐹𝑖 in order to dominate 𝑎 1 , 𝑏 1 , 𝑎 6 , 𝑏 6 in 𝐹𝑖 . Hence, |𝑆𝑖 | ≥ 4, contradicting the supposition that |𝑆𝑖 | ≤ 3. We deduce that 𝛾(𝐻ℓ ) = 4ℓ. Next, we prove that 𝑖(𝐻ℓ ) = 5ℓ. The set {𝑎 1 , 𝑎 4 , 𝑎 6 , 𝑏 2 , 𝑏 4 }, for example, is an ID-set in the graph 𝐹 shown in Figure 15.3, implying that 𝑖(𝐻ℓ ) ≤ 5ℓ. Hence, it suffices to show that 𝑖(𝐻ℓ ) ≥ 5ℓ. Suppose, to the contrary, that 𝑖(𝐻ℓ ) < 5ℓ. Let 𝑆 be an 𝑖-set of 𝐻ℓ , and let 𝑆𝑖 = 𝑆 ∩ 𝑉 (𝐹𝑖 ) for 𝑖 ∈ [ℓ]. By the Pigeonhole Principle, |𝑆𝑖 | ≤ 4 for some 𝑖 ∈ [ℓ]. For clarity, we keep the names of the vertices of 𝐹 for the vertices of 𝐹𝑖 . In particular, 𝑥 = 𝑥𝑖 and 𝑦 = 𝑦 𝑖 . Within the copy of 𝐹𝑖 , let 𝑋𝑖 = {𝑥, 𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑏 1 , 𝑏 2 , 𝑏 3 } and 𝑌𝑖 = {𝑎 4 , 𝑎 5 , 𝑎 6 , 𝑏 4 , 𝑏 5 , 𝑏 6 , 𝑦}. Thus, (𝑋𝑖 , 𝑌𝑖 ) is a partition of 𝑉 (𝐹𝑖 ). In order to dominate the vertices in {𝑎 1 , 𝑎 2 , 𝑏 1 , 𝑏 2 } in 𝐹𝑖 , |𝑋 ∩ 𝑆𝑖 | ≥ 2. Analogously, in order to dominate the vertices in {𝑎 5 , 𝑎 6 , 𝑏 5 , 𝑏 6 }, |𝑌 ∩ 𝑆𝑖 | ≥ 2. Since |𝑋 ∩ 𝑆𝑖 | + |𝑌 ∩ 𝑆𝑖 | = |𝑆𝑖 | = 4, it follows that |𝑋 ∩ 𝑆𝑖 | = 2 and |𝑌 ∩ 𝑆𝑖 | = 2. Since {𝑎 2 , 𝑏 2 } ⊄ 𝑆𝑖 and {𝑎 5 , 𝑏 5 } ⊄ 𝑆𝑖 , the set 𝑆𝑖 must contain a vertex in {𝑎 3 , 𝑎 4 , 𝑏 3 , 𝑏 4 }. By symmetry, we may assume that 𝑎 3 ∈ 𝑆𝑖 , implying that 𝑎 4 , 𝑏 4 ∉ 𝑆𝑖 . Dominating the vertex 𝑏 3 now requires either 𝑏 2 or 𝑏 3 in 𝑆, implying that the vertex 𝑎 1 is not dominated by 𝑆𝑖 , since |𝑋 ∩ 𝑆𝑖 | = 2. We deduce that 𝑖(𝐻ℓ ) = 5ℓ. The following corollary is an immediate consequence of Theorem 15.36. Corollary 15.37 ([618]) There exists an infinite family of 2-connected cubic graphs 𝐺 such that 𝑖(𝐺)/𝛾(𝐺) = 5/4. However, it remains an open question to determine whether the 43 -ratio in Theorem 15.35 can be improved to a 54 -ratio if we forbid finitely many graphs. O and West [618] posed the following question. Question 15.38 ([618]) Does 𝑖(𝐺)/𝛾(𝐺) ≤ 5/4 hold whenever 𝐺 is a connected cubic graph of sufficiently large order?

Section 15.4. Domination Versus Total Domination

451

Recall that 𝑟 3,dom = 34 by Theorem 15.35. It remains an open problem to determine the supremum 𝑟 𝑘,dom for larger regularity 𝑘 ≥ 4. The following conjecture was first posed as a question in 2013 by Goddard and Henning [352]. Conjecture 15.39 𝑟 4,dom = 32 . As remarked in [352], if 𝐺 is the expansion exp(𝐶7 , 2) of a 7-cycle or the expansion exp(𝐶8 , 2) of an 8-cycle, illustrated in Figure 15.5(a) and (b), respectively, then in both cases we have 𝑖(𝐺) = 6 and 𝛾(𝐺) = 4, that is, in both cases we have 𝑖(𝐺)/𝛾(𝐺) = 3/2. Therefore, 𝑟 4,dom ≥ 32 . Hence, to prove Conjecture 15.39, it suffices to show that 𝑟 4,dom ≤ 32 . However, this conjecture has yet to be resolved.

(a) exp(𝐶7 , 2)

(b) exp(𝐶8 , 2)

Figure 15.5 The expansions exp(𝐶7 , 2) and exp(𝐶8 , 2)

15.4

Domination Versus Total Domination

A constructive characterization of trees 𝑇 satisfying 𝛾t (𝑇) = 2𝛾(𝑇) is given in 2001 by Henning [454]. However, it remains an open problem to characterize the graphs 𝐺 satisfying 𝛾t (𝐺) = 2𝛾(𝐺).

15.4.1

Regular Graphs

In this section, we present some results on total domination versus domination in regular graphs. The proof of Theorem 15.2 can be extended slightly to yield the following result for regular graphs. Lemma 15.40 ([209]) For 𝑘 ≥ 1, let 𝐺 be a 𝑘-regular graph of order 𝑛. If 𝛾t (𝐺) = 2𝛾(𝐺), then 𝑛 𝛾(𝐺) = . 𝑘 +1 Proof Let 𝛾t (𝐺) = 2𝛾(𝐺). We adopt the notation used in the proof of Theorem 15.2. In particular, let 𝑆 ∪ 𝑆 ′ be a TD-set of 𝐺, where 𝑆 is a 𝛾-set of 𝐺 and 𝑆 ′ is formed by selecting an arbitrary neighbor 𝑣 ′ of 𝑣 for each vertex 𝑣 ∈ 𝑆. Hence, 2𝛾(𝐺) = 𝛾t (𝐺) ≤ |𝑆 ∪ 𝑆 ′ | ≤ |𝑆| + |𝑆 ′ | ≤ 2|𝑆| = 2𝛾(𝐺). Since 𝛾t (𝐺) = 2𝛾(𝐺), it

Chapter 15. Relating the Core Parameters

452

follows that 𝑆 ∪ 𝑆 ′ is a 𝛾t -set of 𝐺 and |𝑆| + |𝑆 ′ | = 2𝛾(𝐺). Hence, |𝑆| = |𝑆 ′ | = 𝛾(𝐺) and 𝑆 ∩ 𝑆 ′ = ∅. Therefore, for every pair 𝑢 and 𝑣 of vertices in 𝑆, the vertices 𝑢 and 𝑣 are not adjacent and do not have a neighbor in common, that is, N[𝑢] ∩ N[𝑣] = ∅. Thus, the set 𝑆 is a packing in 𝐺. Since 𝑆 is a dominating set of 𝐺, the sets N[𝑣] where 𝑣 ∈ 𝑆 form a partition of 𝑉 (𝐺). Thus, since |N[𝑣] | = deg(𝑣) + 1 = 𝑘 + 1 for every vertex 𝑣 ∈ 𝑆, Ø ∑︁ ∑︁ 𝑛 = |𝑉 (𝐺)| = N[𝑣] = (𝑘 + 1) = |𝑆|(𝑘 + 1) = 𝛾(𝐺) (𝑘 + 1), |N[𝑣] | = 𝑣 ∈𝑆

𝑣 ∈𝑆

𝑣 ∈𝑆

or equivalently, 𝛾(𝐺) = 𝑛/(𝑘 + 1). We wish to determine the connected 𝑘-regular graphs achieving equality in the upper bound of Theorem 15.2 for small values of 𝑘, that is, those connected 𝑘-regular graphs 𝐺 satisfying 𝛾t (𝐺) = 2𝛾(𝐺). If 𝑘 = 1, then 𝐺 = 𝐾2 and 𝛾t (𝐺) = 2 and 𝛾(𝐺) = 1, and so 𝛾t (𝐺)/𝛾(𝐺) = 2. Hence, it is only of interest to consider the cases when 𝑘 ≥ 2. Suppose that 𝑘 = 2. In this case, 𝐺 = 𝐶𝑛 and 𝑛 ≥ 3. Recall that 𝛾t (𝐶𝑛 ) = ⌊𝑛/2⌋ + ⌈𝑛/4⌉ − ⌊𝑛/4⌋ and 𝛾(𝐶𝑛 ) = ⌈𝑛/3⌉. As a consequence of these results, in 2018 Cyman et al. [209] determined the following upper bound on the ratio of the total domination to the domination number. Theorem 15.41 ([209]) If 𝐺 is a cycle 𝐶𝑛 for 𝑛 ≥ 3, then 3    2 +    𝛾t (𝐺) = 32 + 𝛾(𝐺)    3 + 2

and

𝛾t (𝐺) 𝛾 (𝐺)

≤

3 2

3 2(𝑛+1) 3 2𝑛 3 𝑛

if 𝑛 ≡ 2 (mod 12) if 𝑛 ≡ 3, 9 (mod 12) if 𝑛 ≡ 6 (mod 12)

otherwise.

As an immediate consequence of Theorem 15.41, the authors in [209] observed the following. Corollary 15.42 ([209]) If 𝐺 is a cycle, then 𝛾t (𝐺) ≤ 2, 𝛾(𝐺) with equality if and only if 𝐺 ∈ {𝐶3 , 𝐶6 }. Recall that in Chapter 6, we defined the following two infinite families, Gcubic and Hcubic , of cubic graphs as follows. For 𝑘 ≥ 1, let 𝐺 𝑘 be the graph constructed by taking two copies of the path 𝑃2𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑎 2 𝑏 2 . . . 𝑎 𝑘 𝑏 𝑘 and 𝑐 1 𝑑1 𝑐 2 𝑑2 . . . 𝑐 𝑘 𝑑 𝑘 , and for each 𝑖 ∈ [𝑘], join 𝑎 𝑖 to 𝑑𝑖 and 𝑏 𝑖 to 𝑐 𝑖 . To complete the construction of the graph 𝐺 𝑘 join 𝑎 1 to 𝑐 1 and 𝑏 𝑘 to 𝑑 𝑘 . For 𝑘 ≥ 2, let 𝐻 𝑘 be obtained from 𝐺 𝑘 by deleting the two edges 𝑎 1 𝑐 1 and 𝑏 𝑘 𝑑 𝑘 and adding the two edges 𝑎 1 𝑏 𝑘 and 𝑐 1 𝑑 𝑘 . Recall that Gcubic = {𝐺 𝑘 : 𝑘 ≥ 1} and Hcubic = {𝐻 𝑘 : 𝑘 ≥ 2}.

Section 15.4. Domination Versus Total Domination

453

The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic , for example, are illustrated in Figure 15.6 even be the subfamily of H (and shown earlier in Figure 6.21 in Chapter 6). Let Hcubic cubic consisting of all graphs 𝐻 𝑘 , where 𝑘 is even; that is, even Hcubic = {𝐻 𝑘 : 𝑘 ≥ 2 is even}.

𝑎1

𝑐1

𝑏1

𝑑1

𝑎2

𝑐2

𝑏2

𝑑2

𝑎3

𝑐3

𝑏3

𝑑3

𝑎4

𝑐4

𝑏4

𝑑4 (a) 𝐺 4

(b) 𝐻4

Figure 15.6 The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic

Figure 15.7 The generalized Petersen graph 𝑃(8, 3) Cyman et al. [209] characterized the connected cubic graphs 𝐺 satisfying 𝛾t (𝐺)/𝛾(𝐺) = 2. The generalized Petersen graph 𝑃(8, 3) is shown in Figure 15.7 (and shown earlier in Figure 6.20). Theorem 15.43 ([209]) If 𝐺 is a connected cubic graph, then 𝛾t (𝐺)/𝛾(𝐺) ≤ 2, with equality if and only if the following hold: even , or (a) 𝐺 ∈ Gcubic ∪ Hcubic (b) 𝐺 is the generalized Petersen graph 𝑃(8, 3).

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454

15.4.2

Claw-free Graphs

In this section, we present some results on total domination versus domination in claw-free graphs. Recall that 𝑁2 denotes the diamond-necklace with two diamonds, shown in Figure 15.8 (and shown earlier in Figure 10.28).

Figure 15.8 A diamond-necklace 𝑁2 with two diamonds In 2021 Babikir and Henning [44] determined a best possible upper bound on the ratio of the total domination number to the domination number for a connected claw-free cubic graph. Theorem 15.44 ([44]) If 𝐺 ∉ {𝐾4 , 𝑁2 } is a connected claw-free cubic graph, then 𝛾t (𝐺) 12 𝛾 (𝐺) ≤ 7 . Proof Suppose that 𝐺 ∉ {𝐾4 , 𝑁2 } is a connected claw-free cubic graph of order 𝑛. If 𝐺 ∈ {𝐺 18.1 , 𝐺 18.2 }, where 𝐺 18.1 and 𝐺 18.2 are the graphs shown in Figure 10.29(a) (𝐺) and (b), respectively, then 𝛾t (𝐺) = 8 and 𝛾(𝐺) = 5, and so 𝛾𝛾t (𝐺) = 85 < 12 7 . Hence, we may assume that 𝐺 ∉ {𝐺 18.1 , 𝐺 18.2 }. With this assumption, we can apply Theorem 10.80 in Chapter 10 to the graph 𝐺. Since 𝐺 is a cubic graph of order 𝑛, we have 𝛾(𝐺) ≥ 14 𝑛. Hence, 3 𝑛 12 𝛾t (𝐺) ≤ 71 = . 𝛾(𝐺) 7 4𝑛 The bound in Theorem 15.44 is best possible. For example, suppose that 𝐺 is the graph 𝐺 28.1 shown in Figure 15.9 (and shown earlier in Figure 10.30(a)). In this case, 𝛾(𝐺) = 7 and 𝛾t (𝐺) = 12. Further, the red vertices in Figure 15.9(a) form a 𝛾-set of 𝐺 of cardinality 7 and the red vertices in Figure 15.9(b) form a 𝛾t -set of 𝐺 of cardinality 12. Thus, 𝛾t (𝐺)/𝛾(𝐺) = 12/7. We also remark that if 𝐺 is the graph 𝐺 28.2 shown earlier in Figure 10.30(b), then 𝛾t (𝐺)/𝛾(𝐺) = 12/7.

15.5

Upper Domination Versus Independence

In this section, we present relationships between the upper domination number and the independence number. It remains an open problem to characterize the graphs 𝐺 for which 𝛼(𝐺) = Γ(𝐺). If 𝐺 is a graph with 𝛼(𝐺) = 1, then 𝐺 is a complete graph, and hence Γ(𝐺) = 1. For 𝛼(𝐺) ≥ 2, the difference Γ(𝐺) − 𝛼(𝐺) can be made arbitrarily large. For example, for 𝑘 + 2 ≥ 3, if 𝐺 = 𝐾 𝑘+2 □ 𝐾2 , then Γ(𝐺) = 𝑘 + 2 and 𝛼(𝐺) = 2, and so Γ(𝐺) − 𝛼(𝐺) = 𝑘.

Section 15.5. Upper Domination Versus Independence

(a) 𝛾(𝐺) = 7

455

(b) 𝛾t (𝐺) = 12

Figure 15.9 The graph 𝐺 = 𝐺 28.1 with 𝛾(𝐺) = 7 and 𝛾t (𝐺) = 12

As first shown by Cockayne et al. [185] in 1981 and stated formally in Chapter 2, the independence number of a bipartite graph is always equal to its upper domination number. The authors in [185] actually proved that the upper irredundance number is also equal to 𝛼(𝐺) and Γ(𝐺) for bipartite graphs. Theorem 15.45 ([185]) If 𝐺 is a bipartite graph, then 𝛼(𝐺) = Γ(𝐺). Proof Let 𝐺 = (𝑉, 𝐸) be a bipartite graph with partite sets 𝑋 and 𝑌 . Let 𝐷 be a Γ-set of 𝐺 and let 𝐼 be the set of isolated vertices in 𝐺 [𝐷]. Further, let 𝑋1 = 𝑋 ∩ 𝐷, 𝑌1 = 𝐼 ∩ 𝑌 , and 𝑌2 = (𝐷 ∩ 𝑌 ) \ 𝐼, where one or more of these sets may be empty. We note that 𝐷 = 𝑋1 ∪ 𝑌1 ∪ 𝑌2 . Since 𝐷 is a minimal dominating set, by Lemma 2.72 from Chapter 2, we have ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. Let 𝑣 be an arbitrary vertex in 𝑌2 . Since ipn[𝑣, 𝐷] = ∅, we have epn[𝑣, 𝐷] ≠ ∅. Let 𝑣 ′ ∈ epn[𝑣, 𝐷]. Since 𝐺 is a bipartite graph, 𝑣 ′ ∈ 𝑋. Let 𝑋2 = {𝑣 ′ : 𝑣 ∈ 𝑌2 }. We note that |𝑋2 | = |𝑌2 | and 𝑋2 ⊆ 𝑋 \ 𝑋1 . Further, no vertex in 𝑋2 is adjacent to any vertex in 𝑌1 . Therefore, the set (𝐷 \ 𝑌2 ) ∪ 𝑋2 is an independent set in 𝐺. Hence, 𝛼(𝐺) ≥ |𝐷| − |𝑌2 | + |𝑋2 | = |𝐷| = Γ(𝐺). From the Domination Chain, 𝛼(𝐺) ≤ Γ(𝐺), and so 𝛼(𝐺) = Γ(𝐺). Recall that a graph 𝐺 is chordal if every cycle of 𝐺 of length at least 4 contains a chord, that is, an edge joining two non-consecutive vertices of the cycle. It is well known that every chordal graph contains at least one simplicial vertex. Using this property of a chordal graph, in 1990 Jacobson and Peters [506] showed that the independence number of a chordal graph is always equal to its upper domination number (and also equal to its upper irredundance number). Theorem 15.46 ([506]) If 𝐺 is a chordal graph, then 𝛼(𝐺) = Γ(𝐺). Jacobson and Peters [506] showed that forbidding a list of induced subgraphs is sufficient to imply that a graph has equal independence and upper domination numbers. The 2-corona 𝐶3 ◦ 𝑃2 is illustrated in Figure 15.10. Theorem 15.47 ([506]) The following hold: (a) If 𝐺 is a {𝐾1,3 , 𝐶4 , 𝐶3 ◦ 𝑃2 }-free graph, then 𝛼(𝐺) = Γ(𝐺). (b) If 𝐺 is a {𝐶4 , 2𝐾2 }-free graph, then 𝛼(𝐺) = Γ(𝐺).

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456

Figure 15.10 The 2-corona 𝐶3 ◦ 𝑃2

A graph 𝐺 is a circular-arc graph if there exists a set of arcs of a circle with each arc corresponding to a vertex of 𝐺, and two vertices are adjacent if and only if the corresponding arcs have a nonempty intersection on the circle. In 1993 Golumbic and Laskar [361] showed that the independence number of a circular-arc graph is always equal to its upper domination number (and also equal to its upper irredundance number). Theorem 15.48 ([361]) If 𝐺 is a circular-arc graph, then 𝛼(𝐺) = Γ(𝐺). A graph is strongly perfect if each of its induced subgraphs 𝐻 contains an independent set which intersects all the maximal cliques of 𝐻. We remark that every strongly perfect graph is perfect, but the converse is not necessarily true. In 1994 Cheston and Fricke [166] showed that the independence number of a strongly perfect graph is always equal to its upper domination number (and also equal to its upper irredundance number). Theorem 15.49 ([166]) If 𝐺 is a strongly perfect graph, then 𝛼(𝐺) = Γ(𝐺). Proof Let 𝐺 be a strongly perfect graph. From the Domination Chain, 𝛼(𝐺) ≤ Γ(𝐺). Hence, it suffices to show that 𝛼(𝐺) ≥ Γ(𝐺). Let 𝐷 be a Γ-set of 𝐺 and let 𝐴 be the set of isolated vertices in 𝐺 [𝐷]. If 𝐴 = 𝐷, then 𝛼(𝐺) ≥ |𝐷| = Γ(𝐺), as desired. Hence, we may assume that 𝐴 ⊂ 𝐷. Let 𝐵 = 𝐷 \ 𝐴 and so 𝐵 consists of the vertices of 𝐷 that are not isolated in 𝐺 [𝐷]. Since 𝐷 is a minimal dominating set, by Lemma 2.72, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. We note that if 𝑣 is an arbitrary vertex in 𝐵, then ipn[𝑣, 𝐷] = ∅, implying that epn[𝑣, 𝐷] ≠ ∅. For each such vertex 𝑣 ∈ 𝐵, let 𝑣 ′ ∈ epn[𝑣, 𝐷]. Let Ø 𝐵′ = {𝑣 ′ } and 𝑆 = 𝐷 ∪ 𝐵′ , 𝑣∈𝐵

and consider the graph 𝐻 = 𝐺 [𝑆]. We note that the only edges between 𝐷 and 𝐵′ are edges of the form 𝑣𝑣 ′ , where 𝑣 ∈ 𝐵 and 𝑣 ′ ∈ 𝐵′ . Since 𝐺 is strongly perfect, the induced subgraph 𝐻 of 𝐺 contains an independent set 𝐼 which meets all the maximal cliques of 𝐻. In particular, since every vertex in 𝐴 is an isolated vertex in 𝐻, the set 𝐼 must contain the set 𝐴. We now consider an arbitrary edge 𝑣𝑣 ′ in 𝐻, where 𝑣 ∈ 𝐵 and 𝑣 ′ ∈ 𝐵′ . No vertex of 𝐵′ \ {𝑣 ′ } is adjacent to 𝑣 and no vertex of 𝐷 \ {𝑣} is adjacent to 𝑣 ′ , implying that the set {𝑣, 𝑣 ′ } forms a maximal clique in 𝐻. Therefore, 𝐼 contains exactly one of 𝑣 and 𝑣 ′ for every vertex 𝑣 ∈ 𝐵, implying that 𝛼(𝐺) ≥ 𝛼(𝐻) ≥ |𝐼 | ≥ | 𝐴| + |𝐵| = |𝐷| = Γ(𝐺). Consequently, 𝛼(𝐺) = Γ(𝐺).

Section 15.5. Upper Domination Versus Independence

457

There are many important classes of strongly perfect graphs, including 𝑃4 -free graphs, triangulated graphs, comparability graphs, bipartite graphs, Meyniel graphs, cographs, chordal graphs, parity graphs, perfectly orderable graphs, and peripheral graphs. (We do not define all these graph classes here, which can be found, for example, in [102].) Hence, for any graph 𝐺 in any one of these classes of graphs, 𝛼(𝐺) = Γ(𝐺) = IR(𝐺). In a sense dual to the class of domination perfect graphs is the class of upper domination perfect graphs. A graph 𝐺 is upper domination perfect, or Γ-perfect, if 𝛼(𝐻) = Γ(𝐻) for every induced subgraph 𝐻 of 𝐺. By our earlier observations, circular-arc graphs and strongly perfect graphs are examples of classes of Γ-perfect. A graph 𝐺 is minimal Γ-perfect if 𝐺 is not Γ-perfect but 𝛼(𝐻) = Γ(𝐻) for every proper induced subgraph 𝐻 of 𝐺. A graph 𝐺 is upper irredundance perfect, or IR-perfect, if Γ(𝐻) = IR(𝐻) for every induced subgraph 𝐻 of 𝐺. In 1998 Gutin and Zverovich [370] showed that every Γ-perfect graph is IR-perfect. In order to characterize the class of Γ-perfect graphs, they defined the family W to consist of all connected graphs 𝐺 of order at least 10 with 𝛿(𝐺) ≥ 2, and its vertex set 𝑉 (𝐺) has a partition 𝑉 (𝐺) = 𝐴 ∪ 𝐵 such that | 𝐴| = |𝐵| = 𝛼(𝐺) + 1 and the only edges between 𝐴 and 𝐵 form a perfect matching. To illustrate an infinite subclass of the class W, recall that for 𝑛1 , 𝑛2 ≥ 3 and 𝑘 ≥ 1, a dumbbell 𝐷 𝑏 (𝑛1 , 𝑛2 , 𝑘) is the graph obtained from two disjoint cycles 𝐶𝑛1 and 𝐶𝑛2 by adding an edge joining the two cycles and subdividing this edge 𝑘 − 1 times. As observed in [370], if 𝑛1 = 𝑛2 ≥ 5 and 𝑛1 ≡ 1 (mod 4) and if 𝑘 ≥ 1 is odd, then the dumbbell 𝐷 𝑏 (𝑛1 , 𝑛2 , 𝑘) belongs to the class W. The dumbbells 𝐷 𝑏 (5, 5, 1) and 𝐷 𝑏 (5, 5, 3) are shown in Figure 15.11, where the vertices in 𝐴 are highlighted red. We note that if 𝐺 = 𝐷 𝑏 (5, 5, 1), then 𝛼(𝐺) = 4 and | 𝐴| = |𝐵| = 5, while if 𝐺 = 𝐷 𝑏 (5, 5, 3), then 𝛼(𝐺) = 5 and | 𝐴| = |𝐵| = 6. Further, in both cases, the only edges between 𝐴 and 𝐵 form a perfect matching.

(a) 𝐷 𝑏 (5, 5, 1)

(b) 𝐷 𝑏 (5, 5, 3)

Figure 15.11 The dumbbells 𝐷 𝑏 (5, 5, 1) and 𝐷 𝑏 (5, 5, 3) As observed in [370], if 𝐺 ∈ W, then Γ(𝐺) = 𝛼(𝐺) + 1. Let Wsmall = {𝐺 1 , 𝐺 2 , . . . , 𝐺 15 } be the family of fifteen graphs (one of order six and fourteen of order eight) shown in Figure 15.12. We are now in a position to give the forbidden induced subgraph characterization of Gutin and Zverovich [370] of Γ-perfect graphs. Theorem 15.50 ([370]) A graph 𝐺 is Γ-perfect if and only if 𝐺 does not contain a graph in the family W ∪ Wsmall as an induced subgraph.

Chapter 15. Relating the Core Parameters

458

(a) 𝐺 1

(b) 𝐺 2

(c) 𝐺 3

(d) 𝐺 4

(e) 𝐺 5

(f) 𝐺 6

(g) 𝐺 7

(h) 𝐺 8

(i) 𝐺 9

(j) 𝐺 10

(k) 𝐺 11

(l) 𝐺 12

(m) 𝐺 13

(n) 𝐺 14

(o) 𝐺 15

Figure 15.12 The family Wsmall of fifteen minimal Γ-imperfect graphs

In 1999 Zverovich and Zverovich [791] gave a semi-induced subgraph characterization of upper domination perfect graphs. However, we do not define their concept of a semi-induced subgraph here.

15.6

Upper Domination Versus Upper Total Domination

In this section, we study how the upper domination and upper total domination numbers of a graph are related. For this purpose, recall the fundamental property of minimal total dominating sets established in 1980 by Cockayne et al. [182], which was presented as Lemma 4.25 in Chapter 4 and which we restate here. Lemma 15.51 ([182]) A TD-set 𝑆 in a graph 𝐺 is a minimal TD-set if and only if for every vertex 𝑣 ∈ 𝑆, |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1. In order to establish a relationship between the upper domination number of a graph and its upper total domination number, in 2008 Dorbec et al. [237] proved the following property of a Γt -set of a graph. Theorem 15.52 ([237]) Every Γt -set of an isolate-free graph 𝐺 contains as a subset a minimal dominating set 𝑆 such that the following hold: (a) |𝑆| ≥ 12 Γt (𝐺), and (b) |epn(𝑣, 𝑆)| ≥ 1 for every vertex 𝑣 ∈ 𝑆. Proof Let 𝐷 be a Γt -set of 𝐺. Thus, 𝐷 is a minimal TD-set of 𝐺 of maximum cardinality, namely Γt (𝐺). Let 𝐴 be the set of vertices in 𝐷 that have a 𝐷-external

Section 15.6. Upper Domination Versus Upper Total Domination

459

private neighbor and let 𝐵 be the set of vertices in 𝐷 \ 𝐴 that have a neighbor in 𝐴. Further, let 𝐶 be the set of remaining vertices in 𝐷. Thus, 𝐷 = 𝐴 ∪ 𝐵 ∪ 𝐶 where  𝐴 = 𝑣 ∈ 𝐷 : |epn(𝑣, 𝐷)| ≥ 1 ,  𝐵 = 𝑣 ∈ 𝐷 \ 𝐴 : deg 𝐴 (𝑣) ≥ 1 , and 𝐶 = 𝐷 \ ( 𝐴 ∪ 𝐵). Consider a vertex 𝑣 ∈ 𝐵 ∪ 𝐶. Since epn(𝑣, 𝐷) = ∅, by Lemma 15.51, we have |ipn(𝑣, 𝐷)| ≥ 1. Let 𝑣 ′ ∈ ipn(𝑣, 𝐷) and so 𝑣 ′ ∈ 𝐷 and N(𝑣 ′ ) ∩ 𝐷 = {𝑣}. Thus, the vertex 𝑣 is the unique neighbor of 𝑣 ′ in 𝐺 [𝐷] and so deg𝐷 (𝑣 ′ ) = 1. Since 𝑣 ∉ 𝐴, we have that 𝑣 ′ ∉ 𝐵. We consider two cases, depending on whether 𝑣 ∈ 𝐵 or 𝑣 ∈ 𝐶. Suppose that 𝑣 ∈ 𝐶. Then, deg 𝐴 (𝑣) = 0, implying that 𝑣 ′ ∉ 𝐴 and 𝑣 ′ ∈ 𝐶. Hence, epn(𝑣 ′ , 𝐷) = ∅ and so by Lemma 15.51, |ipn(𝑣 ′ , 𝐷)| ≥ 1. However, the vertex 𝑣 is the only neighbor of 𝑣 ′ in 𝐺 [𝐷], implying that ipn(𝑣 ′ , 𝐷) = {𝑣}, that is, N(𝑣) ∩ 𝐷 = {𝑣 ′ } and so deg𝐷 (𝑣) = 1. Hence, if 𝐶 ≠ ∅, then 𝐺 [𝐶] = |𝐶2 | 𝐾2 and for each 𝑣 ∈ 𝐶, we have deg𝐷 (𝑣) = 1. We call two adjacent vertices in 𝐺 [𝐶] partners in 𝐶. Let 𝑋 and 𝑌 be partite sets in the graph 𝐺 [𝐶] and so each vertex in 𝑋 (respectively, in 𝑌 ) is adjacent in 𝐺 [𝐷] only to its partner in 𝑌 (respectively, in 𝑋). For each 𝑥 ∈ 𝑋, let 𝑦 𝑥 be the partner of 𝑥 in 𝐶. Suppose next that 𝑣 ∈ 𝐵. Since 𝐺 [𝐶] = |𝐶2 | 𝐾2 and each vertex of 𝐶 has degree 1 in 𝐺 [𝐷], the vertex 𝑣 is not adjacent to any vertex of 𝐶. Therefore, 𝑣 ′ ∈ 𝐴 ∪ 𝐵. As shown earlier, 𝑣 ′ ∉ 𝐵. Hence, 𝑣 ′ ∈ 𝐴. Thus, for each 𝑣 ∈ 𝐵, we have epn(𝑣, 𝐷) = ∅ and |ipn(𝑣, 𝐷)| ≥ 1, where ipn(𝑣, 𝐷) ⊆ 𝐴. This in turn implies that ∑︁ | 𝐴| ≥ |ipn(𝑣, 𝐷)| ≥ |𝐵|. (15.3) 𝑣∈𝐵

By our earlier observations, the set 𝐴 dominates the set 𝐵, and the set 𝑋 dominates the set 𝑌 . Thus, 𝐴 ∪ 𝑋 is a dominating set in 𝐺 [𝐷]. Let 𝑈 be the set of
vertices in 𝐺 not dominated by 𝐴 or 𝑋, and so 𝑈 = 𝑉 (𝐺) \ 𝐷 ∪ N( 𝐴) ∪ N(𝑋) . Since 𝐷 is a TD-set of 𝐺, the set 𝑈 is dominated by 𝐵 ∪ 𝑌 . Let 𝐵𝑌 be a minimum subset of 𝐵 ∪ 𝑌 that dominates 𝑈. Thus, for each vertex 𝑣 ∈ 𝐵𝑌 , we have epn(𝑣, 𝐵𝑌 ) ∩ 𝑈 ≥ 1. We now consider the set 𝑆★ = 𝐴 ∪ 𝐵𝑌 ∪ 𝑋. Since 𝐵 ⊆ N( 𝐴) and 𝑌 ⊆ N(𝑋), the set 𝑆★ dominates 𝐷. By construction, the set 𝑆★ also dominates 𝑉 (𝐺) \ 𝐷. Thus, the set 𝑆★ is a dominating set of 𝐺. However, 𝑆★ is not necessarily a minimal dominating set of 𝐺. We now construct a minimal dominating set 𝑆 of 𝐺 from the dominating set 𝑆★ as follows. We consider the vertices of 𝑋 in turn, and for each vertex 𝑥 ∈ 𝑋, we systematically delete 𝑥 from 𝑆★ if epn(𝑥, 𝑆★) = ∅ at each stage in the resulting set 𝑆★. Observe that if the partner 𝑦 𝑥 ∈ 𝑌 of 𝑥 in 𝐶 is not in 𝑆★, then 𝑦 𝑥 ∈ epn(𝑥, 𝑆★), and so 𝑥 is not deleted from 𝑆★. Let 𝑋 ★ be the resulting subset of vertices of 𝑋 that belong to the set 𝑆★ upon the completion of this process, and let 𝑆 = 𝐴 ∪ 𝐵𝑌 ∪ 𝑋 ★. By construction, |epn(𝑣, 𝑆)| ≥ 1 for each 𝑣 ∈ 𝑆. If 𝑥 ∈ 𝑋 \ 𝑋 ★, then the partner of 𝑥 in 𝐶 is in the set 𝑆, implying that 𝑆 dominates 𝐶. Since 𝐵 ⊆ N( 𝐴), the set 𝐵 is dominated by 𝑆. Therefore, the set 𝑆 dominates 𝐷. By construction, the set 𝑆 also dominates 𝑉 (𝐺) \ 𝐷. Hence, 𝑆 is a minimal dominating set of 𝐺, and so Γ(𝐺) ≥ |𝑆|.

460

Chapter 15. Relating the Core Parameters

It remains to show that |𝑆| ≥ 21 Γt (𝐺). For every vertex 𝑥 ∈ 𝑋, the set 𝑆 contains at least one of 𝑥 and its partner in 𝐶, implying that |𝑆 ∩ 𝐶 | ≥ |𝑋 | = 12 |𝐶 |. By Inequality (15.3), we have | 𝐴| ≥ |𝐵|. Hence, Γ(𝐺) ≥ |𝑆| = |𝑆 ∩ 𝐴| + |𝑆 ∩ 𝐵| + |𝑆 ∩ 𝐶 | ≥ | 𝐴| + |𝑆 ∩ 𝐶 |
≥ 12 2| 𝐴| + |𝐶 |
≥ 12 | 𝐴| + |𝐵| + |𝐶 |

(15.4)

= 12 |𝐷 | = 12 Γt (𝐺). Thus, properties (a) and (b) in the statement of Theorem 15.52 hold. We are now in a position to present the following inequalities between the upper domination number and the upper total domination number, given in [237]. Theorem 15.53 ([237]) If 𝐺 is an isolate-free graph of order 𝑛, then
2 𝑛−1 Γ(𝐺) ≤ Γt (𝐺) ≤ 2 Γ(𝐺). Proof Let 𝐷 be a Γt -set of 𝐺. By Theorem 15.52, there is a minimal dominating set 𝑆 of 𝐺 such that 𝑆 ⊆ 𝐷 and |𝑆| ≥ 12 Γt (𝐺). Hence, Γ(𝐺) ≥ |𝑆| ≥ 12 Γt (𝐺). This establishes the upper bound. The lower bound follows from Observation 14.3 in Chapter 14, which states that Γ(𝐺) ≤ 𝑛 − 1, and the fact that Γt (𝐺) ≥ 2. It remains an open problem to find a characterization of graphs 𝐺 satisfying equality in the upper bound of Theorem 15.53, that is, to characterize graphs 𝐺 satisfying Γt (𝐺) = 2 Γ(𝐺). The following structural properties of such graphs are implicitly implied in [237]. The proof we give of this result can be found in [209]. Lemma 15.54 ([209, 237]) If 𝐺 is a graph with Γt (𝐺) = 2 Γ(𝐺), then the following three properties hold: (a) Γ(𝐺) = 𝛼(𝐺). (b) Every Γt -set of 𝐺 induces a subgraph that consists of disjoint copies of 𝐾2 . (c) Every vertex 𝑣 that does not belong to a given Γt -set 𝐷 of 𝐺 is contained in a triangle with two vertices from 𝐷, that is, 𝑣 is adjacent to two adjacent vertices of 𝐷. Proof Suppose that 𝐺 is an isolate-free graph satisfying Γ(𝐺) = 12 Γt (𝐺). Let 𝐷 be an arbitrary Γt -set of 𝐺. We adopt the notation introduced in the proof of Theorem 15.52. We must have equality throughout Inequality (15.4) in the proof of Theorem 15.52. Hence, 𝑆 = 𝐴 ∪ 𝐵𝑌 ∪ 𝑋 ★ is a Γ-set of 𝐺 and so Γ(𝐺) = |𝑆|. Further, | 𝐴| = |𝐵|, 𝑆 ∩ 𝐵 = ∅, and |𝑆 ∩ 𝐶 | = 12 |𝐶 |. Since | 𝐴| = |𝐵|, we must have equality throughout Inequality (15.3), implying that |ipn(𝑣, 𝐷)| = 1 for every 𝑣 ∈ 𝐵 and every vertex of 𝐴 is the (internal) 𝐷-private neighbor of some vertex of 𝐵. In particular, deg𝐷 (𝑣) = 1 for every 𝑣 ∈ 𝐴 and every vertex in 𝐵 is adjacent to a unique vertex of 𝐴, namely to its (internal) private neighbor in 𝑆. Hence, 𝐴 is an independent set

Section 15.6. Upper Domination Versus Upper Total Domination

461

and the subgraph 𝐺 [ 𝐴 ∪ 𝐵] induced by 𝐴 ∪ 𝐵 contains a perfect matching. Note that it is possible that there may be some edges joining vertices of 𝐵. As observed in the proof of Theorem 15.52, for every vertex 𝑥 ∈ 𝑋, the set 𝑆 contains at least one of 𝑥 and its partner 𝑦 𝑥 in 𝐶, and so |𝑆 ∩ 𝐶 | ≥ 21 |𝐶 |. However, |𝑆 ∩ 𝐶 | = 12 |𝐶 |, implying that the set 𝑆 contains exactly one of 𝑥 and its partner 𝑦 𝑥 in 𝐶. Renaming vertices of 𝐶 if necessary, we may assume that 𝑆 ∩ 𝐶 = 𝑋. Since 𝐴 is an independent set and there is no edge between 𝐴 and 𝐶, the set 𝑆 = 𝐴 ∪ 𝑋 is an independent set and so 𝛼(𝐺) ≥ |𝑆| = | 𝐴| + 12 |𝐶 |. By Theorem 15.1 and our earlier observations, |𝑆| ≤ 𝛼(𝐺) ≤ Γ(𝐺) = |𝑆|. Hence, we must have equality throughout this inequality chain, implying that Γ(𝐺) = 𝛼(𝐺) = |𝑆|. This proves (a). Suppose that 𝐴 ≠ ∅ and consider an arbitrary vertex 𝑢 ∈ 𝐴. Let 𝑣 be the neighbor of 𝑢 in 𝐺 [𝐷], and so 𝑣 ∈ 𝐵. Thus, ipn(𝑣, 𝐷) = {𝑢} and epn(𝑣, 𝐷) = ∅. Further, let 𝑤 ∈ epn(𝑢, 𝑆). The vertex 𝑤 exists since 𝑆 ⊆ 𝐷 and 𝑢 ∈ 𝐴 implies that 𝑢 has a 𝐷-external private neighbor. Let |𝐶 | = 2𝑐 and recall that 𝐺 [𝐶] = 𝑐𝐾2 and each vertex of 𝐶 has degree 1 in 𝐺 [𝐷]. In
particular, 𝑣 is adjacent to no vertex of 𝐴 ∪ 𝐶 different from 𝑢. Thus, the set 𝑆 \ {𝑢} ∪ {𝑣, 𝑤} is an independent set in 𝐺, implying that 𝛼(𝐺) ≥ |𝑆| + 1, a contradiction. Hence, 𝐴 = ∅, which in turn implies that 𝐵 = ∅. Thus, 𝐷 = 𝐶 and 𝐺 [𝐷] = 𝐺 [𝐶] = 𝑐𝐾2 , where |𝐶 | = 2𝑐. Since 𝐷 is an arbitrary Γt -set of 𝐺, every Γt -set of 𝐺 induces a subgraph isomorphic to 𝑐𝐾2 . This proves (b). By our earlier observations, Γ(𝐺) = 𝛼(𝐺) = 𝑐. To prove (c), let 𝑣 be an arbitrary vertex in 𝑉 (𝐺) \ 𝐷. As observed above, 𝐷 = 𝐶 and 𝐺 [𝐶] = 𝑐𝐾2 . We show that 𝑣 is contained in a triangle with two vertices of 𝐶. Suppose, to the contrary, that 𝑣 does not belong to a triangle with two vertices of 𝐶. This implies that 𝑣 is adjacent to at most one of 𝑥 and its partner 𝑦 𝑥 for every vertex 𝑥 ∈ 𝐶. Let 𝐶 ′ be a subset of 𝐶 constructed as follows. Initially, let 𝐶 ′ = ∅. For each pair 𝑥 and 𝑦 𝑥 where 𝑥 ∈ 𝐶 (and 𝑥𝑦 𝑥 is an edge), select one vertex that is not adjacent to 𝑣 and add it to the set 𝐶 ′ . Thus, |𝐶 ′ | = 𝑐 and 𝐶 ′ is an independent set. By construction, no vertex of 𝐶 ′ is adjacent to 𝑣. Hence, 𝐶 ′ ∪ {𝑣} is an independent set in 𝐺 and so 𝛼(𝐺) ≥ |𝐶 ′ | + 1 = 𝑐 + 1 > 𝛼(𝐺), a contradiction. Therefore, the vertex 𝑣 is adjacent to both 𝑥 and its partner 𝑦 𝑥 in 𝐶 for at least one vertex 𝑥 ∈ 𝐶. This proves (c). As remarked earlier, it remains an open problem to characterize the graphs 𝐺 satisfying Γt (𝐺) = 2 Γ(𝐺).

15.6.1

Regular Graphs

In 2018 Cyman et al. [209] determined the connected 𝑟-regular graphs 𝐺 with Γt (𝐺) = 2 Γ(𝐺) for small values of 𝑟. If 𝑟 = 1, then 𝐺 = 𝐾2 and Γ(𝐺) = 1 and Γt (𝐺) = 2. Hence, it is only of interest to consider the cases when 𝑟 ≥ 2. Suppose that 𝑟 = 2 and so 𝐺 is a cycle. The upper domination number of a cycle is given as follows.   Proposition 15.55 For 𝑛 ≥ 3, Γ(𝐶𝑛 ) = 12 𝑛 . The upper total domination number of a cycle was determined in [209]. We omit the proof.

Chapter 15. Relating the Core Parameters

462

Proposition 15.56 ([209]) For 𝑛 ≥ 3, ( 𝑛 2 + 1 if 𝑛 ≡ 5 (mod 6) Γt (𝐶𝑛 ) =  3  𝑛 2 3 otherwise. As a consequence of Propositions 15.55 and 15.56, we have the following result on the ratio of the upper total domination to the upper domination number of a cycle. Theorem 15.57 ([209]) If 𝐺 is a cycle 𝐶𝑛 for 𝑛 ≥ 3, then Γt (𝐺) 4 4 ≤ + . Γ(𝐺) 3 3(𝑛 − 1) Thus,

Γt (𝐺) Γ (𝐺)

≤ 2, with equality if and only if 𝐺 = 𝐶3 .

In the case when 𝐺 is a connected 3-regular graph, it is shown in [209] that equality occurs in the upper bound of Theorem 15.53 if and only if 𝐺 = 𝐾4 . We remark that their proof heavily relies on structural properties given in the statement of Lemma 15.54. Theorem 15.58 ([209]) If 𝐺 is a connected cubic graph, then Γt (𝐺) ≤ 2, Γ(𝐺) with equality if and only if 𝐺 = 𝐾4 . In 2019 Zhu et al. [787] characterized the (infinite) family of connected subcubic isolate-free graphs 𝐺 satisfying Γt (𝐺) = 2 Γ(𝐺). The authors in [787] posed the following conjecture. Conjecture 15.59 ([787]) If 𝐺 ≠ 𝐾4 is a connected cubic graph, then Γt (𝐺) 7 ≤ . Γ(𝐺) 4 If Conjecture 15.59 is true, then the bound is best possible as may be seen by considering the cubic graph 𝐺 shown in Figure 15.13. The set of vertices highlighted in red forms a Γt -set of 𝐺 of cardinality 7. It is shown in [787] that Γ(𝐺) = 4. From Theorems 15.53, 15.57, and 15.58, we know that for 𝑘 ∈ [3], if 𝐺 is a connected 𝑘-regular graph, then Γt (𝐺) ≤ 2 Γ(𝐺), with equality if and only if 𝐺 is the complete graph 𝐾 𝑘+1 . Zhu et al. [787] showed that for 𝑘 ≥ 4, there are 𝑘-regular graphs 𝐺 with Γt (𝐺) = 2 Γ(𝐺) that are different from the complete graph 𝐾 𝑘+1 . Circulant graphs are defined in Appendix A and in Chapter 9. Proposition 15.60 ([787]) For 𝑛 ≥ 8 an even integer, if 𝐺 is the circulant graph 𝐶𝑛 ⟨𝐿⟩ with list 𝐿 = {1, 2, . . . , 12 𝑛 − 2}, then 𝐺 is an (𝑛 − 4)-regular graph with Γt (𝐺) = 4 and Γ(𝐺) = 2.

Section 15.7. Independence Versus Total Domination

463

Figure 15.13 A cubic graph 𝐺 with Γt (𝐺) = 7 and Γ(𝐺) = 4

15.7

Independence Versus Total Domination

In this section, we briefly discuss the independent domination number versus the total domination number, and the independence number versus the total domination number.

15.7.1

Independent Domination Versus Total Domination

In general the independent domination number 𝑖(𝐺) and the total domination number 𝛾t (𝐺) of an isolate-free graph are incomparable. For 𝑟 ≥ 2, if we take the generalized corona 𝐺 = cor(𝐾𝑟 , 𝑟 − 1), then 𝐺 has order 𝑛 = 𝑟 2 , 𝑖(𝐺) = (𝑟 − 1) 2 + 1, and 𝛾t (𝐺) = 𝑟, showing that the difference 𝑖(𝐺) − 𝛾t (𝐺) = (𝑟 − 1) (𝑟 − 2) can be arbitrarily large. Thus, it is not necessarily true that 𝑖(𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺. It is also not necessarily true that 𝛾t (𝐺) ≤ 𝑖(𝐺) for all isolate-free graphs 𝐺. For example, for 𝑘 ≥ 1 if 𝐺 is the 2-corona of a connected graph 𝐹 of order 𝑘, that is, if 𝐺 = 𝐹 ◦ 𝑃2 , then 𝐺 has order 𝑛 = 3𝑘, 𝑖(𝐺) = 𝑘, and 𝛾t (𝐺) = 2𝑘, showing that the difference 𝛾t (𝐺) − 𝑖(𝐺) = 𝑘 can be arbitrarily large. If we impose a regularity condition on the graph, then due to the relationship 𝛾(𝐺) ≤ 𝛾t (𝐺), we have the following immediate corollary of Theorem 15.34 due to Knor et al. [533]. Corollary 15.61 ([533]) For all integers 𝑘 ≥ 3, if 𝐺 is a connected 𝑘-regular graph, then 𝑖(𝐺) 𝑘 ≤ , 𝛾t (𝐺) 2 with equality if and only if 𝐺 = 𝐾 𝑘,𝑘 . A natural problem is to determine, for each value of 𝑘 ≥ 3, an improved upper bound on the ratio 𝑖(𝐺)/𝛾t (𝐺) given in Corollary 15.61 if we forbid the exceptional 𝑘 denote graph 𝐺 = 𝐾 𝑘,𝑘 . As defined earlier (see Section 15.3.3), for 𝑘 ≥ 3, let R reg the family of all connected 𝑘-regular graphs different from 𝐾 𝑘,𝑘 . We now define 𝑟 𝑘,tdom = sup 𝑘 𝐺 ∈ R reg

𝑖(𝐺) . 𝛾t (𝐺)

Chapter 15. Relating the Core Parameters

464

To date, the value 𝑟 𝑘,tdom is not yet known for any 𝑘 ≥ 3. Surprisingly, even the value 𝑟 3,tdom has yet to be determined. Recall that in Section 6.4.3 we defined an 1 infinite family Fcubic of connected cubic graphs. We illustrate an example of a graph 1 in the family Fcubic in Figure 15.14. If 𝐺 is an arbitrary graph of order 𝑛 that belongs 1 , then 𝑖(𝐺) = 𝛾 (𝐺) = 3 𝑛, showing that there are infinitely many to the family Fcubic t 8 connected cubic graphs satisfying 𝑖(𝐺) = 𝛾t (𝐺). Indeed, there are many other such infinite families. In particular, we note that 𝑟 3,tdom ≥ 1.

1 Figure 15.14 A graph in the family Fcubic

Recall that in Theorem 6.93 in Chapter 6, Lam et al. [552] proved that if 𝐺 ≠ 𝐾3,3 is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 25 𝑛. Further recall that if 𝐺 is an isolate-free graph of order 𝑛, then we have the elementary lower bound 𝛾t (𝐺) ≥ 𝑛/Δ(𝐺). In particular, if 𝐺 is a cubic graph of order 𝑛, then 𝛾t (𝐺) ≥ 𝑛/3. Thus, the ratio of 𝑖(𝐺) to 𝛾t (𝐺) for connected cubic graphs 𝐺 of order 𝑛 different from 𝐾3,3 is at most 25 𝑛/ 13 𝑛 = 65 , yielding the upper bound 𝑟 3,tdom ≤ 65 . The above observations yield the following lower and upper bounds on 𝑟 3,tdom . Observation 15.62 1 ≤ 𝑟 3,tdom ≤ 65 . Henning posed the conjecture at several conferences that for a connected cubic graph 𝐺 different from 𝐾3,3 , the parameters 𝑖(𝐺) and 𝛾t (𝐺) are related by the inequality 𝑖(𝐺) ≤ 𝛾t (𝐺). We state this conjecture formally as follows. Conjecture 15.63 (Henning) 𝑟 3,tdom = 1. We also mention the following conjecture posed at several conferences by Henning. Conjecture 15.64 (Henning) 𝑟 4,tdom = 32 . If 𝐺 is the expansion exp(𝐶7 , 2) of a 7-cycle or the expansion exp(𝐶8 , 2) of an 8-cycle, illustrated in Figure 15.5(a) and (b), respectively, then in both cases we have 𝑖(𝐺) = 6 and 𝛾t (𝐺) = 4, that is, in both cases we have 𝑖(𝐺)/𝛾t (𝐺) = 3/2. Therefore, 𝑟 4,tdom ≥ 3/2. Hence, to prove Conjecture 15.64 it suffices to prove that 𝑟 4,tdom ≤ 3/2. We remark that if Conjecture 15.39 is true, then this would immediately imply the truth of Conjecture 15.64. However, even this weaker Conjecture 15.64 has yet to be resolved.

Section 15.8. Summary

465

15.7.2 Independence Versus Total Domination We next briefly discuss the independence number versus the total domination number. In general the independence number 𝛼(𝐺) and the total domination number 𝛾t (𝐺) of an isolate-free graph are incomparable. For 𝑟 ≥ 2, if we take 𝐺 = cor(𝐾𝑟 , 𝑟 − 1), then 𝐺 has order 𝑛 = 𝑟 2 , 𝛼(𝐺) = 𝑟 (𝑟 − 1), and 𝛾t (𝐺) = 𝑟, showing that the difference 𝛼(𝐺) − 𝛾t (𝐺) = 𝑟 (𝑟 − 2) can be arbitrarily large. Thus, it is not necessarily true that 𝛼(𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺. It is also not necessarily true that 𝛾t (𝐺) ≤ 𝛼(𝐺) for all isolate-free graphs 𝐺. For example, for 𝑘 ≥ 2 if 𝐺 is the 2-corona of a clique 𝐾 𝑘 of order 𝑘, that is, if 𝐺 = 𝐾 𝑘 ◦ 𝑃2 , then 𝐺 has order 𝑛 = 3𝑘, 𝛼(𝐺) = 𝑘 + 1, and 𝛾t (𝐺) = 2𝑘, showing that the difference 𝛾t (𝐺) − 𝛼(𝐺) = 𝑘 − 1 can be arbitrarily large. If we impose a regularity condition on the graph, then we can determine a tight upper bound on the ratio of the independence number to the total domination number. Recall that by Theorem 6.89 due to Rosenfeld [659] in 1964, if 𝐺 is an 𝑟-regular graph, for 𝑟 ≥ 1, of order 𝑛, then 𝛼(𝐺) ≤ 21 𝑛, and by Corollary 14.10, equality holds if and only if every component of 𝐺 belongs to the family Breg . By the elementary lower bound 𝛾t (𝐺) ≥ 𝑛/Δ(𝐺) for an isolate-free graph 𝐺 of order 𝑛, if 𝐺 is an 𝑟-regular graph, for 𝑟 ≥ 1, of order 𝑛, then 𝛾t (𝐺) ≥ 𝑛/𝑟. Thus, the ratio of 𝛼(𝐺) to 𝛾t (𝐺) for 𝑟-regular graphs 𝐺 is at most 𝑟/2. We state this formally as follows. Observation 15.65 For all integers 𝑘 ≥ 3, if 𝐺 is a 𝑘-regular graph, then 𝛼(𝐺) 𝑘 ≤ . 𝛾t (𝐺) 2 The bound in Observation 15.65 is tight. For example, consider the following subfamily of the family Breg . For 𝑝 ≥ 1 and 𝑘 ≥ 2, consider 𝑝 vertex-disjoint copies of 𝐾 𝑘,𝑘 , where the 𝑖 th copy of 𝐾 𝑘,𝑘 has partite sets 𝑈𝑖 and 𝑉𝑖 for 𝑖 ∈ [ 𝑝]. Further, let {𝑢 𝑖 , 𝑧𝑖 } ⊆ 𝑈𝑖 and {𝑣 𝑖 , 𝑤 𝑖 } ⊆ 𝑉𝑖 for 𝑖 ∈ [ 𝑝]. Let 𝐺 = 𝐺 𝑝,𝑘 be the 𝑘-regular graph of order 𝑛 = 2𝑘 𝑝 obtained from the disjoint union of these 𝑝 copies of 𝐾 𝑘,𝑘 by deleting the 𝑝 edges 𝑢 𝑖 𝑣 𝑖 for 𝑖 ∈ [ 𝑝] and adding the 𝑝 edges 𝑣 𝑖 𝑢 𝑖+1 where addition is taken Ð𝑝 modulo 𝑝. The set 𝑖=1 {𝑤 𝑖 , 𝑧𝑖 }, for example, is both a TD-set of 𝐺 and an open Ð𝑝 packing of 𝐺, implying that 𝛾 (𝐺) = 2𝑝. Moreover, the set 𝑈 is an independent t 𝑖 𝑖=1 Í𝑝 set of 𝐺, and so 𝛼(𝐺) ≥ 𝑖=1 |𝑈𝑖 | = 𝑘 𝑝. Since 𝛼(𝐺) ≤ 𝑛/2 = 𝑘 𝑝, we therefore have that 𝛼(𝐺) = 𝑘 𝑝. Thus, for every 𝑝 ≥ 1 and 𝑘 ≥ 2, 𝛼(𝐺) 𝑘 = . 𝛾t (𝐺) 2

15.8

Summary

In this chapter, we investigated relationships among several pairs of the six core parameters (𝛾, 𝛾t , 𝑖, Γ, Γt , 𝛼) that have been studied in the literature. We presented selected results based on the most studied relationships in the literature, which focus on the pairs (𝛾, 𝑖), (𝛾, 𝛾t ), (Γ, 𝛼), and (Γ, Γt ). It is natural that relationships between

466

Chapter 15. Relating the Core Parameters

certain pairs of the six core parameters are not found in the literature, since either they are not very much related or the relationships follow readily from known results. For example, consider domination versus upper domination. If 𝐺 is nontrivial a star 𝐾1,𝑛−1 , then Γ(𝐺) = 𝑛 − 1 and 𝛾(𝐺) = 1, and so the trivial upper bound Γ(𝐺)/𝛾(𝐺) ≤ 𝑛 − 1 is achievable. Moreover if 𝐺 is a 𝑘-regular graph, for 𝑘 ≥ 1, of 1 order 𝑛, then recall by Theorem 14.9
that Γ(𝐺) ≤ 2 𝑛. Together with the elementary lower bound 𝛾(𝐺) ≥ 𝑛/ Δ(𝐺) + 1 = 𝑛/(𝑘 + 1), this yields the upper bound Γ(𝐺) 𝑘 +1 ≤ 𝛾(𝐺) 2 for the class of 𝑘-regular graphs. As a further example, consider total domination versus upper total domination. If 𝐺 is a subdivided star 𝑆(𝐾1,𝑛−1 ), then Γt (𝐺) = 𝑛 − 1 and 𝛾t (𝐺) = 2, and so the trivial upper bound Γt (𝐺)/𝛾t (𝐺) ≤ (𝑛 − 1)/2 is achievable. Moreover if 𝐺 is a 𝑘-regular graph, for 𝑘 ≥ 1, of order 𝑛, then recall by Theorem 14.11 that Γt (𝐺) ≤

𝑛 . 2−

1 𝑘

Together with the elementary lower bound 𝛾t (𝐺) ≥ 𝑛/Δ(𝐺) = 𝑛/𝑘, this yields the upper bound Γt (𝐺) 𝑘2 ≤ 𝛾t (𝐺) 2𝑘 − 1 for the class of 𝑘-regular graphs. As discussed in this chapter, many open problems and conjectures on the six core parameters have yet to be settled and provide opportunities for further research on relationships among these core parameters.

Chapter 16

Nordhaus-Gaddum Bounds 16.1 Introduction In this chapter, we consider bounds on the sum and product of the domination numbers of a graph 𝐺 and its complement 𝐺. Similar sum and product bounds for the total domination and independent domination numbers are also presented. In 1956 Nordhaus and Gaddum [615] established the following classic bounds for the chromatic numbers of a graph 𝐺 and its complement 𝐺. Theorem 16.1 ([615]) For any graph 𝐺 of order 𝑛 and its complement 𝐺, the following √ hold: (a) 2 𝑛 ≤ 𝜒(𝐺) + 𝜒(𝐺) ≤ 𝑛 + 1. (b) 𝑛 ≤ 𝜒(𝐺) 𝜒(𝐺) ≤ 14 (𝑛 + 1) 2 . Furthermore, these bounds are tight. Because Nordhaus and Gaddum were the first to consider (tight) lower and upper bounds on the sum and product of a parameter of a graph and its complement, bounds of this form are called Nordhaus-Gaddum bounds. For more details on Nordhaus-Gaddum bounds, we refer the reader to the 2013 survey by Aouchiche and Hansen [32].

16.2

Domination Number

In this section, we present Nordhaus-Gaddum bounds involving the domination numbers of a graph and its complement. Several of the proofs in this chapter rely on Ore’s Theorem given in Theorem 4.21 in Chapter 4, so we repeat it here. Theorem 16.2 (Ore’s Theorem [622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 12 𝑛. In 1972 Jaeger and Payan [507] presented the first Nordhaus-Gaddum bounds involving domination. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_16

467

Chapter 16. Nordhaus-Gaddum Bounds

468

Theorem 16.3 ([507]) If 𝐺 is a graph of order 𝑛 ≥ 2, then the following hold: (a) 3 ≤ 𝛾(𝐺) + 𝛾( 𝐺) ≤ 𝑛 + 1. (b) 2 ≤ 𝛾(𝐺)𝛾(𝐺) ≤ 𝑛. Proof The proof for the lower bounds follows from the observation that if 𝛾(𝐺) = 1 or 𝛾(𝐺) = 1, then 𝛾(𝐺) ≥ 2 or 𝛾(𝐺) ≥ 2, respectively. Both lower bounds are achieved, for example, by the graph 𝐺 = 𝐾1,𝑛−1 with 𝛾(𝐺) = 1 and 𝛾(𝐺) = 2. Next, we verify the upper bound on the sum. If 𝐺 has an isolated vertex, then 𝛾(𝐺) = 1. Since 𝛾(𝐺) ≤ 𝑛 for any graph 𝐺, we therefore have 𝛾(𝐺) + 𝛾(𝐺) ≤ 𝑛 + 1. Similarly, if 𝐺 has an isolated vertex, then 𝛾(𝐺) = 1 and 𝛾(𝐺) ≤ 𝑛, and so 𝛾(𝐺) + 𝛾(𝐺) ≤ 𝑛 + 1. If both 𝐺 and 𝐺 are isolate-free, then by Theorem 16.2, 𝛾(𝐺) ≤ 12 𝑛 and 𝛾(𝐺) ≤ 12 𝑛 and so 𝛾(𝐺) + 𝛾(𝐺) ≤ 𝑛. Next, we show that 𝛾(𝐺)𝛾(𝐺) ≤ 𝑛. For any set 𝑋 of vertices, let 𝐷 (𝑋) be the vertices in 𝑋 that dominate 𝑋, that is, 𝑥 ∈ 𝐷 (𝑋) if 𝑥 ∈ 𝑋 and 𝑥 is adjacent to every other vertex of 𝑋. Let 𝑆 = {𝑥 1 , 𝑥2 , . . . , 𝑥 𝛾 } be a 𝛾-set of 𝐺, where 𝛾 = 𝛾(𝐺). Partition the vertices of 𝐺 into 𝛾 subsets 𝑋1 , 𝑋2 , . . . , 𝑋𝛾 such that 𝑥𝑖 ∈ 𝐷 (𝑋𝑖 ) for all 𝑖 ∈ [𝛾]. Among all such partitions, choose P to maximize the sum 𝜗 P (𝐺; 𝑋) =

𝛾 ∑︁

|𝐷 (𝑋𝑖 )|.

𝑖=1

Note that since 𝑥 𝑖 ∈ 𝑋𝑖 , we have that |𝐷 (𝑋𝑖 )| ≥ 1 for all 𝑖 ∈ [𝛾]. We show that no vertex in 𝑉 \ 𝑋𝑖 dominates 𝑋𝑖 . Suppose that there exists an 𝑥 ∈ 𝑋 𝑗 , 𝑗
≠ 𝑖, such that 𝑥 is adjacent to every vertex in 𝑋𝑖 . If 𝑥 ∈ 𝐷 (𝑋 𝑗 ), then 𝑆 \ {𝑥 𝑖 , 𝑥 𝑗 } ∪ {𝑥} is a dominating set of 𝐺 having cardinality less than 𝛾(𝐺), a contradiction. Hence, 𝑥 ∉ 𝐷 (𝑋 𝑗 ). But now the partition P ′ with 𝑋ℓ′ = 𝑋ℓ for ℓ ∈ [𝛾] \ {𝑖, 𝑗 }, 𝑋𝑖′ = 𝑋𝑖 ∪ {𝑥}, and 𝑋 ′𝑗 = 𝑋 𝑗 \ {𝑥}, for which |𝐷 (𝑋ℓ′ )| = |𝐷 (𝑋ℓ )|, |𝐷 (𝑋𝑖′ )| = |𝐷 (𝑋𝑖 )| + 1, and |𝐷 (𝑋 ′𝑗 )| ≥ |𝐷 (𝑋 𝑗 )| contradicts our choice of P. Hence, for each 𝑋𝑖 ∈ P, no vertex in 𝑉 \ 𝑋𝑖 dominates 𝑋𝑖 in 𝐺, which implies that every vertex in 𝑉 \ 𝑋𝑖 is adjacent to at least one vertex in 𝑋𝑖 in 𝐺. In other words, each set 𝑋𝑖 in P is a dominating set in 𝐺, and so 𝛾(𝐺) ≤ |𝑋𝑖 | for all 𝑖 ∈ [𝛾]. Thus, 𝑛=

𝛾∑︁ (𝐺)

|𝑋𝑖 | ≥ 𝛾(𝐺)𝛾(𝐺).

𝑖=1

The proof of Theorem 16.3 shows that each set 𝑋 𝑗 of the partition P is a dominating set of 𝐺. Thus, P is a domatic partition of 𝐺 and we have the following corollary. Corollary 16.4 For every graph 𝐺, dom(𝐺) ≥ 𝛾(𝐺). We note that Theorem 16.3, differing only in the lower bounds, was proven independently by Borowiecki [88] in 1976. The difference in the lower bounds is because Borowiecki’s result included the trivial graph 𝐾1 , while Theorem 16.3 does not. Borowiecki [88] characterized the graphs attaining the lower bounds on the sum and product of Theorem 16.3 as follows.

Section 16.2. Domination Number

469

Theorem 16.5 ([88]) If 𝐺 is a graph of order 𝑛 ≥ 2, then 𝛾(𝐺) + 𝛾( 𝐺) = 3 if and only if 𝐺 has a vertex of degree 1 and a vertex of degree 𝑛 − 1, or 𝐺 has a vertex of degree 0 and a vertex of degree 𝑛 − 2. The graphs 𝐺 for which 𝛾(𝐺) + 𝛾(𝐺) = 3, characterized in Theorem 16.5, also attain the lower bound of 2 on the product 𝛾(𝐺)𝛾(𝐺). Extremal graphs for the upper bound on the sum in Theorem 16.3 were characterized independently by Borowiecki [88] in 1976 and by Cockayne and Hedetniemi [194] in 1977. Theorem 16.6 ([88, 194]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) + 𝛾(𝐺) = 𝑛 + 1 if and only {𝐺, 𝐺} = {𝐾𝑛 , 𝐾 𝑛 }. We note that the Cartesian product 𝐺 = 𝐾3 □ 𝐾3 is a self-complementary graph, as illustrated in Figure 16.1 with 𝑉 (𝐺) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 9 }. Further, we note that 𝛾(𝐺) = 𝛾(𝐺) = 3, where the vertices highlighted in red in Figure 16.1 form a 𝛾-set of the graph.

𝑣3

𝑣6

𝑣9

𝑣3

𝑣4

𝑣8

𝑣2

𝑣5

𝑣8

𝑣5

𝑣9

𝑣1

𝑣1

𝑣4

𝑣7

𝑣7

𝑣2

𝑣6

(a) 𝐾3 □ 𝐾3

(b) 𝐾3 □ 𝐾3

Figure 16.1 The self-complementary Cartesian product 𝐾3 □ 𝐾3 In 1982 Payan and Xuong [633] characterized the extremal graphs for the upper bound on the product of Theorem 16.3. Theorem 16.7 ([633]) For every   graph of order 𝑛, 𝛾(𝐺)𝛾(𝐺) = 𝑛 if and only if 𝐺 = 𝐾3 □ 𝐾3 or 𝛾(𝐺), 𝛾(𝐺) ∈ {1, 𝑛}, {2, 12 𝑛} . We remark that Chambers et al. [140] in 2009 gave a different proof of Theorem 16.7 for graphs of order 𝑛 ≥ 184. In Chapter 4, we presented the characterization due to Payan and Xuong [633] in 1982 (and proven independently by Fink et al. [297] in 1985) of isolate-free graphs with largest possible domination number. We repeat this characterization here. Theorem 16.8 ([633]) If 𝐺 is an isolate-free graph of even order 𝑛, then 𝛾(𝐺) = 12 𝑛 if and only if every component of 𝐺 is a 4-cycle or 𝐺 = 𝐻 ◦ 𝐾1 for some graph 𝐻. As observed in the proof of Theorem 16.3, if 𝐺 has an isolated vertex, then 𝛾(𝐺) = 1. The complete graph 𝐾𝑛 and its complement 𝐾 𝑛 are the only graphs

Chapter 16. Nordhaus-Gaddum Bounds

470

 having 𝛾(𝐺), 𝛾( 𝐺) = {1, 𝑛}. Moreover, every extremal graph 𝐺 described in Theorem 16.8 has 𝛾(𝐺) = 2. Thus, Theorem 16.7 can be restated equivalently as follows. Theorem 16.9 For any graph 𝐺 of order 𝑛, 𝛾(𝐺)𝛾(𝐺) = 𝑛 if and only if 𝐺 or 𝐺 is one of the following: (a) a complete graph 𝐾𝑛 , (b) the Cartesian product 𝐾3 □ 𝐾3 , (c) a graph for which each component is a 4-cycle or a corona of a connected graph.

16.2.1

Minimum Degree One

In 1995 Joseph and Arumugam [513] significantly improved the upper bound in Theorem 16.3(a) on the sum of the domination number of a graph and its complement in the case when both graphs are isolate-free. Note that if a graph and its complement are both isolate-free, then neither graph has a dominating vertex. Theorem 16.10 ([513])  If 𝐺 and its complement 𝐺 are isolate-free graphs order 𝑛, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 12 𝑛 + 2, with equality if and only if    (a) 𝛾(𝐺), 𝛾(𝐺) ∈ 12 𝑛 , 2 , or (b) 𝑛 = 9 and 𝛾(𝐺) = 𝛾(𝐺) = 3. Proof Since 𝐺 is isolate-free and has no dominating vertex, 𝛾(𝐺) ≥ 2 and  𝛾(𝐺) ≥ 2. Since 𝛿(𝐺) ≥ 1 and 𝛿(𝐺) ≥ 1, by Theorem 16.2, we have 𝛾(𝐺) ≤ 21 𝑛 and 𝛾(𝐺) ≤ 1  1  2 𝑛 . This implies that if 𝛾(𝐺) = 2 or 𝛾(𝐺) = 2, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 2 𝑛 + 2, as desired. Therefore, we may assume that 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) ≥ 3. By Theorem 16.3, we have 𝛾(𝐺)𝛾(𝐺) ≤ 𝑛, implying that if 𝛾(𝐺) ≥ 4 and 𝛾(𝐺) ≥ 4, then 𝛾(𝐺) ≤ 14 𝑛   and 𝛾(𝐺) ≤ 14 𝑛, and so 𝛾(𝐺) +𝛾(𝐺) ≤ 14 𝑛+ 14 𝑛 = 21 𝑛 < 12 𝑛 +2. The only remaining possibility is that either 𝛾(𝐺) = 3 and 𝛾(𝐺) ≥ 3 or 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) = 3. We may assume, without loss of generality,   that 𝛾(𝐺) ≥  3 and 𝛾(𝐺) = 3. By Theorem 16.3, we therefore have 𝛾(𝐺) ≤ 13 𝑛 and 𝛾(𝐺) ≤ 13 𝑛 . Since 𝛾(𝐺) = 3 ≤ 13 𝑛, we note     that 𝑛 ≥ 9. Therefore, 𝛾(𝐺) + 𝛾(𝐺) ≤ 13 𝑛 + 3 ≤ 12 𝑛 + 2, with strict inequality if 𝑛 ≥ 10. This establishes the desired upper bound.   Suppose there is equality in this bound, that is, 𝛾(𝐺) + 𝛾(𝐺) = 12 𝑛 + 2. By our   earlier observations, this implies that 𝛾(𝐺) = 2 and 𝛾(𝐺) = 12 𝑛 , or 𝛾(𝐺) = 2 and 1  𝛾(𝐺) = 2 𝑛 , or 𝑛 = 9 and 𝛾(𝐺) = 𝛾(𝐺) = 3. Thus, (a) or (b) holds. The sufficiency   is immediate since if (a) or (b) holds, then 𝛾(𝐺) + 𝛾(𝐺) = 12 𝑛 + 2. Payan and Xuong [633] proved that the only graph 𝐺 of order 9 having 𝛾(𝐺) = 𝛾(𝐺) = 3 is the Cartesian product 𝐺 = 𝐾3 □ 𝐾3 , which as observed earlier is self-complementary. Theorem 16.11 ([633]) If 𝐺 is an isolate-free graph of order 𝑛 = 9, then 𝛾(𝐺) = 𝛾(𝐺) = 3 if and only if 𝐺 = 𝐾3 □ 𝐾3 .

Section 16.2. Domination Number

471

Proof If 𝐺 = 𝐾3 □ 𝐾3 , then as observed earlier, 𝛾(𝐺) = 𝛾( 𝐺) = 3. To prove the necessity, let 𝐺 be an isolate-free graph of order 𝑛 = 9 having 𝛾(𝐺) = 𝛾(𝐺) = 3. Let 𝑥 be a vertex of maximum degree Δ(𝐺) in 𝐺. If Δ(𝐺) = 8, then 𝑥 is a dominating vertex of 𝐺, and so 𝛾(𝐺) = 1, a contradiction. If Δ(𝐺) = 7, then 𝑥 and its (unique) non-neighbor in 𝐺 form a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, Δ(𝐺) ≤ 6. Let 𝑋 = N𝐺 (𝑥) and let 𝑌 = 𝑉 (𝐺) \ N𝐺 [𝑥]. Suppose Δ(𝐺) = 6, and so |𝑌 | = 2. If a neighbor 𝑦 of 𝑥 is adjacent to both vertices of 𝑌 , then {𝑥, 𝑦} is a dominating set of 𝐺. If no neighbor of 𝑥 is adjacent to both vertices of 𝑌 , then 𝑌 is a dominating set of 𝐺. In both cases, we produce a dominating set of cardinality 2, a contradiction. Hence, Δ(𝐺) ≤ 5. Similarly, Δ(𝐺) ≤ 5. Therefore, 𝛿(𝐺) ≥ 3 and 𝛿(𝐺) ≥ 3. Suppose Δ(𝐺) = 5, and so 𝛿(𝐺) = 3. In this case, |𝑋 | = 5 and |𝑌 | = 3. Let 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 } and let 𝑌 = {𝑦 1 , 𝑦 2 , 𝑦 3 }. If a vertex 𝑥 ′ ∈ 𝑋 is adjacent to every vertex of 𝑌 , then {𝑥, 𝑥 ′ } is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, each vertex of 𝑋 is adjacent in 𝐺 to at most two vertices in 𝑌 . If there is a vertex 𝑦 ∈ 𝑌 adjacent in 𝐺 to the other two vertices in 𝑌 , then {𝑥, 𝑦} is a dominating set of 𝐺, a contradiction. Therefore, 𝐺 [𝑌 ] has maximum degree at most 1, implying that 𝐺 [𝑌 ] contains an isolated vertex. This in turn implies that since {𝑦 𝑖 , 𝑦 𝑗 } is not a dominating set of 𝐺 for 1 ≤ 𝑖 < 𝑗 ≤ 3, there is a vertex 𝑥𝑖 𝑗 ∈ 𝑋 adjacent in 𝐺 to both 𝑦 𝑖 and 𝑦 𝑗 . Renaming vertices if necessary, we may assume that 𝑥1 = 𝑥23 , 𝑥2 = 𝑥13 , and 𝑥3 = 𝑥12 . We note that in 𝐺, the vertex 𝑥 is adjacent to 𝑦 1 , 𝑦 2 , and 𝑦 3 . Further in 𝐺, the only edges between {𝑥1 , 𝑥2 , 𝑥3 } and {𝑦 1 , 𝑦 2 , 𝑦 3 } are the edges 𝑥𝑖 𝑦 𝑖 for 𝑖 ∈ [3]. Since {𝑥𝑖 , 𝑦 𝑖 }, for all 𝑖 ∈ [3], is not a dominating set of 𝐺, {𝑥𝑖 , 𝑦 𝑖 } does not dominate 𝑥4 or 𝑥 5 in 𝐺. Thus, 𝑥4 or 𝑥5 is adjacent to both 𝑥 𝑖 and 𝑦 𝑖 in 𝛾(𝐺) for all 𝑖 ∈ [3]. Since Δ(𝐺) ≤ 5, renaming 𝑥4 and 𝑥5 if necessary, we assume that 𝑥 4 is adjacent in 𝐺 to all four vertices in {𝑥1 , 𝑥2 , 𝑦 1 , 𝑦 2 } and that 𝑥5 is adjacent in 𝐺 to both vertices 𝑥3 and 𝑦 3 . But then {𝑥4 , 𝑦 3 } is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Therefore, the graph 𝐺, and hence also its complement 𝐺, is 4-regular. Thus, |𝑋 | = |𝑌 | = 4. Let 𝑋 = {𝑥 1 , 𝑥2 , 𝑥3 , 𝑥4 } and let 𝑌 = {𝑦 1 , 𝑦 2 , 𝑦 3 , 𝑦 4 }. If a vertex 𝑥 ′ ∈ 𝑋 is adjacent to no other vertex of 𝑋 in 𝐺, then {𝑥, 𝑥 ′ } is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, 𝐺 [𝑋] is isolate-free, implying that every vertex of 𝑋 has at least one neighbor in 𝑋 and at most two neighbors in 𝑌 in 𝐺. For the set [𝑋, 𝑌 ] of edges between 𝑋 and 𝑌 in 𝐺, we have | [𝑋, 𝑌 ] | ≤ 2|𝑋 | = 8. If a vertex 𝑦 ∈ 𝑌 is adjacent to every other vertex of 𝑌 in 𝐺, then {𝑥, 𝑦} is a dominating set of 𝐺 and so 𝛾(𝐺) = 2, a contradiction. Hence, every vertex in 𝑌 is adjacent to at most two other vertices of 𝑌 , and therefore to at least two vertices of 𝑋 in 𝐺, and so | [𝑋, 𝑌 ] | ≥ 2|𝑌 | = 8. Consequently, | [𝑋, 𝑌 ] | = 8, implying by our earlier observations that 𝐺 [𝑋] is a 1-regular graph and every vertex in 𝑋 has exactly two neighbors in 𝑌 . Further, 𝐺 [𝑌 ] is a 2-regular graph and every vertex in 𝑌 has exactly two neighbors in 𝑋. Thus, 𝐺 [𝑋] = 2𝐾2 and 𝐺 [𝑌 ] = 𝐶4 . Renaming vertices if necessary, we may assume that 𝑥 1 𝑥2 and 𝑥3 𝑥4 are the two edges in 𝐺 [𝑋], and 𝐶 : 𝑦 1 𝑦 2 𝑦 3 𝑦 4 𝑦 1 is the 4-cycle 𝐺 [𝑌 ].

Chapter 16. Nordhaus-Gaddum Bounds

472

We now consider an arbitrary edge 𝑦 𝑖 𝑦 𝑖+1 in 𝐶, where 𝑖 ∈ [4] and addition is taken modulo 4. Since {𝑦 𝑖 , 𝑦 𝑖+1 } is not a dominating set in 𝐺, there is a vertex in 𝑋 adjacent in 𝐺 to both 𝑦 𝑖 and 𝑦 𝑖+1 . This is true for all 𝑖 ∈ [4]. Renaming vertices if necessary, we may assume that 𝑥 1 is adjacent to 𝑦 1 and 𝑦 2 . We now consider the vertex in 𝑋 adjacent to both 𝑦 3 and 𝑦 4 . If 𝑥3 (respectively, 𝑥4 ) is such a vertex, then {𝑥 1 , 𝑥3 } (respectively, {𝑥1 , 𝑥4 }) is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, 𝑥2 is the vertex of 𝑋 adjacent to both 𝑦 3 and 𝑦 4 . Renaming 𝑥3 and 𝑥4 if necessary, analogous arguments show that 𝑥 3 is adjacent to both 𝑦 1 and 𝑦 4 , and 𝑥4 is adjacent to both 𝑦 2 and 𝑦 3 . The graph 𝐺 is now determined and 𝐺 = 𝐾3 □ 𝐾3 . Combining the results of Theorems 16.10 and 16.11, we have the following theorem. Theorem 16.12 ([513, 633]) If𝐺 and graphs of   its complement 𝐺 are isolate-free order 𝑛 ≥ 2, then 𝛾(𝐺)+𝛾(𝐺) ≤ 12 𝑛 +2, with equality if and only if 𝛾(𝐺), 𝛾(𝐺) ∈ 1  2 𝑛 , 2 or 𝐺 = 𝐾3 □ 𝐾3 . The graphs of even order 𝑛 having 𝛾(𝐺) = 12 𝑛 are examples of extremal graphs for both the upper bound of Theorem 16.3(b) on the product and the upper bound of Theorem 16.12 on the sum. Using completely different methods, independently around the same time, Randerath and Volkmann [648] in 1998 and Baogen et al. [57]  in 2000 characterized the connected graphs 𝐺 having order 𝑛 and 𝛾(𝐺) = 12 𝑛 . Both characterizations yield the same family of extremal graphs but the descriptions of the graph families are different. We give the description from [57] here. Let Bdom = {𝐵1 , 𝐵2 , . . . , 𝐵7 } be the family of seven graphs 𝐺 of order 𝑛 satisfying 𝛾(𝐺) > 25 𝑛. This family was illustrated in Figure 6.1 in Chapter 6, which we repeat in Figure 16.2.

(a) 𝐵1

(d) 𝐵4

(b) 𝐵2

(e) 𝐵5

(c) 𝐵3

(f) 𝐵6

(g) 𝐵7

Figure 16.2 The family Bdom of seven graphs 𝐺 satisfying 𝛾(𝐺) > 25 𝑛 Let Adom = {𝐴1 , 𝐴2 , . . . , 𝐴5 } be the family of five graphs illustrated in Figure 16.3.

Section 16.2. Domination Number

(a) 𝐴1

(b) 𝐴2

473

(c) 𝐴3

(d) 𝐴4

(e) 𝐴5

Figure 16.3 The family Adom of five graphs

We next define six families of graphs. For this purpose, we introduce the following additional notation. For any graph 𝐻, let S(𝐻) denote the set of connected graphs which may be formed from the corona 𝐻 ◦ 𝐾1 by adding a new vertex 𝑥 and edges joining 𝑥 to one or more vertices of 𝐻. Further, let 𝐺 ★ be the graph obtained from a graph 𝐺 ∈ S(𝐻) by adding a 4-cycle and joining the new vertex 𝑥 added in forming 𝐺 to one vertex of the added 4-cycle. For any graph 𝐻, let P (𝐻) be the set of connected graphs which may be formed from the corona 𝐻 ◦ 𝐾1 by adding a path 𝑃3 and joining both leaves of the 𝑃3 to one or more vertices of 𝐻. For any graph 𝐻 and any graph 𝑋 ∈ Adom , let R (𝐻, 𝑋) be the set of connected graphs which may be formed from the corona 𝐻 ◦ 𝐾1 by joining each vertex of 𝑈 ⊆ 𝑉 (𝑋) to one or more vertices of 𝐻 such that 𝛾(𝑋) vertices of 𝑋 are required to dominate 𝑉 (𝑋) \ 𝑈. Let G1 , G2 , . . . , G6 be the six families of graphs defined as follows: • G1 = {𝐺 : 𝐺 = 𝐻 ◦ 𝐾1 , where 𝐻 is a connected graph}, • G2 = Ð Adom ∪ Bdom , • G3 = 𝐻 S(𝐻), where the union is taken over all graphs 𝐻, • G4 = Ð {𝐺 ★ : 𝐺 ∈ G3 }, • G5 = 𝐻 P (𝐻), where the union is taken over all graphs 𝐻, Ð • G6 = 𝐻,𝑋 R (𝐻, 𝑋), where the union is taken over all graphs 𝐻 and all graphs 𝑋 ∈ Adom . We are now in a position to present the characterization given in [57].   Theorem 16.13 ([57, 648]) If 𝐺 is a connected graph of order 𝑛, then 𝛾(𝐺) = 12 𝑛 Ð6 if and only if 𝐺 ∈ G = 𝑖=1 G𝑖 . Using Theorem 16.13, Baogen et al. [57] characterized the family of extremal graphs attaining the bound of Theorem 16.10. Theorem 16.14 ([57]) If 𝐺 and its complement 𝐺 are isolate-free graphs of order 𝑛 ≥ 2, then the following hold:   (a) If 𝐺 is connected, then 𝛾(𝐺) + 𝛾(𝐺) = 12 𝑛 + 2 if and only if 𝐺 = 𝐾3 □ 𝐾3 or one of 𝐺 and its complement 𝐺 is in G.   (b) If 𝐺 is disconnected, then 𝛾(𝐺) + 𝛾(𝐺) = 12 𝑛 + 2 if and only if at most one component of 𝐺 has odd order and all components of 𝐺 belong to G.

Chapter 16. Nordhaus-Gaddum Bounds

474

Proof (a) This is immediate from Theorems 16.12 and 16.13. (b) Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑡 be the components of 𝐺, where 𝑡 ≥ 2. For each 𝑖 ∈ [𝑡], let 𝐺 𝑖 have order 𝑛𝑖 and 𝛾𝑖 = 𝛾(𝐺 𝑖 ). Since 𝐺 and 𝐺 are isolate-free graphs and 𝑡 ≥ 2, Í𝑡 we have 𝛾(𝐺) = 2 and 𝛾(𝐺) = 𝑖=1 𝛾𝑖 . Therefore, by Theorem 16.12, the bound is attained if and only if   𝑡 ∑︁ 𝑛 𝛾𝑖 = . (16.1) 2 𝑖=1 Suppose that two or more components of 𝐺 have odd order. Renaming components if necessary, we may assume that 𝑛1 = 2𝑘 1 + 1 and 𝑛2 = 2𝑘 2 + 1 for some integers 𝑘 1 , 𝑘 2 ≥ 1. For each 𝑖 ∈ [𝑡], 𝐺 𝑖 has no isolated vertices. By Theorem 16.2, 𝛾1 ≤ 𝑘 1 , 𝛾2 ≤ 𝑘 2 , and for 𝑖 ∈ [𝑡] \ {1, 2}, we have 𝛾𝑖 ≤ 12 𝑛𝑖 . Hence, 𝑡 ∑︁ 𝑖=1

  𝑡 1 ∑︁ 𝑛 𝑛 𝛾𝑖 ≤ 𝑘 1 + 𝑘 2 + 𝑛𝑖 = − 1 < , 2 𝑖=3 2 2

contrary to Equation (16.1). at most at most one component of 𝐺 has odd order.  Thus,  Suppose next that 𝛾𝑖 < 12 𝑛𝑖 for some 𝑖 ∈ [𝑡]. Renaming components if necessary,   we may assume that 𝛾1 < 12 𝑛1 . Hence, 𝑡 ∑︁ 𝑖=1

 𝛾𝑖 ≤

    𝑡  ∑︁ 𝑛1 𝑛𝑖 𝑛 𝑛 −1+ ≤ −1 < , 2 2 2 2 𝑖=2

  contrary to Equation (16.1). Hence, 𝛾𝑖 = 12 𝑛𝑖 for all 𝑖 ∈ [𝑡], implying by Theorem 16.13 that 𝐺 𝑖 ∈ G for all 𝑖 ∈ [𝑡]. Conversely, suppose that at most one component of 𝐺 has  odd  order and all components of 𝐺 belong to G. Since 𝐺 𝑖 ∈ G, we have 𝛾𝑖 = 12 𝑛𝑖 for all 𝑖 ∈ [𝑡] by Theorem 16.13. Renaming components if necessary, we may assume that 𝑛2 , 𝑛3 , . . . , 𝑛𝑡 are even and so 𝛾𝑖 = 12 𝑛𝑖 for all 𝑖 ∈ [𝑡] \ {1}. Thus, 𝑡 ∑︁ 𝑖=1

16.2.2

 𝛾𝑖 =

 ∑︁   𝑡 𝑛1 𝑛𝑖 𝑛 + = . 2 2 2 𝑖=2

Minimum Degree Two

As we have seen in Chapter 6, if the minimum degree of a graph 𝐺 is at least 2, the upper bound on 𝛾(𝐺) in Theorem 16.2 can be improved from one-half its order to two-fifths its order, except for seven exceptional graphs in the family Bdom illustrated in Figure 16.2. We restate this 1989 result of McCuaig and Shepherd [586], which is Theorem 6.18 in Chapter 6. Theorem 16.15 ([586]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 is one of the seven exceptional graphs in the family Bdom . Since the family Bdom consists of one graph of order 4 and six of order 7, as an immediate consequence of Theorem 16.15, we have the following result.

Section 16.2. Domination Number

475

Corollary 16.16 If 𝐺 is a connected graph of order 𝑛 ≥ 8 with 𝛿(𝐺) ≥ 2,  ([586])  then 𝛾(𝐺) ≤ 52 𝑛 . As an application of Corollary 16.16, Dunbar et al. [249] in 2005 proved the following result using a proof technique similar to the one used to prove Theorem 16.10. Recall that the graph 𝐵1 ∈ Bdom shown in Figure 16.2 is the 4-cycle. Theorem 16.17 ([249]) If 𝐺 and its complement 𝐺 are connected graphs of order   𝑛 ≥ 2 with 𝛿(𝐺) ≥ 2 and 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 25 𝑛 + 3, with equality if and only if 𝐺 = 𝐾3 □ 𝐾3 or one of 𝐺 and its complement 𝐺 is in Bdom \ {𝐵1 }. Proof Let 𝐺 and 𝐺 be connected graphs of order 𝑛 for which 𝛿(𝐺) ≥ 2 and 𝛿(𝐺) ≥ 2. In particular, we note that neither 𝐺 nor 𝐺 is the graph 𝐵1 ∈ Bdom since 𝐶 4 does not have minimum degree   at least 2. If either 𝐺 or 𝐺 is in Bdom \ {𝐵1 }, then 𝑛 = 7 and 𝛾(𝐺) + 𝛾(𝐺) = 5 = 25 𝑛 + 3. Hence, we may assume that neither 𝐺 nor 𝐺 is in Bdom \ {𝐵1 }, for otherwise the desired result is immediate. Thus, by Theorem 16.15, we have 𝛾(𝐺) ≤ 25 𝑛 and 𝛾(𝐺) ≤ 25 𝑛. Hence, if 𝛾(𝐺) ≤ 3 or 𝛾(𝐺) ≤ 3, then the desired result follows. Therefore, we may assume that 𝛾(𝐺) ≥ 4 and 𝛾(𝐺) ≥ 4. By Theorem 16.3(b), we have 𝑛 ≥ 16. If 𝛾(𝐺) ≥ 5 and 𝛾(𝐺) ≥ 5, then Theorem 16.3(b)  implies that 𝛾(𝐺) ≤ 15 𝑛 and 𝛾(𝐺) ≤ 15 𝑛. In this case, 𝛾(𝐺) + 𝛾(𝐺) ≤ 25 𝑛 < 25 𝑛 + 3. Hence, we may assume that 𝛾(𝐺) = 4 and 𝛾(𝐺) ≥ 4 or 𝛾(𝐺) ≥ 4 and 𝛾(𝐺) = 4. Without loss of generality, Theorem   we may assume that 𝛾(𝐺) = 4. Again,    16.3(b) implies that 𝛾(𝐺) ≤ 14 𝑛 . Therefore, 𝛾(𝐺) + 𝛾(𝐺) ≤ 14 𝑛 + 4 < 25 𝑛 + 3 since 𝑛 ≥ 16, yielding the desired bound (with strict inequality in this case).   To determine the extremal graphs, assume that 𝛾(𝐺) + 𝛾(𝐺) = 25 𝑛 + 3 for a graph 𝐺 and is complement 𝐺 satisfying the conditions in the statement of the theorem. Based on the proof of the upper bound presented in the previous paragraph, equality is only possible if 𝐺 or 𝐺 is in Bdom \ {𝐵1 } or if 𝐺 or 𝐺 has domination number at most 3. As observed earlier, neither  𝐺 nor 𝐺 is the graph 𝐵1 , and if 𝐺 or 𝐺 is in Bdom \ {𝐵1 }, then 𝛾(𝐺) + 𝛾(𝐺) = 25 𝑛 + 3. Hence, we may assume that 𝐺 ∉ Bdom . Thus, Theorem 16.15 implies that 𝛾(𝐺) ≤ 25 𝑛 and 𝛾(𝐺) ≤ 25 𝑛.     If 𝛾(𝐺) ≤ 2 or 𝛾(𝐺) ≤ 2, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 25 𝑛 + 2 < 25 𝑛 + 3, a contradiction. Hence, 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) ≥ 3. By our earlier observations, 𝛾(𝐺) ≤ 3 or 𝛾(𝐺) ≤ 3. Without loss of generality, we may assume that 𝛾(𝐺) ≤ 3. Consequently, 𝛾(𝐺) = 3. As observed earlier,   𝛾(𝐺) ≥ 3. By Theorem 16.3(b),   we have   𝑛 ≥ 9 and 𝛾(𝐺) ≤ 31 𝑛 . Thus, 25 𝑛 + 3 = 𝛾(𝐺) + 𝛾(𝐺) ≤ 3 + 13 𝑛 ≤ 3 + 25 𝑛 . We must therefore have equality throughout this equality chain, implying that either 𝑛 = 9 and 𝛾(𝐺) = 3 or 𝑛 = 12 and 𝛾(𝐺) = 4. If 𝑛 = 12 and 𝛾(𝐺) = 4, then 𝛾(𝐺)𝛾(𝐺) = 𝑛, contradicting Theorem 16.7. Hence, 𝑛 = 9 and 𝛾(𝐺) = 𝛾(𝐺) = 3. Thus, by Theorem 16.11, we have 𝐺 = 𝐾3 □ 𝐾3 . As an immediate consequence of Theorem 16.17, we have the following result. Theorem 16.18 ([249]) If 𝐺 and its complement 𝐺 are connected graphs of order 𝑛 ≥ 2 with 𝛿(𝐺) ≥ 2 and 𝛿(𝐺) ≥ 2, and if neither   𝐺 nor 𝐺 is the Cartesian product 𝐾3 □ 𝐾3 or is in Bdom , then 𝛾(𝐺) + 𝛾(𝐺) ≤ 25 𝑛 + 2.

Chapter 16. Nordhaus-Gaddum Bounds

476

The extremal graphs achieving equality in the bound of Theorem 16.18 with order 𝑛 ≥ 23 were characterized in [249]. Theorem 16.19 ([249]) If 𝐺 and its complement 𝐺 are connected   graphs of order 𝑛 ≥ 23 with 𝛿(𝐺) ≥ 2 and 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) + 𝛾(𝐺) = 25 𝑛 + 2 if and only if    𝛾(𝐺), 𝛾(𝐺) = 25 𝑛 , 2 . For connected graphs 𝐺 and 𝐺 with minimum degree at least 2 and order 𝑛 ≤ 22, Dunbar et al. [249] noted that Theorem 16.19 holds for 𝑛 ≤ 9 if neither 𝐺 nor 𝐺 is one of the eight graphs in Bdom ∪ {𝐾3 □ 𝐾3 }. They also showed that Theorem 16.19 holds for such graphs 𝐺 of order 𝑛 = 17 with 𝛾(𝐺) = 𝛾(𝐺) = 4, and for graphs of order 𝑛 ∈ {13, 15, 18, 20, 21}. Volkmann [737] in 2010 showed that the theorem holds for graphs 𝐺 of order 𝑛 ≥ 18 with 𝛾(𝐺) = 3. On the other hand, Dunbar et al. [249] gave the following counterexamples for graphs of order 𝑛 ∈ {10, 11, 12}. Form a graph 𝐺 𝑣 from a copy of 𝐾3 □ 𝐾3 by adding a new vertex 𝑣 and edges from 𝑣 to three vertices in a common triangle   of 𝐾3 □ 𝐾3 . Then, 𝐺 𝑣 has order 𝑛 = 10 and 𝛾(𝐺 𝑣 ) + 𝛾(𝐾𝐺𝑣 ) = 3 + 3 = 6 = 25 𝑛 + 2, 2  but 𝛾(𝐺 𝑣 ) ∉ 5 𝑛 , 2 . For 𝑛 = 11, construct a graph 𝐺 𝑢 from the graph 𝐺 𝑣 by adding a new vertex 𝑢 and edges such that 𝑢 is adjacent to exactly one vertex in N(𝑣) and three additional   2 triangle   vertices in a common  of 𝐺 𝑣 − N(𝑣). Again, 𝛾(𝐺 𝑢 ) + 𝛾(𝐺 𝑢 ) = 6 = 25 𝑛 + 2 and 𝛾(𝐺 𝑢 ) ∉ 5 𝑛 , 2 . For 𝑛 = 12, the graph 𝐾3 □ 𝐾4 is a counterexample to the theorem. Considering the implications of Theorem 16.3(b), Dunbar  et al. [249] noted that the problem remained open for the cases (i) 𝑛 = 14 and 𝛾(𝐺), 𝛾(𝐺) = {3, 4},   (ii) 𝑛 ∈ {16, 17} and 𝛾(𝐺), 𝛾(𝐺) = {3, 5}, (iii) 𝑛 = 19 and 𝛾(𝐺), 𝛾(𝐺) = {3, 6},  and (iv) 𝑛 = 22 and 𝛾(𝐺), 𝛾(𝐺) = {3, 7}. Volkmann [737] showed that Theorem 16.19 holds for graphs 𝐺 satisfying the necessary conditions and having 𝑛 ≥ 18 and 𝛾(𝐺) = 3. Thus, Volkmann [737] solved the open cases for 𝑛 = 19 and 𝑛 = 22. The only remaining open cases are: (i) 𝑛 = 14 and 𝛾(𝐺), 𝛾(𝐺) = {3, 4} and  (ii) 𝑛 ∈ {16, 17} and 𝛾(𝐺), 𝛾(𝐺) = {3, 5}.

16.2.3

Minimum Degree Three

In this section, we restrict the minimum degree of 𝐺 and 𝐺 to be at least 3. Recall Reed’s result, which is Theorem 6.20 in Chapter 6. Theorem 16.20 ([655]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 38 𝑛. Recall, as stated in Chapter 4 and first observed in [118], that if 𝐺 is a graph with diam(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 2. Equivalently, this result can be stated as follows. Observation 16.21 ([118]) If 𝐺 is a graph with 𝛾(𝐺) ≥ 3, then diam(𝐺) ≤ 2. We shall also need the following result due to Hellwig and Volkmann [452] in 2006.

Section 16.2. Domination Number

477

Theorem 16.22 ([452]) If 𝐺 is a diameter-2 graph of order 𝑛, then   𝛾(𝐺) ≤ 41 𝑛 + 1. We remark that using a similar proof to that used to prove Lemma 7.19 in Chapter 7, the following result can be proved. However, for 𝑛 ≤ 51, the bound in Theorem 16.22 yields a better bound. √︁ Lemma 16.23 If 𝐺 is a diameter-2 graph of order 𝑛 ≥ 24, then 𝛾(𝐺) < 𝑛 ln(𝑛). We prove next the following result, which is a slight strengthening of results due to Dunbar et al. [249] and Volkmann [737]. Theorem 16.24 ([249, 737]) If 𝐺 ≠ 𝐾3□ 𝐾3 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3 and 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 38 𝑛 + 2, unless 𝑛 = 13, in which case   𝛾(𝐺) + 𝛾(𝐺) ≤ 3𝑛+1 + 2. 8 Proof Let 𝐺 and 𝐺 be graphs satisfying the conditions of the theorem. By Theorem 16.20, 𝛾(𝐺) ≤ 38 𝑛 and 𝛾(𝐺) ≤ 38 𝑛. Hence, if 𝛾(𝐺) = 2 or 𝛾(𝐺) = 2, then the desired result follows. Therefore, we assume that 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) ≥ 3 and so both 𝐺 and 𝐺 are connected. It follows from Theorem 16.3(b) that 𝑛 ≥ 9. 1 If 𝛾(𝐺) ≥ 6 and 𝛾(𝐺) ≥ 6, then Theorem 16.3(b)  3  implies that 𝛾(𝐺) ≤ 6 𝑛 and 1 1 𝛾(𝐺) ≤ 6 𝑛. Therefore, 𝛾(𝐺) + 𝛾(𝐺) ≤ 3 𝑛 < 8 𝑛 + 2. Hence, we may assume, without loss of generality, that 3 ≤ 𝛾(𝐺) ≤ 5 and 𝛾(𝐺) ≤ 𝛾(𝐺).   If 𝛾(𝐺) = 5, then by Theorem 16.3(b), we have 𝑛 ≥ 25 and 𝛾(𝐺) ≤ 15 𝑛 , and 1  3  so 𝛾(𝐺) + 𝛾(𝐺) ≤ 5 + 5 𝑛 < 8 𝑛 + 2. If 𝛾(𝐺) = 4, then by Theorem 16.3(b),       𝑛 ≥ 16 and 𝛾(𝐺) ≤ 14 𝑛 , and so 𝛾(𝐺) + 𝛾(𝐺) ≤ 4 + 14 𝑛 ≤ 38 𝑛 + 2. Hence, we may assume that 𝛾(𝐺) = 3, for otherwise the desired result follows. By our earlier assumptions, 𝛾(𝐺) ≥ 3. If 𝑛 = 9, then 𝛾(𝐺) = 𝛾(𝐺) = 3 and, by Theorem 16.11, we have 𝐺 = 𝐾3 □ 𝐾3 , a contradiction. Hence, 𝑛 ≥ 10. If 𝑛 = 10, then byTheorem 16.3(b) and our earlier assumptions, 𝛾(𝐺) = 3 and so  𝛾(𝐺) + 𝛾(𝐺) ≤ 6 = 38 𝑛 + 2. Hence, we may assume that 𝑛 ≥ 11. Since 𝛾(𝐺) = 3, by Observation we have diam(𝐺) = 2. This in turn implies by Theorem 16.22  16.21  that 𝛾(𝐺) ≤ 14 𝑛 + 1. By our earlier assumptions, 𝛾(𝐺) = 3. If 𝑛 ∉ {12, 13}, then     𝛾(𝐺) + 𝛾(𝐺) ≤ 3 + 14 𝑛 + 1 ≤ 38 𝑛 + 2. Hence, we may assume that 𝑛 ∈ {12, 13}. If 𝑛 = 12, then by Theorem 16.3(b), 𝛾(𝐺) ≤ 4. If 𝛾(𝐺) = 4, then 𝛾(𝐺)𝛾(𝐺)  = 𝑛, contradicting Theorem 16.7. Therefore, 𝛾(𝐺) = 3, and so 𝛾(𝐺)+𝛾(𝐺) ≤ 6 = 38 𝑛 +2. Finally,  3𝑛+1  if 𝑛 = 13, then by Theorem 16.3(b), 𝛾(𝐺) ≤ 4, and so 𝛾(𝐺) + 𝛾(𝐺) ≤ 7 = + 2. 8 We remark that the case for 𝑛 = 13 in Theorem 16.24 open, in the sense  remains  that it is not yet known if the bound 𝛾(𝐺) + 𝛾(𝐺) ≤ 3𝑛+1 is achievable. As + 2 8 shown in the proof of Theorem 16.24, this would imply the existence of a graph 𝐺 of order 𝑛 = 13 with 𝛾(𝐺) = 3 and 𝛾(𝐺) = 4. Volkmann [737] in 2010 proved the following result about the extremal graphs for Theorem 16.24.

Chapter 16. Nordhaus-Gaddum Bounds

478

Theorem 16.25 ([737]) If 𝐺 ≠ 𝐾3 □ 𝐾3 is a graph of order 𝑛 ∉ {11, 12, . . . , 18} ∪ {20, 21} with 𝛿(𝐺) ≥ 3 and 𝛿( 𝐺) ≥ 3, then 𝛾(𝐺) + 𝛾(𝐺) = 38 𝑛 + 2 if and only if    𝛾(𝐺), 𝛾(𝐺) = 38 𝑛 , 2 . Volkmann [737] noted that Theorem 16.25 does not hold for 𝑛 ∈ {11, 12}.  The graph 𝐺 = 𝐾3 □ 𝐾4 is a counterexample for 𝑛 = 12 as 𝛾(𝐺) + 𝛾(𝐺) = 6 = 38 𝑛 + 2   and 𝛾(𝐺) ∉ 38 𝑛 , 2 . For 𝑛 = 11, consider the graph 𝐾3 □ 𝐾3 , where the vertices of one 𝐾3 are labeled 𝑥1 , 𝑥2 , 𝑥3 and the vertices of the other are 𝑦 1 , 𝑦 2 , 𝑦 3 . Form the graph 𝐺 from 𝐾3 □ 𝐾3 by adding two new vertices  𝑢 and 𝑣 and edges such that 𝑢 and 𝑣 are adjacent, 𝑢 is adjacent to each vertex in (𝑥1 , 𝑦 1 ), (𝑥2 , 𝑦 1 ), (𝑥3 , 𝑦 1 ), (𝑥 3 , 𝑦 3 ) ,  and 𝑣 is adjacent to each vertex in (𝑥1 , 𝑦 2 ), (𝑥2 , 𝑦 2 ), (𝑥3 , 𝑦 2 ), (𝑥3 , 𝑦 3 ) . The resulting   3  graph 𝐺 has order 𝑛 = 11 and 𝛾(𝐺)+𝛾(𝐺) = 3+3 = 6 = 8 𝑛 , but 𝛾(𝐺) ∉ 38 𝑛 , 2 . We remark that if 𝐺 is the graph of order 𝑛 = 13 obtained from a copy of 𝐾3 □ 𝐾4 by adding a new vertex 𝑣 and edges from 𝑣 to four vertices in a common   copy of 𝐾4 in 𝐾3 □ 𝐾4 , then 𝛾(𝐺) = 𝛾(𝐺) = 3, and so 𝛾(𝐺) + 𝛾(𝐺) = 6 = 38 𝑛 + 2, but   𝛾(𝐺) ∉ 38 𝑛 , 2 . Hence, Theorem 16.25 does not hold for 𝑛 = 13. The cases for 𝑛 when 14 ≤ 𝑛 ≤ 21 and 𝑛 ≠ 19 remain open.

16.2.4

Minimum Degree Four

Volkmann [738] in 2011 considered graphs and their complements in which both have minimum degree 4. Recall the result of Sohn and Xudong [682], which is Theorem 6.29 in Chapter 6 and repeated below. Theorem 16.26 ([682]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤

4 11 𝑛.

As an application of Theorem 16.26, Volkmann [738] proved the following result. We omit a proof of this result, which is along similar lines to that of Theorem 16.24. Theorem 16.27 ([738]) If 𝐺 ≠ 𝐾3 □ 𝐾3 is a graph  4  of order 𝑛 ∉ {10, 13, 16} with 𝛿(𝐺) ≥ 4 and 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 11 𝑛 + 2. The cases for 𝑛 ∉ {10, 13, 16} remain open. Volkmann [738] determined the extremal graphs achieving the bound of Theorem 16.24 as follows. Theorem 16.28 ([738]) If 𝐺 ≠ 𝐾3 □ 𝐾3 is a graph of order 𝑛 ∉{11, 12, . . . , 19} ∪ 4 {21} with 𝛿(𝐺) ≥ 4 and 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) + 𝛾(𝐺) = 11 𝑛 + 2 if and only if   4  𝛾(𝐺), 𝛾(𝐺) = 11 𝑛 , 2 . The same graphs used to show Theorem 16.25 does not hold for 𝑛 ∈ {11, 12} are also counterexamples for Theorem 16.28. Volkmann [738] gave counterexamples to Theorem 16.28 for 𝑛 ∈ {13, 16}. The cases for 𝑛 ∈ {14, 15, 17, 18, 19, 21} remain open.

16.2.5

Minimum Degree Five

We repeat below a result from Chapter 6 due to Bujtás [123].

Section 16.2. Domination Number

479

Theorem 16.29 ([123]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤ 31 𝑛. As a consequence of Theorems 16.3(b) and 16.29, we can readily establish a Nordhaus-Gaddum bound involving the sum of the domination numbers of a graph and its complement where both have minimum degree at least 5. For simplicity, we restrict the order of the graph to be at least 21. Theorem 16.30 If 𝐺  isa graph of order 𝑛 ≥ 21 with 𝛿(𝐺) ≥ 5 and 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 13 𝑛 + 2. Proof Let 𝐺 be a graph of order 𝑛 ≥ 21 with 𝛿(𝐺) ≥ 5 and 𝛿(𝐺) ≥ 5. By Theorem 16.29, 𝛾(𝐺) ≤ 13 𝑛 and 𝛾(𝐺) ≤ 13 𝑛. Hence, if 𝛾(𝐺) = 2 or 𝛾(𝐺) = 2, then the desired result follows. Therefore, we assume that 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) ≥ 3, and so both 𝐺 and 𝐺 are connected. If 𝛾(𝐺) ≥ 6 and 𝛾(𝐺) ≥ 6, then Theorem implies  16.3(b)  that 𝛾(𝐺) ≤ 16 𝑛 and 𝛾(𝐺) ≤ 16 𝑛. Therefore, 𝛾(𝐺) + 𝛾(𝐺) ≤ 13 𝑛 < 13 𝑛 + 2. Hence, we may assume, without loss of generality, that 3 ≤ 𝛾(𝐺) ≤ 5 and 𝛾(𝐺) ≤ 𝛾(𝐺). If 𝛾(𝐺) = 5, then by Theorem 16.3(b), we have 𝑛 ≥ 25 and 𝛾(𝐺) ≤ 15 𝑛 , and     so 𝛾(𝐺) + 𝛾(𝐺) ≤ 5 + 15 𝑛 ≤ 13 𝑛 + 2. If 𝛾(𝐺) = 4, then by Theorem 16.3(b), 1  𝛾(𝐺) ≤ 4 𝑛 . If 𝛾(𝐺) = 3, then by Observation 16.21, we have diam(𝐺) = 2. This   in turn implies by Theorem 16.22 that 𝛾(𝐺) ≤ 14 𝑛 + 1. Hence, if 𝛾(𝐺) ∈ {3, 4}, 1  1  then 𝛾(𝐺) + 𝛾(𝐺) ≤ 4 + 4 𝑛 ≤ 3 𝑛 + 2, noting that 𝑛 ≥ 21.

16.2.6

Minimum Degree Six

Recall the result due to Bujtás and Henning [124] in 2021 that the domination number of a graph with minimum degree at least 6 is strictly less than 13 𝑛. We repeat this theorem here, which is Theorem 6.37 in Chapter 6. Theorem 16.31 ([124]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤ 127 418 𝑛. As a consequence of Theorems 16.3(b) and 16.31, we can readily establish a Nordhaus-Gaddum bound involving the sum of the domination numbers of a graph and its complement, both of which have minimum degree at least 6. For simplicity, we restrict the order of the graph to be at least 37. Theorem 16.32 If 𝐺  is a graph of order 𝑛 ≥ 37 with 𝛿(𝐺) ≥ 6 and 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) + 𝛾(𝐺) ≤ 127 418 𝑛 + 2. Proof Let 𝐺 be a graph of order 𝑛 ≥ 37 with 𝛿(𝐺) ≥ 6 and 𝛿(𝐺) ≥ 6. By 127 Theorem 16.31, 𝛾(𝐺) ≤ 127 418 𝑛 and 𝛾(𝐺) ≤ 418 𝑛. Hence, if 𝛾(𝐺) = 2 or 𝛾(𝐺) = 2, then the desired result follows. Therefore, we assume that 𝛾(𝐺) ≥ 3 and 𝛾(𝐺) ≥ 3 and so both 𝐺 and 𝐺 are connected. We may assume, without loss of generality, that 𝛾(𝐺)  ≤ 𝛾(𝐺). If 𝛾(𝐺)  ≥ 7, then Theorem 16.3(b) implies   that  𝑛 ≥ 49, 𝛾(𝐺) ≤ 17 𝑛 , and 𝛾(𝐺) ≤ 17 𝑛 . Therefore, 𝛾(𝐺) + 𝛾(𝐺) ≤ 2 17 𝑛 < 127 418 𝑛 + 2. Hence, we may assume that 3 ≤ 𝛾(𝐺) ≤ 6.

Chapter 16. Nordhaus-Gaddum Bounds

480

  If 𝛾(𝐺) = 6, then by Theorem 16.3(b), we have 𝛾( 𝐺) ≤ 16 𝑛 , implying that 1   127  𝛾(𝐺) + 𝛾(𝐺) ≤ 6 + 6 𝑛 ≤ 418 𝑛 + 2 since 𝑛 ≥ 37. If 𝛾(𝐺) = 5, then by Theo    rem 16.3(b), we have 𝛾(𝐺) ≤ 15 𝑛 , implying that 𝛾(𝐺) + 𝛾(𝐺) ≤ 5 + 15 𝑛 ≤ 1   127  418 𝑛 + 2 since 𝑛 ≥ 37. If 𝛾(𝐺) = 4, then by Theorem 16.3(b), 𝛾(𝐺) ≤ 4 𝑛 . If 𝛾(𝐺) = 3, then, by Observation   16.21, diam(𝐺) = 2. This in turn implies by Theorem 16.22 that 𝛾(𝐺) ≤ 14 𝑛 + 1. Hence, if 𝛾(𝐺) ∈ {3, 4}, then 𝛾(𝐺) + 𝛾(𝐺) ≤     4 + 14 𝑛 ≤ 127 418 𝑛 + 2 since 𝑛 ≥ 37.

16.2.7

Summary of Bounds with Specified Minimum Degree

 For a graph 𝐺, let 𝛿★ (𝐺) = min 𝛿(𝐺), 𝛿(𝐺) . We summarize selected NordhausGaddum bounds for the sum 𝛾(𝐺) + 𝛾(𝐺) of graphs 𝐺 and 𝐺 with specified 𝛿★ (𝐺) in Table 16.1.

𝛿★ (𝐺)

Order of 𝐺

𝛿★ (𝐺) ≥ 0 𝛿★ (𝐺) ≥ 1

𝑛≥2 𝑛≥2

𝛿★ (𝐺) ≥ 2

𝑛 ≥ 10

𝛿★ (𝐺)

≥3

𝑛 ≥ 14

𝛿★ (𝐺)

≥4

𝑛 ≥ 17

𝛿★ (𝐺)

≥5

𝑛 ≥ 21

𝛿★ (𝐺)

≥6

𝑛 ≥ 37

Upper bound on 𝛾(𝐺) + 𝛾(𝐺) 𝑛+1 1  𝑛 +2  22  𝑛 +2  53  𝑛 +2  84  𝑛 +2 11  1 3𝑛 + 2  127  418 𝑛 + 2

Reference [507] [507] [249] [249, 737] [738] Proven in this chapter Proven in this chapter

Table 16.1 Upper bounds on 𝛾(𝐺) + 𝛾(𝐺) when 𝛿★ (𝐺) ≥ 𝑘, for 𝑘 ∈ [6]

16.2.8

Multiple Factors

A factor of a graph is a spanning subgraph of the graph. A graph 𝐺 is factorable into the factors 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑡 if these factors are pairwise edge-disjoint and 𝑡 Ø

𝐸 (𝐺 𝑖 ) = 𝐸 (𝐺).

𝑖=1

If 𝐺 is factored into 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑡 , then we write 𝐺 = 𝐺 1 ⊕ 𝐺 2 ⊕ · · · ⊕ 𝐺 𝑡 , which is called a factorization of 𝐺. Thus, if 𝐺 and 𝐻 are graphs on the same vertex set but with disjoint edge sets, then 𝐺 ⊕ 𝐻 denotes the graph whose edge set is the union of their edge sets. The upper bounds of Theorem 16.1 can be restated as follows.

Section 16.2. Domination Number

481

Theorem 16.33 ([615]) For 𝑛 ≥ 2, if 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 , then the following hold: (a) 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) ≤ 𝑛 + 1. (b) 𝛾(𝐺 1 )𝛾(𝐺 2 ) ≤ 𝑛. In 1992 Goddard et al. [359] extended Nordhaus-Gaddum type results to the general case where the complete graph is factored into more than two edge-disjoint factors. In particular, they considered factoring 𝐾𝑛 into three factors 𝐺 1 , 𝐺 2 , and 𝐺 3 , and determined upper bounds on the sum and product of the domination numbers of the factors. We shall need the following lemma. Lemma 16.34 ([359]) If 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 , then 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 𝛾(𝐺 2 ⊕ 𝐺 3 ) + 𝑛. Proof Let 𝐺 = 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . Let 𝐷 be a 𝛾-set of the graph 𝐺 2 ⊕ 𝐺 3 , and so |𝐷| = 𝛾(𝐺 2 ⊕ 𝐺 3 ). For 𝑖 ∈ {2, 3}, let 𝐷 𝑖 be the set of vertices not dominated by the set 𝐷 in 𝐺 𝑖 . The set 𝐷∪𝐷 𝑖 is a dominating set of 𝐺 𝑖 , and so 𝛾(𝐺 𝑖 ) ≤ |𝐷|+|𝐷 𝑖 | for each 𝑖 ∈ {2, 3}. We note that 𝐷 2 ∪ 𝐷 3 ⊆ 𝑉 (𝐺) \ 𝐷. Since 𝐷 is a dominating set of 𝐺 2 ⊕ 𝐺 3 , the set of vertices in 𝐷 𝑖 is dominated by the set 𝐷 in the factor 𝐺 5−𝑖 for 𝑖 ∈ {2, 3}, implying that 𝐷 2 ∩ 𝐷 3 = ∅. Hence, |𝐷 2 | + |𝐷 3 | = |𝐷 2 ∪ 𝐷 3 | ≤ 𝑉 (𝐺) \ 𝐷 = 𝑛 − |𝐷| and so |𝐷 | + |𝐷 2 | + |𝐷 3 | ≤ 𝑛. Thus, 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ |𝐷 ∪ 𝐷 2 | + |𝐷 ∪ 𝐷 3 | = 2|𝐷| + |𝐷 2 | + |𝐷 3 | ≤ |𝐷 | + 𝑛 = 𝛾(𝐺 2 ⊕ 𝐺 3 ) + 𝑛. As a consequence of Lemma 16.34, we have the following result. Theorem 16.35 ([359]) If 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 , then 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 2𝑛 + 1. Proof Let 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . By Lemma 16.34, we have 𝛾(𝐺 2 ) +𝛾(𝐺 3 ) ≤ 𝛾(𝐺 2 ⊕ 𝐺 3 )+𝑛. Since 𝐺 2 ⊕𝐺 3 = 𝐺 1 , by Theorem 16.33(a), we have that 𝛾(𝐺 1 )+𝛾(𝐺 2 ⊕𝐺 3 ) ≤ 𝑛 + 1. Therefore, 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ⊕ 𝐺 3 ) + 𝑛 ≤ 2𝑛 + 1. We consider next an upper bound on the triple product of the domination numbers of the factors when 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . Since 𝐺 2 ⊕ 𝐺 3 = 𝐺 1 , by Theorem 16.33(b), we have that 𝛾(𝐺 1 )𝛾(𝐺 2 ⊕ 𝐺 3 ) ≤ 𝑛, or equivalently, 𝛾(𝐺 2 ⊕ 𝐺 3 ) ≤ 𝑛/𝛾(𝐺 1 ). Therefore, by Lemma 16.34, 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 𝑛 + 𝛾(𝐺 2 ⊕ 𝐺 3 ) ≤ 𝑛 +

𝑛 . 𝛾(𝐺 1 )

The maximum value of the triple product 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) is at most the value of maximize 𝑥1 𝑥2 𝑥3 subject to 1 ≤ 𝑥1 ≤ 𝑥2 ≤ 𝑥 3 , and 𝑥2 + 𝑥 3 ≤ 𝑛 + 𝑥𝑛1 , where 𝑥1 , 𝑥2 , 𝑥3 are integers. For small values of 𝑛, the integer optimization gives the maxima summarized in Table 16.2.

Chapter 16. Nordhaus-Gaddum Bounds

482 𝑛

1

2

3

4

5

6

7

8

maximum product

1

4

9

18

27

40

64

80

Table 16.2 Optimal values of the triple product for small 𝑛

The values in Table 16.2 are realized as follows. When 𝑛 = 1 and 𝑛 = 2, a realization is trivial. For 𝑛 ∈ {3, 4, 5, 6, 7}, a realization of the upper bound is given in Table 16.3. For 𝑛 = 8, we give a more general construction after the proof of Theorem 16.38. 𝑛

𝐺1

𝐺2

𝐺3

3 4 5 6

𝐾3 𝐶4 𝐶4 ∪ 𝐾1 𝐾 (2, 2, 2)
(𝐾1 ∪ 𝐾2 ) + 2𝐾1 ∪ 2𝐾1

𝐾3 𝐾2 ∪ 2𝐾1 𝐾3 ∪ 2𝐾1 𝐾2 ∪ 4𝐾1
(𝐾1 ∪ 𝐾2 ) + 2𝐾1 ∪ 2𝐾1

𝐾3 𝐾2 ∪ 2𝐾1 𝐾3 ∪ 2𝐾1 2𝐾2 ∪ 2𝐾1
(𝐾1 ∪ 𝐾2 ) + 2𝐾1 ∪ 2𝐾1

7

Table 16.3 Realization of the triple product for 𝑛 ≤ 7

To determine an upper bound on the triple product for larger values of 𝑛, we recall the probabilistic bound on the domination number given in Theorem 7.2 in Chapter 7, which we repeat here. Theorem 16.36 ([36, 632]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   1 + ln(𝛿 + 1) 𝛾(𝐺) ≤ 𝑛. 𝛿+1 We shall need the following lemma. Lemma 16.37 ([359]) Let 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . If 𝑣 𝑖 is a vertex of degree 𝑑𝑖 in 𝐺 𝑖 for 𝑖 ∈ [3], where the vertices 𝑣 1 , 𝑣 2 , 𝑣 3 are not necessarily distinct, then 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 𝑛 + 6 + 𝑑1 + 𝑑2 + 𝑑3 . Proof Let 𝐺 = 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . Let 𝐷 = {𝑣 1 , 𝑣 2 , 𝑣 3 }. Let 𝐴𝑖 be the set of vertices adjacent to all vertices of 𝐷 in the factor 𝐺 𝑖 for 𝑖 ∈ [3]. Let 𝐴 = 𝐴1 ∪ 𝐴2 ∪ 𝐴3 . We note that the sets ( 𝐴1 , 𝐴2 , 𝐴3 ) form a partition of the set 𝐴. Moreover, we note that | 𝐴𝑖 | ≤ 𝑑𝑖 for 𝑖 ∈ [3], and so | 𝐴| = | 𝐴1 | + | 𝐴2 | + | 𝐴3 | ≤ 𝑑1 + 𝑑2 + 𝑑3 . Let 𝐵 = 𝑉 (𝐺) \ ( 𝐴 ∪ 𝐷) and so 𝑛 = | 𝐴| + |𝐵| + |𝐷|. By definition of the set 𝐴, if 𝑣 ∈ 𝐵, then 𝑣 is not adjacent to every vertex of 𝐷 in the factor 𝐺 𝑖 for all 𝑖 ∈ [3]. Thus, in at least two of the three factors 𝐺 1 , 𝐺 2 , and 𝐺 3 , the set 𝐷 dominates the vertex 𝑣. Equivalently, the vertex 𝑣 is not dominated by 𝐷 in at most one of the three factors 𝐺 1 ,

Section 16.2. Domination Number

483

𝐺 2 , and 𝐺 3 . Let 𝐵𝑖 be the set of vertices in 𝐵 that are not dominated by the set 𝐷 in the factor 𝐺 𝑖 for 𝑖 ∈ [3]. By our earlier observations, the sets 𝐵1 , 𝐵2 , and 𝐵3 are vertex-disjoint and so |𝐵1 | + |𝐵2 | + |𝐵3 | ≤ |𝐵|. Let 𝐷 1 = 𝐷 ∪ 𝐴2 ∪ 𝐴3 ∪ 𝐵 1 , 𝐷 2 = 𝐷 ∪ 𝐴1 ∪ 𝐴3 ∪ 𝐵 2 , 𝐷 3 = 𝐷 ∪ 𝐴1 ∪ 𝐴2 ∪ 𝐵 3 . By construction, the set 𝐷 𝑖 is a dominating set in the factor 𝐺 𝑖 for all 𝑖 ∈ [3], implying that 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ |𝐷 1 | + |𝐷 2 | + |𝐷 3 |
= 3|𝐷 | + 2 | 𝐴1 | + | 𝐴2 | + | 𝐴3 | + |𝐵1 | + |𝐵2 | + |𝐵3 | ≤ 3|𝐷| + 2| 𝐴| + |𝐵| = 𝑛 + 2|𝐷| + | 𝐴| ≤ 𝑛 + 6 + 𝑑1 + 𝑑2 + 𝑑3 . As a consequence of Lemma 16.37 and Theorem 16.36, we have the following result. 1 3 Theorem 16.38 ([359]) If 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 , then 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) ≤ 27 𝑛 + 2 Θ(𝑛 ), that is, there exist constants 𝑐 1 and 𝑐 2 where 0 ≤ 𝑐 1 ≤ 𝑐 2 such that the 1 3 1 3 maximum triple product always lies between 27 𝑛 + 𝑐 1 𝑛2 and 27 𝑛 + 𝑐 2 𝑛2 . 1 3 Proof Let 𝐾𝑛 = 𝐺 1 ⊕ 𝐺 2 ⊕ 𝐺 3 . If 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) ≤ 27 𝑛 , then the desired 1 3 result is immediate. Hence, we may assume that 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) > 27 𝑛 . We 1 note that 𝛾(𝐺 𝑖 ) ≤ 𝑛 for 𝑖 ∈ [3]. Thus, if 𝛾(𝐺 𝑖 ) ≤ 27 𝑛 for some 𝑖 ∈ [3], then 1 3 1 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) ≤ 27 𝑛 , a contradiction. Therefore, 𝛾(𝐺 𝑖 ) > 27 𝑛 for all 𝑖 ∈ [3]. 1 By Theorem 16.36, if 𝛿(𝐺 𝑖 ) ≥ 164, then 𝛾(𝐺 𝑖 ) ≤ 27 𝑛 for 𝑖 ∈ [3], a contradiction. Hence, 𝛿(𝐺 𝑖 ) ≤ 163 for all 𝑖 ∈ [3]. Let 𝑑 = max{𝛿1 , 𝛿2 , 𝛿3 }, and so 𝑑 is a constant and 𝑑 ≤ 163. Let 𝑣 𝑖 be a vertex of minimum degree 𝛿(𝐺 𝑖 ) in the factor 𝐺 𝑖 for 𝑖 ∈ [3], and let 𝑑𝑖 = deg𝐺𝑖 (𝑣 𝑖 ) = 𝛿(𝐺 𝑖 ). By Lemma 16.37, 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) + 𝛾(𝐺 3 ) ≤ 𝑛+6+𝑑1 +𝑑2 +𝑑3 ≤ 𝑛+6+3𝑑 ≤ 𝑛+495. The maximum product of three positive integers 1 summing to 𝑛 + 495 is 27 (𝑛 + 495) 3 , which occurs when each of the three terms in the 1 1 3 product is equal to 3 (𝑛+495). This implies that 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) ≤ 27 𝑛 +Θ(𝑛2 ).

To illustrate a realization of the upper bound in Theorem 16.38 via a general construction, let 𝑛 ≥ 12 be a multiple of 3 and let {𝐴, 𝐵, 𝐶} denote a partition of the vertex set of 𝐺 = 𝐾𝑛 with | 𝐴| = |𝐵| = |𝐶 | = 13 𝑛 ≥ 4. We now construct factors 𝐺 1 , 𝐺 2 , and 𝐺 3 as follows. The sets 𝐴, 𝐵, and 𝐶 are the sets of vertices isolated in 𝐺 1 , 𝐺 2 , and 𝐺 3 , respectively. Further, 𝐺 1 has all the edges between 𝐵 and 𝐶, 𝐺 2 has all the edges between 𝐴 and 𝐶, and 𝐺 3 has all the edges between 𝐴 and 𝐵. Finally, each of 𝐺 1 [𝐵], 𝐺 1 [𝐶], 𝐺 2 [ 𝐴], 𝐺 2 [𝐶], 𝐺 3 [ 𝐴], and 𝐺 3 [𝐵] is isolate-free. Thus, 𝛾(𝐺 1 ) = | 𝐴|+2 = 13 (𝑛+6), 𝛾(𝐺 2 ) = |𝐵|+2 = 13 (𝑛+6), and 𝛾(𝐺 3 ) = |𝐶 |+2 = 13 (𝑛+6). Therefore, this construction yields, as a lower bound, the maximum product of three

Chapter 16. Nordhaus-Gaddum Bounds

484

positive integers summing to 𝑛 + 6. This shows that the maximum product is at least 1 3 2 2 27 𝑛 + 3 𝑛 . For smaller 𝑛 the construction is not so straightforward. For example, for 𝑛 = 8 the maximum triple product can be achieved by taking | 𝐴| = |𝐵| = 2 and |𝐶 | = 4 in the above construction, and letting 𝐺 3 [ 𝐴] = 𝐺 3 [𝐵] = 𝐾2 . Further, the edges in 𝐺 [𝐶] are distributed between 𝐺 1 and 𝐺 2 in such a way that both 𝐺 1 [𝐶] and 𝐺 2 [𝐶] are isolate-free. The resulting construction yields a realization of the maximum triple product of 𝛾(𝐺 1 )𝛾(𝐺 2 )𝛾(𝐺 3 ) = 80 listed in Table 16.2.

16.2.9

Relative Complement

Goddard et al. [359] also considered another variation on Nordhaus-Gaddum type results and extended the concept by considering 𝐺 1 ⊕ 𝐺 2 = 𝐾𝑠,𝑠 rather than 𝐺 1 ⊕ 𝐺 2 = 𝐾𝑛 . In general, if 𝐺 1 ⊕ 𝐺 2 = 𝐻, where 𝐻 ≠ 𝐾𝑛 , we say that 𝐺 1 and 𝐺 2 are complements of each other relative to 𝐻, or that 𝐺 1 and 𝐺 2 are relative complements with respect to 𝐻. In [359] the authors determined the graphs 𝐻 with respect to which complements are always unique in the following sense: if 𝐺 1 and 𝐺 2 are isomorphic subgraphs of 𝐻, then their complements 𝐻 − 𝐸 (𝐺 1 ) and 𝐻 − 𝐸 (𝐺 2 ) are isomorphic. Theorem 16.39 ([359]) If 𝐻 is an isolate-free graph with respect to which complements are always unique, then 𝐻 is one of the following: (a) 𝑟𝐾1,𝑠 , (b) 𝑟𝐾3 , (c) 𝐾𝑠 , (d) 𝐶5 , or (e) 𝐾𝑠,𝑠 , for some integers 𝑟 and/or 𝑠. Theorem 16.39 suggests that the complete bipartite graph 𝐾𝑠,𝑠 is an obvious alternate to 𝐾𝑛 in Nordhaus-Gaddum results. In this section, we consider bounds on the sums and products of the domination numbers of 𝐺 1 and 𝐺 2 , where 𝐺 1 ⊕ 𝐺 2 = 𝐾𝑠,𝑠 , and both 𝐺 1 and 𝐺 2 are isolate-free graphs. The following lower and upper bounds on the sum of the domination numbers of the two factors is determined in [359]. Theorem 16.40 ([359]) For 𝑠 ≥ 3, if 𝐺 1 ⊕ 𝐺 2 = 𝐾𝑠,𝑠 , where both 𝐺 1 and 𝐺 2 are isolate-free, then the following hold: (a) 5 ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) ≤ 2𝑠 + 2. 
2 (b) 6 ≤ 𝛾(𝐺 1 )𝛾(𝐺 2 ) ≤ 12 𝑠 + 2 . The bounds in Theorem 16.40 are tight. For the lower bounds, equality is attained, for example, if 𝐺 1 = 2𝐾1,𝑠−1 and 𝐺 2 = 𝐾2 ∪ 𝐾𝑠−1,𝑠−1 . In this case, 𝛾(𝐺 1 ) = 2 and 𝛾(𝐺 2 ) = 3, yielding equality in the lower bounds on the sum and product. Equality in the upper bound on the sum is attained, for example, by taking 𝐺 1 as the empty graph and 𝐺 2 = 𝐾𝑠,𝑠 . In this case, 𝛾(𝐺 1 ) = 2𝑠 and 𝛾(𝐺 2 ) = 2.

Section 16.3. Total Domination Number

485

In the special case of the upper bound on the product when 𝑠 = 3, we take 𝐺 1 as the empty graph and 𝐺 2 = 𝐾𝑠,𝑠 , yielding 𝛾(𝐺 1 ) = 6 and 𝛾(𝐺 2 ) = 2. Thus,    1
2 𝛾(𝐺 1 )𝛾(𝐺 2 ) = 12 = 49 . 4 = 2𝑠 + 2 For 𝑠 ≥ 4, there are at least two constructions that realize the upper bound on the product. Let  𝐾𝑠,𝑠 have partite  sets  𝑋 and 𝑌 . Partition 𝑋 into two sets 𝑋1 and 𝑋2 , where |𝑋1 | = 12 𝑠 and |𝑋2 | = 12 𝑠 . Similarly, partition 𝑌 into sets 𝑌1 and 𝑌2 , where     |𝑌1 | = 12 𝑠 and |𝑌2 | = 12 𝑠 . For the first construction, let 𝐺 1 have all edges between 𝑋 and 𝑌1 and let 𝐺 2 have all edges between 𝑋 and 𝑌2 . Since the vertices in 𝑌2 are isolated in 𝐺 1 , every dominating set of 𝐺 1 contains the set 𝑌2 . Since |𝑌1 | ≥ 2, two vertices are needed  to dominate the remaining vertices in 𝑋 ∪ 𝑌1 , implying that 𝛾(𝐺 1 ) = |𝑌2 | + 2 = 12 𝑠 + 2. Similarly,    
 

2 𝛾(𝐺 2 ) = |𝑌1 | +2 = 12 𝑠 +2. Thus, 𝛾(𝐺 1 )𝛾(𝐺 2 ) = 12 𝑠 +2 12 𝑠 +2 = 12 𝑠+2 . For the second construction, let 𝐺 1 contain all edges between 𝑋 and 𝑌1 , but with the edges of a perfect matching between 𝑋1 and 𝑌1 removed, and with the edges of a perfect matching between 𝑋2 and 𝑌2 added. Thus, each vertex in 𝑌2 has degree 1 in 𝐺 1 and therefore in a 𝛾-set of 𝐺 1 , we insert the (unique) neighbor of each vertex of 𝑌2 in 𝐺 1 into the dominating set. Thus, the 𝛾-set contains the set 𝑋2 . The only vertices not dominated by 𝑋2 are vertices in the set 𝑋1 . However,   two vertices in 𝑌1 are needed to dominate 𝑋1 , implying that 𝛾(𝐺 1 ) = |𝑌2 | + 2 = 12 𝑠 + 2. By symmetry,   𝛾(𝐺 2 ) = |𝑌1 | + 2 = 12 𝑠 + 2. This yields the same domination numbers as in the first 
2 construction, and again, 𝛾(𝐺 1 )𝛾(𝐺 2 ) = 12 𝑠 + 2 .

16.3

Total Domination Number

The first Nordhaus-Gaddum type result for the total domination numbers appeared in the introductory paper on total domination by Cockayne et al. [182] in 1980. Before we present their result, we restate the upper bounds on 𝛾t (𝐺) in terms of the maximum degree presented in Chapter 4. Theorem 16.41 ([182]) If 𝐺 is an isolate-free graph of order 𝑛, then the following hold: (a) If 𝐺 is connected and does not contain a dominating vertex, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺). (b) If 𝐺 is disconnected or if 𝐺 is connected but contains a dominating vertex, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1. If 𝐺 and 𝐺 are isolate-free graphs of order 𝑛, then we note that 2 ≤ 𝛾t (𝐺) and 2 ≤ 𝛾t (𝐺), yielding the trivial lower bound 4 ≤ 𝛾t (𝐺) + 𝛾t (𝐺). As a consequence of Theorem 16.41, Cockayne et al. [182] established the following NordhausGaddum upper bound on the sum of the total domination numbers of a graph and its complement. Theorem 16.42 ([182]) If 𝐺 and its complement 𝐺 are isolate-free graphs of order 𝑛, then 𝛾t (𝐺) + 𝛾t (𝐺) ≤ 𝑛 + 2, with equality if and only if 𝐺 or 𝐺 consists of disjoint copies of 𝐾2 .

Chapter 16. Nordhaus-Gaddum Bounds

486

Proof Let 𝐺 and 𝐺 be isolate-free graphs of order 𝑛. Thus, neither 𝐺 nor 𝐺 has a dominating vertex. Since 𝐺 or 𝐺 is connected, we may assume without loss of generality that 𝐺 is connected. By Theorem 16.41, 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) and 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1. If 𝑣 is a vertex of maximum degree in 𝐺, then 𝑣 is a vertex of minimum degree in 𝐺. Thus, choosing 𝑣 to have maximum degree Δ(𝐺) in 𝐺, we have Δ(𝐺) + Δ(𝐺) ≥ Δ(𝐺) + 𝛿(𝐺) = deg𝐺 (𝑣) + deg𝐺 (𝑣) = 𝑛 − 1. Therefore,

𝛾t (𝐺) + 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1 + 𝑛 − Δ(𝐺)
= 2𝑛 − Δ(𝐺) + Δ(𝐺) + 1 ≤ 2𝑛 − (𝑛 − 1) + 1 = 𝑛 + 2, establishing the desired upper bound on the sum. Suppose next that 𝛾t (𝐺) + 𝛾t (𝐺) = 𝑛 + 2. Thus, we must have equality throughout the previous inequality chain. In particular, this implies that Δ(𝐺) = 𝛿(𝐺) and therefore 𝐺 and 𝐺 are regular graphs. Thus, Δ(𝐺) = 𝛿(𝐺). Further, 𝛾t (𝐺) = 𝑛−Δ(𝐺) and 𝛾t (𝐺) = 𝑛 − Δ(𝐺) + 1, implying that 𝐺 is a disconnected graph. Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 be the components of 𝐺, where 𝑘 ≥ 2. By linearity and applying Theorem 16.41 to each component of 𝐺, 𝑛−Δ(𝐺)+1 = 𝛾t (𝐺) =

𝑘 ∑︁ 𝑖=1

𝛾t (𝐺 𝑖 ) ≤

𝑘 ∑︁


𝑛(𝐺 𝑖 )−Δ(𝐺 𝑖 )+1 = 𝑛−𝑘Δ(𝐺)+𝑘, (16.2)

𝑖=1

or equivalently, Δ(𝐺) (𝑘 − 1) ≤ 𝑘 − 1. Since 𝐺 is isolate-free, Δ(𝐺) ≥ 1 and so Δ(𝐺) (𝑘 − 1) ≥ 𝑘 − 1. Consequently, we must have equality throughout Inequality (16.2). This implies that 𝛾t (𝐺 𝑖 ) = 𝑛(𝐺 𝑖 ) − Δ(𝐺 𝑖 ) + 1 for all 𝑖 ∈ [𝑘]. By Theorem 16.41, the component 𝐺 𝑖 therefore has a dominating vertex, that is, Δ(𝐺 𝑖 ) = 𝑛(𝐺 𝑖 ) − 1. By our earlier observation, the graph 𝐺 is regular, implying that 𝐺 𝑖 = 𝐾Δ(𝐺)+1 for all 𝑖 ∈ [𝑘], that is, 𝐺 = 𝑘𝐾Δ(𝐺)+1 . This in turn implies that 𝛾t (𝐺) = 2𝑘 and 𝛾(𝐺) = 2, and therefore 𝛾t (𝐺) + 𝛾(𝐺) = 2𝑘 + 2. By supposition, 𝛾t (𝐺) + 𝛾t (𝐺) = 𝑛 +2. Consequently, 2𝑘 +2 = 𝑛 +2, and so the number of components in 𝐺 is given by 𝑘 = 12 𝑛. Therefore, Δ(𝐺) = 1 and 𝐺 = 𝑘𝐾2 , that is, 𝐺 consists of disjoint copies of 𝐾2 . In order to establish a Nordhaus-Gaddum upper bound on the product of the total domination numbers of a graph we present some preliminary  and its complement, lemmas. Let 𝛿★ (𝐺) = min 𝛿(𝐺), 𝛿(𝐺) . The proof of the first lemma is due to Karami et al. [516] in 2012. We remark that they proved a stronger result and showed that the lemma holds for the connected domination number. Lemma 16.43 ([516]) If 𝐺 and its complement 𝐺 are connected graphs, then

𝛾t (𝐺) + 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 4 − 𝛾t (𝐺) − 3 𝛾t (𝐺) − 3 . (16.3) Proof By symmetry, we may assume that 𝛿★ (𝐺) = 𝛿(𝐺). If 𝛾t (𝐺) = 2 or 𝛾t (𝐺) = 2, then Inequality (16.3) reduces to 𝛿★ (𝐺) ≥ 1, which holds since both 𝐺 and 𝐺 are

Section 16.3. Total Domination Number

487

connected of order at least 4. Hence, we may assume that 𝛾t (𝐺) ≥ 3 and 𝛾t ( 𝐺) ≥ 3. Since any two vertices at distance 3 or more apart in 𝐺 form a total dominating set in 𝐺, it follows that diam(𝐺) = diam(𝐺) = 2. Let 𝑥 be a vertex of minimum degree 𝛿(𝐺) in 𝐺. Let 𝑋 = N𝐺 (𝑥) and let 𝑌 = 𝑉 (𝐺) \ N𝐺 [𝑥]. Thus, 𝑉 (𝐺) = 𝑋 ∪ 𝑌 ∪ {𝑥}. Since 𝛾t (𝐺) ≥ 3, we have 𝑌 ≠ ∅. Also, since diam(𝐺) = 2, the set 𝑋 dominates the set 𝑌 in 𝐺. We successively select disjoint sets 𝐷 0 , 𝐷 1 , . . . , 𝐷 𝑘 in 𝑋 as follows. Let 𝑋0 = 𝑋 and let 𝐷 0 be a largest subset of 𝑋0 that does not dominate 𝑌 . Let 𝑋1 = 𝑋0 \ 𝐷 0 . By the maximality of 𝐷 0 , every vertex 𝑣 of 𝑋1 dominates 𝑌 \ N𝐺 (𝐷 0 ). However, the set 𝑋1 may or may not dominate the set 𝑌 . If 𝑋1 dominates 𝑌 , then let 𝐷 1 be a largest subset of 𝑋1 that does not dominate 𝑌 . Let 𝑋2 = 𝑋1 \ 𝐷 1 and so 𝑋 = 𝑋0 ⊃ 𝑋1 ⊃ 𝑋2 . We continue this process, constructing sets 𝑋0 , 𝑋1 , . . . , 𝑋 𝑘 with 𝑋0 ⊃ 𝑋1 ⊃ · · · ⊃ 𝑋 𝑘 (where 𝑘 ≥ 1) and sets 𝐷 0 , 𝐷 1 , . . . , 𝐷 𝑘−1 such that (a) for 𝑖 < 𝑘, the set 𝑋𝑖 dominates the set 𝑌 , (b) for 𝑖 < 𝑘, the set 𝐷 𝑖 is a largest subset of 𝑋𝑖 that does not dominate 𝑌 , and 𝑋𝑖+1 = 𝑋𝑖 \ 𝐷 𝑖 , and (c) the set 𝑋 𝑘 does not dominate the set 𝑌 . By (a) and (b), for 𝑖 < 𝑘, the set 𝑋𝑖 dominates the set 𝑌 but the set 𝐷 𝑖 does not dominate 𝑌 . Hence, the sets 𝑋0 , 𝑋1 , . . . , 𝑋 𝑘 are nonempty. By the maximality of 𝐷 𝑖 , the set 𝐷 𝑖 ∪ {𝑣} dominates 𝑌 for every vertex 𝑣 ∈ 𝑋1 . Thus, 𝐷 𝑖 ∪ {𝑥, 𝑣} is a TD-set of 𝐺 and so 𝛾t (𝐺) ≤ |𝐷 𝑖 | + 2 (16.4) for all 𝑖 ∈ [𝑘 − 1] 0 . For 𝑖 ∈ [𝑘 − 1] 0 , let 𝑦 𝑖 be a vertex of 𝑌 that is not dominated by 𝐷 𝑖 and let 𝑦 𝑘 be a vertex of 𝑌 that is not dominated by 𝑋 𝑘 . By construction, the sets 𝐷 0 , 𝐷 1 , . . . , 𝐷 𝑘−1 , 𝑋 𝑘 form a partition of 𝑋. Since the vertex 𝑦 𝑖 of 𝑌 dominates 𝐷 𝑖 in the complement 𝐺 for 𝑖 ∈ [𝑘 − 1] 0 , and the vertex 𝑦 𝑘 of 𝑌 dominates 𝑋 𝑘 in the complement 𝐺, the set {𝑦 0 , 𝑦 1 , . . . , 𝑦 𝑘 } dominates 𝑋 ∪ {𝑥} in 𝐺. Since the vertex 𝑥 dominates 𝑌 in 𝐺, this yields a TD-set {𝑥, 𝑦 0 , 𝑦 1 , . . . , 𝑦 𝑘 } in 𝐺 and so 𝛾t (𝐺) ≤ 𝑘 + 2.

(16.5)

We note that |𝑋 | = 𝛿(𝐺) and |𝐷 0 | = |𝑋 | − |𝑋 𝑘 | −

𝑘−1 ∑︁

|𝐷 𝑖 |.

(16.6)

𝑖=1

From Inequalities (16.4) and (16.5), and Equation (16.6) and noting that |𝑋 𝑘 | ≥ 1, we have
𝛾t (𝐺) + 𝛾t (𝐺) ≤ |𝐷 0 | + 2 + (𝑘 + 2) ! 𝑘−1 ∑︁ = 𝛿(𝐺) + 2 − |𝑋 𝑘 | − |𝐷 𝑖 | + (𝑘 + 2) 𝑖=1


≤ 𝛿(𝐺) + 4 − |𝑋 𝑘 | − (𝑘 − 1) 𝛾t (𝐺) − 2 + 𝑘
≤ 𝛿(𝐺) + 4 − (𝑘 − 1) 𝛾t (𝐺) − 3

≤ 𝛿(𝐺) + 4 − 𝛾t (𝐺) − 3 𝛾t (𝐺) − 3 .

488

Chapter 16. Nordhaus-Gaddum Bounds



By symmetry, 𝛾t (𝐺) +𝛾t ( 𝐺) ≤ 𝛿(𝐺) +4− 𝛾t (𝐺) −3 𝛾t (𝐺) −3 . This completes the proof of the lemma. As an immediate consequence of Lemma 16.43, we have the following NordhausGaddum upper bound on the sum of the total domination numbers of a graph and its complement. Corollary 16.44 ([516]) If 𝐺 and its complement 𝐺 are connected graphs with 𝛾t (𝐺) ≥ 3 and 𝛾t (𝐺) ≥ 3, then 𝛾t (𝐺) + 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 4. The proof of the second lemma is due to Henning et al. [464] in 2011. Lemma 16.45 ([464]) If 𝐺 is a graph with diam(𝐺) = diam(𝐺) = 2, then 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1 and 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1. Proof By symmetry, we may assume that 𝛿★ (𝐺) = 𝛿(𝐺). The diameter constraint on 𝐺 and 𝐺 implies that 𝛿★ (𝐺) ≥ 1. Let 𝑣 be a vertex of minimum degree 𝛿★ (𝐺) in 𝐺. Let 𝑋 = N𝐺 (𝑣) and let 𝑌 = 𝑉 (𝐺) \ N𝐺 [𝑣]. Thus, 𝑉 (𝐺) = 𝑋 ∪ 𝑌 ∪ {𝑣}. Since diam(𝐺) = 2, the set 𝑋 ∪ {𝑣} is a TD-set in 𝐺 and so 𝛾t (𝐺) ≤ |𝑋 | + 1 = 𝛿★ (𝐺) + 1. Consider the set 𝑋 in 𝐺. Since |𝑋 | = 𝛿★ ≤ 𝛿(𝐺) and no vertex in 𝑋 is adjacent to 𝑣 in 𝐺, it follows that every vertex in 𝑋 has at least one neighbor in 𝑌 in 𝐺. For each vertex 𝑥 ∈Ð𝑋, let 𝑥 ′ be an arbitrary neighbor of 𝑥 in 𝐺 such that 𝑥 ′ belongs to 𝑌 . Let 𝑋 ′ = 𝑥 ∈𝑋 {𝑥 ′ }. By our choice of the set 𝑋 ′ , the set 𝑋 ′ ⊆ 𝑌 and 𝑋 ′ dominates 𝑋 ∪ {𝑣} in 𝐺. Further, vertex 𝑣 dominates the set 𝑌 in 𝐺. Therefore, 𝑋 ′ ∪ {𝑣} is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ |𝑋 ′ | + 1 ≤ |𝑋 | + 1 = 𝛿★ (𝐺) + 1. A corollary of the next lemma will be used to prove Theorem 16.48. Lemma 16.46 If 𝐺 is a graph of order 𝑛 with 𝛾t (𝐺) ≥ 3 and 𝛾t (𝐺) ≥ 3, then
𝛾t (𝐺)𝛾t (𝐺) ≤ 3 𝛿★ (𝐺) + 1 . Proof Let 𝐺 be a graph of order 𝑛 with 𝛾t (𝐺) ≥ 3 and 𝛾t (𝐺) ≥ 3. This requires that both 𝐺 and 𝐺 are connected and diam(𝐺) = diam(𝐺) = 2, since if 𝐺 is disconnected or if 𝐺 is connected and diam(𝐺) ≥ 3, then 𝛾t (𝐺) = 2, a contradiction. We may assume that 𝛾t (𝐺) ≥ 𝛾t (𝐺) ≥ 3. We note that 𝑛 ≥ 5. By Lemma 16.45, we have
2 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1 and 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1. Thus, 𝛾t (𝐺)𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1 . Since diam(𝐺) = diam(𝐺) = 2, we note that 𝛿★ (𝐺) ≥ 1. If 𝛿★ (𝐺) = 1, then
2 ★ 𝛾t (𝐺)𝛾t (𝐺)
≤ 2 < 6 = 3 𝛿 (𝐺) + 1 . If 𝛿★ (𝐺) = 2, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 32 = 3 𝛿★ (𝐺) + 1 . Hence, we may assume that 𝛿★ (𝐺) ≥ 3, for otherwise the desired result holds. By supposition, 𝛾t (𝐺) ≥ 𝛾t (𝐺) ≥ 3. If 𝛾t (𝐺) = 3, then
by Lemma 16.45, we have 𝛾t (𝐺) ≤ 𝛿★ (𝐺) + 1, and so 𝛾t (𝐺)𝛾t (𝐺) ≤ 3 𝛿★ (𝐺) + 1 , as desired. Hence, we may assume that 𝛾t (𝐺) ≥ 𝛾t (𝐺) ≥ 4. In this case,
𝛾t (𝐺)𝛾t (𝐺) ≤ 𝛿★ (𝐺) − 5 + 2 𝛾t (𝐺) + 𝛾t (𝐺)
≤ 𝛿★ (𝐺) − 5 + 2 𝛿★ (𝐺) + 4 = 3𝛿★ (𝐺) + 3,

Section 16.4. Independent Domination Number

489

where the first inequality follows from Lemma 16.43 and the second from Corollary 16.44. Once again this yields the desired bound. If 𝐺 is a graph of order 𝑛, then 𝛿★ (𝐺) ≤ 12 (𝑛 − 1). Hence, as an immediate consequence of Lemma 16.46, we have the following result. Corollary 16.47 If 𝐺 is a graph of order 𝑛 with 𝛾t (𝐺) ≥ 3 and 𝛾t (𝐺) ≥ 3, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 32 (𝑛 + 1). We are now in a position to present a proof of the following Nordhaus-Gaddum upper bound on the product of the total domination numbers of a graph and its complement due to Henning et al. [464]. Theorem 16.48 ([464]) If 𝐺 and its complement 𝐺 are isolate-free graphs of order 𝑛, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 2𝑛, with equality if and only if 𝐺 or 𝐺 consists of disjoint copies of 𝐾2 . Proof Let 𝐺 and 𝐺 be isolate-free graphs of order 𝑛. Thus, 𝑛 ≥ 4. By symmetry, we may assume that 𝛾t (𝐺) ≥ 𝛾t (𝐺) ≥ 2. If 𝛾t (𝐺) = 2, then the upper bound is immediate since 𝛾t (𝐺) ≤ 𝑛, with equality if and only if 𝐺 = 𝑛2 𝐾2 . Hence, we may assume that 𝛾t (𝐺) ≥ 𝛾t (𝐺) ≥ 3. Thus, 𝑛 ≥ 5. By Corollary 16.47, we therefore have 𝛾t (𝐺)𝛾t (𝐺) ≤ 32 (𝑛 + 1) < 2𝑛. The upper bound on the sum in Theorem 16.42 and the upper bound on the product in Theorem 16.48 can be improved if we impose a lower bound on the minimum degree of 𝐺 and 𝐺. For example, the following selected results are shown in [464]: (a) If 𝛿★ (𝐺) = 2 and 𝑛 ≥ 7, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 43 𝑛. (b) If 𝛿★ (𝐺) = 3 and 𝑛 ≥ 12, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 𝑛. (c) If 𝛿★ (𝐺) = 4 and 𝑛 ≥ 18, then 𝛾t (𝐺)𝛾t (𝐺) ≤ 67 𝑛. A more detailed discussion of such improved bounds can be found in [490].

16.4

Independent Domination Number

In this section, we present Nordhaus-Gaddum bounds involving the independent domination numbers of a graph and its complement. Theorem 16.49 If 𝐺 is a graph of order 𝑛 ≥ 2, then 3 ≤ 𝑖(𝐺) + 𝑖(𝐺) ≤ 𝑛 + 1, and these bounds are tight. Proof The proof of the lower bound follows immediately from the observation that if 𝑖(𝐺) = 1 or 𝑖(𝐺) = 1, then 𝑖(𝐺) ≥ 2 or 𝑖(𝐺) ≥ 2, respectively. Thus, 𝑖(𝐺) + 𝑖(𝐺) ≥ 3. That this lower bound is tight may be seen by considering the graph 𝐺 = 𝐾1,𝑛−1 with 𝑖(𝐺) = 1 and 𝑖(𝐺) = 2. Next, we verify the upper bound on the sum. By Theorem 4.18, we have 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺) and 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺). As observed in the proof of Theorem 16.42, Δ(𝐺) + Δ(𝐺) ≥ 𝑛 − 1. Therefore,

490

Chapter 16. Nordhaus-Gaddum Bounds

𝑖(𝐺) + 𝑖( 𝐺) ≤ 𝑛 − Δ(𝐺) + 𝑛 − Δ(𝐺)
= 2𝑛 − Δ(𝐺) + Δ(𝐺) ≤ 2𝑛 − (𝑛 − 1) = 𝑛 + 1.

That this upper bound on the sum is tight may be seen by taking 𝐺 = 𝐾𝑛 or 𝐺 = 𝐾𝑛 . As we have seen in the proof of Theorem 16.49, establishing a Nordhaus-Gaddum upper bound on the sum of the independent domination numbers of a graph and its complement is straightforward. However, unlike the relatively simple proof of Theorem 16.10 which establishes a Nordhaus-Gaddum upper bound on the sum of the domination numbers of a graph and its complement, where both graphs are isolate-free, the situation is considerably more complex for the independent domination number. Moreover, determining a Nordhaus-Gaddum upper bound on the product of the independent domination numbers of a graph and its complement is also much more challenging than for the domination numbers. In 2003 Goddard and Henning [350] determined the maximum possible value for the sum 𝑖(𝐺) + 𝑖(𝐺) when both 𝐺 and 𝐺 are isolate-free. They also determined the maximum possible value for the product 𝑖(𝐺)𝑖(𝐺) (where isolated vertices are permitted). In order to present their results and proofs, we introduce some additional notation. For an integer 𝑝 ≥ 2, recall that a 𝑝-clique is a complete graph of order 𝑝. Definition 16.50 For integers 𝑝, 𝑞 ≥ 2, let 𝑓 ( 𝑝, 𝑞) equal the minimum order of a graph in which every vertex is in a 𝑝-clique and in an independent set of cardinality 𝑞. Definition 16.51 For 𝑝 ≥ 2, a graph is 𝑝-clique-minimal if every vertex is in a 𝑝-clique, but upon the removal of any edge 𝑒 some vertex 𝑣 no longer belongs to a 𝑝-clique. We say that such an edge 𝑒 is critical to the vertex 𝑣. We proceed further with the following preliminary lemma given by Entringer et al. [257] in 1997. Lemma 16.52 ([257]) For 𝑝 ≥ 2, if 𝐺 is a 𝑝-clique-minimal graph and 𝑒 is an arbitrary edge of 𝐺, then there is a vertex 𝑣 𝑒 that is in only one 𝑝-clique 𝐾 𝑝 , and 𝐾 𝑝 contains 𝑒. Proof Let 𝑒 = 𝑣𝑤 be an arbitrary edge of 𝐺. Since 𝐺 is a 𝑝-clique-minimal graph, the edge 𝑒 is critical to some vertex. Let 𝐹 be a 𝑝-clique containing 𝑒. If 𝐹 is the only clique to which 𝑒 belongs, then let 𝑣 𝑒 = 𝑣. Hence, we may assume that the edge 𝑒 belongs to at least two distinct 𝑝-cliques. We show that some vertex 𝑣 1 not incident with 𝑒 is such that 𝑒 is critical to 𝑣 1 . If 𝑒 is not critical to one of the vertices incident to it, then we select for 𝑣 1 any vertex such that 𝑒 is critical to that vertex. Hence, we may assume that 𝑒 is critical to one its vertices. Renaming 𝑣 and 𝑤 if necessary, we may assume that 𝑒 is critical to 𝑣. Let 𝑒 1 = 𝑣𝑢 1 be an edge not in 𝐹. The edge 𝑒 1 is critical to some vertex 𝑢, where possibly 𝑢 = 𝑢 1 . We note that 𝑢 ∉ 𝑉 (𝐹). In particular, 𝑢 ∉ {𝑣, 𝑤}. Every 𝑝-clique containing 𝑒 1 contains 𝑣 and hence contains 𝑒 since 𝑒 is critical to 𝑣. Thus, 𝑒 is

Section 16.4. Independent Domination Number

491

critical to 𝑢. Thus, in this case, we can take as 𝑣 1 the vertex 𝑢. In both cases, there exists a vertex 𝑣 1 not incident with 𝑒 such that 𝑒 is critical to 𝑣 1 . Let 𝐹1 be a 𝑝-clique containing 𝑣 1 . Necessarily, 𝐹1 contains 𝑒 and hence 𝑣 and 𝑤. If 𝐹1 is unique, then we take 𝑣 𝑒 = 𝑣 1 and we are finished. Otherwise, there is an edge 𝑒 2 = 𝑣 1 𝑢 2 such that 𝑢 2 ∉ 𝑉 (𝐹1 ). The edge 𝑒 2 is critical to some vertex 𝑣 2 , where possibly 𝑣 2 = 𝑢 2 . We note that 𝑣 2 ∉ {𝑣, 𝑣 1 , 𝑤}. Every 𝑝-clique that contains 𝑣 1 𝑢 2 contains 𝑣 1 and therefore also contains 𝑒. Hence, 𝑒 is critical to 𝑣 2 . Indeed, every 𝑝-clique containing 𝑣 2 contains 𝑣 1 , 𝑣, and 𝑤 as well. Continuing this process, we produce a sequence of distinct vertices 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑖 and a sequence of 𝑝-cliques 𝐹𝑖 such that every 𝑝-clique containing 𝑣 𝑖 contains 𝑣 𝑖−1 , 𝑣 𝑖−2 , . . . , 𝑣 1 , 𝑣, and 𝑤. Since 𝑝 is finite, this process must terminate. Hence, for some 𝑖 ≤ 𝑝 − 2, the 𝑝-clique must be unique. We now take 𝑣 𝑒 = 𝑣 𝑖 , yielding the desired result. We are now in a position to prove the following result due to Entringer et al. [257]. Theorem 16.53 ([257]) For 𝑝, 𝑞 ≥ 2, 𝑓 ( 𝑝, 𝑞) =

l√︁

m  2m l √︁ √︁ 𝑝−1+ 𝑞−1 = 𝑝 + 𝑞 − 2 + 2 ( 𝑝 − 1)(𝑞 − 1) .

Proof Let 𝐺 be a graph in which every vertex is in a 𝑝-clique and in an independent set of cardinality 𝑞. We first prove the following lower bound on 𝑓 ( 𝑝, 𝑞): l √︁ m 𝑓 ( 𝑝, 𝑞) ≥ 𝑝 + 𝑞 − 2 + 2 ( 𝑝 − 1)(𝑞 − 1) (16.7) Let 𝐺 = (𝑉, 𝐸) be a 𝑝-clique-minimal graph and let 𝑆 𝑝−1 be the set of vertices of 𝐺 of smallest possible degree 𝑝 − 1. Thus, 𝑆 𝑝−1 is the set of vertices of 𝐺 that belong to a unique 𝑝-clique. By Lemma 16.52, we note that 𝑆 𝑝−1 ≠ ∅. Let 𝑆 be a maximum independent subset of 𝑆 𝑝−1 , let 𝑋 = 𝑉 \ 𝑆, let 𝐺 have order 𝑛, and let
|𝑋 | = 𝑥. We note that 𝑥 ≥ 𝑝 − 1. For every vertex 𝑣 of 𝐺, let 𝐼 𝑣 = {𝑣} ∪ 𝑆 \ N𝐺 (𝑣) . We proceed further with the following two claims. Claim 16.53.1 If 𝑣 is a vertex of 𝐺, then 𝐼 𝑣 is an independent set of maximum cardinality that contains 𝑣. Proof By construction, the set 𝐼 𝑣 is independent and contains 𝑣. Hence, it suffices to prove that 𝐼 𝑣 has maximum cardinality among all such sets. Among all independent sets containing 𝑣 of maximum cardinality, let 𝐼 𝑣′ be chosen to contain as many vertices in 𝐼 𝑣 as possible. We show that 𝐼 𝑣′ = 𝐼 𝑣 . Suppose 𝐼 𝑣′ contains a vertex 𝑤 of 𝑋 \ {𝑣}. By Lemma 16.52, there is a vertex 𝑢 ∈ 𝑆 𝑝−1 such that 𝑤 is in the (unique) 𝑝-clique containing 𝑢, where possibly 𝑢 = 𝑤. If 𝑢 ∉ 𝑆, then by the maximality of 𝑆, the vertex 𝑢 has a neighbor 𝑢 ′ in 𝑆, implying that 𝑤 is in the 𝑝-clique
containing 𝑢 ′ . Hence, we can choose the vertex 𝑢 so that 𝑢 ∈ 𝑆. The set 𝐼 𝑣′ \ {𝑤} ∪ {𝑢} is an independent set containing 𝑣 that has the same cardinality as 𝐼 𝑣′ , contradicting our choice of the set 𝐼 𝑣′ . Hence, 𝐼 𝑣′ contains no vertex of 𝑋 \ {𝑣}. By maximality of the set 𝑆, it follows that 𝐼 𝑣′ = 𝐼 𝑣 .

Chapter 16. Nordhaus-Gaddum Bounds

492


Claim 16.53.2 𝑥 + 𝑥 − ( 𝑝 − 1) (𝑛 − 𝑥) ≥ 𝑞𝑥. Proof For a vertex 𝑣 of 𝐺, let 𝛼(𝐺, 𝑣) denote the maximum cardinality of an independent set of vertices of 𝐺 that contains 𝑣. Every vertex of 𝐺 is in an independent set of cardinality 𝑞. By Claim 16.53.1, 𝛼(𝐺, 𝑣) = |𝐼 𝑣 | ≥ 𝑞 for all vertices 𝑣 in 𝐺. Hence, ∑︁ ∑︁ 𝑞 = 𝑞𝑥. 𝛼(𝐺, 𝑣) ≥ (16.8) 𝑣 ∈𝑋

𝑣 ∈𝑋

By Claim 16.53.1, ∑︁

𝛼(𝐺, 𝑣) =

𝑣 ∈𝑋

∑︁ 𝑣 ∈𝑋

|𝐼 𝑣 | =

∑︁ 𝑣 ∈𝑋

|{𝑣}| +

∑︁

|𝑆 \ N𝐺 (𝑣)|.

(16.9)

𝑣 ∈𝑋

Every vertex of 𝑆 is adjacent to exactly 𝑝 − 1 of the vertices in 𝑋. Thus, for the set [𝑆, 𝑋] of edges between 𝑆 and 𝑋, we have | [𝑆, 𝑋] | = |𝑆|( 𝑝 − 1). For a vertex 𝑣 ∈ 𝑋, let deg𝑆 (𝑣) denote the number of vertices in 𝑆 adjacent to 𝑣, and so ∑︁ deg𝑆 (𝑣) = | [𝑆, 𝑋] |. 𝑣 ∈𝑋

By our earlier observations, ∑︁ ∑︁
|𝑆 \ N(𝑣)| = |𝑆| − deg𝑆 (𝑣) 𝑣 ∈𝑋

𝑣 ∈𝑋

= |𝑆| × |𝑋 | − | [𝑆, 𝑋] |
= |𝑆| |𝑋 | − ( 𝑝 − 1)
= (𝑛 − 𝑥) 𝑥 − ( 𝑝 − 1) . Equation (16.9) therefore simplifies to ∑︁
𝛼(𝐺, 𝑣) = 𝑥 + (𝑛 − 𝑥) 𝑥 − ( 𝑝 − 1) .

(16.10)

𝑣 ∈𝑋

The inequality in the hypothesis of the claim follows from Inequality (16.8) and Equation (16.10). Rearranged, the inequality of Claim 16.53.2 says that 𝑛 ≥ ( 𝑝 − 1) + (𝑞 − 1) +

( 𝑝 − 1) (𝑞 − 1) + 𝛼, 𝛼

where 𝛼 = 𝑥 − ( 𝑝 − 1). √︁ Since the function 𝑓 (𝛼) = 𝛼 + ( 𝑝 − 1) (𝑞 − 1)/𝛼 assumes its minimum value at 𝛼 ( 𝑝 − 1)(𝑞 − 1), it follows that a graph, which is minimal with respect to the property that every vertex is in a 𝑝-clique and in an independent set of cardinality 𝑞, has order at least l √︁ m 𝑝 + 𝑞 − 2 + 2 ( 𝑝 − 1) (𝑞 − 1) .

Section 16.4. Independent Domination Number This proves Inequality (16.7). We next prove the following: l √︁ m 𝑓 ( 𝑝, 𝑞) ≤ 𝑝 + 𝑞 − 2 + 2 ( 𝑝 − 1)(𝑞 − 1) .

493

(16.11)

For positive integers 𝑝, 𝑥, and 𝑠, where 𝑠( 𝑝 − 1) ≥ 𝑥 and 𝑥 ≥ 𝑝 − 1, we define a graph 𝐺 ( 𝑝, 𝑥, 𝑠) as follows. This graph has vertex set 𝑈 ∪ 𝑉, where 𝑈 = {𝑢 0 , 𝑢 1 , . . . , 𝑢 𝑥−1 } and 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑠 }. The vertices in 𝑈 form a clique and the vertices in 𝑉 an independent set. Then for 𝑖 ∈ [𝑠], join 𝑣 𝑖 to the 𝑝 − 1 vertices 𝑢 𝑗 for ( 𝑝 − 1) (𝑖 − 1) ≤ 𝑗 ≤ ( 𝑝 − 1)𝑖 − 1, where subscripts are taken modulo 𝑥. Hence, there are 𝑠( 𝑝 − 1) edges with one end in 𝑈 and the other end in 𝑉, and these are distributed evenly among the vertices of 𝑈. By construction, the resulting graph 𝐺 ( 𝑝, 𝑥, 𝑠) has the following property. Claim 16.53.3 Every  vertex of 𝐺  ( 𝑝, 𝑥, 𝑠) is in a 𝑝-clique and is in an independent set of cardinality 𝑠 − 𝑠( 𝑝 − 1)/𝑥 + 1. For a real number 𝑥, we denote by v𝑥w the integer nearest to 𝑥, where if 𝑥 is halfway√︁between two consecutive integers, then it is rounded to the even one. Now let 𝛼 = ( 𝑝 − 1)(𝑞 − 1) and consider the graph 𝐺 isomorphic to 𝐺 ( 𝑝, 𝑥, 𝑠), where 𝑥 = 𝑝 − 1 + v𝛼w and 𝑠 = 𝑞 − 1 + ⌈𝛼⌉. We note that 𝑥 ≥ 𝑝 − 1. Further, since 𝑝, 𝑞 ≥ 2, we have 𝑠( 𝑝 − 1) ≥ 𝑥. Claim 16.53.4 The following properties hold √︁ in the graph 𝐺:  (a) The graph 𝐺 has order 𝑝 + 𝑞 − 2 + 2 ( 𝑝 − 1)(𝑞 − 1) . (b) Every vertex of 𝐺 is in a 𝑝-clique. (c) Every vertex of 𝐺 is in an independent set of cardinality 𝑞. Proof Since v𝛽w + ⌈𝛽⌉ = ⌈2𝛽⌉ for any positive real number 𝛽, the graph 𝐺 has order ( 𝑝 − 1) + (𝑞 − 1) + ⌈2𝛼⌉. This proves (a). By Claim 16.53.3, every vertex of 𝐺 is in a 𝑝-clique and so   (b) holds. To prove (c), it suffices by Claim 16.53.3 to show that 𝑠 − 𝑠( 𝑝 − 1)/𝑥 + 1 ≥ 𝑞, or equivalently, that & '
𝑞 − 1 + ⌈𝛼⌉ ( 𝑝 − 1) ⌈𝛼⌉ ≥ . (16.12) 𝑝 − 1 + v𝛼w We now set 𝑦 = v𝛼w, 𝑎 = 𝑝 − 1, and 𝑏 = 𝑞 − 1. If ⌈𝛼⌉ = 𝑦, then Inequality (16.12) is true if 𝑦 ≥ (𝑏 + 𝑦)𝑎/(𝑎 + 𝑦) (since 𝑦 is an integer) which is equivalent to 𝑦 2 ≥ 𝑎𝑏 and hence to 𝑦 ≥ 𝛼, which is true. On the other hand, if ⌈𝛼⌉ = 𝑦 + 1, then Inequality (16.12) is true if 𝑦 + 1 ≥ (𝑏 + 𝑦 + 1)𝑎/(𝑎 + 𝑦) (since 𝑦 + 1 ∈ Z) which is equivalent to 𝑦 2 + 𝑦 ≥ 𝑎𝑏. But 𝑦 ≥ 𝛼 − 12 and so 𝑦 2 + 𝑦 ≥ 𝛼2 − 14 = 𝑎𝑏 − 14 . However, since 𝑦, 𝑎, and 𝑏 are integers, it follows that 𝑦 2 + 𝑦 ≥ 𝑎𝑏, as required. Inequality (16.11) follows from Claim 16.53.4. Together with Inequality (16.7), this proves the equation in the statement of the theorem. Applying Theorem 16.53, Goddard and Henning [350] proved the following Nordhaus-Gaddum upper bound on the sum of the independent domination numbers of a graph and its complement.

494

Chapter 16. Nordhaus-Gaddum Bounds

Theorem 16.54 ([350]) If 𝐺 and its complement 𝐺 are isolate-free graphs of order 𝑛, then √ 𝑖(𝐺) + 𝑖(𝐺) ≤ 𝑛 + 4 − 2 𝑛, and this is best possible (if one rounds down) for all 𝑛 (necessarily, 𝑛 ≥ 4). Proof Let 𝑖(𝐺) = 𝑝 and 𝑖(𝐺) = 𝑞. Since 𝐺 and 𝐺 are isolate-free, we know that 𝑝 ≥ 2 and 𝑞 ≥ 2. It follows that every vertex of 𝐺 is in a 𝑝-clique and is in an independent set of cardinality 𝑞. By Theorem 16.53, √︁ 2 √︁ 𝑛≥ 𝑝−1+ 𝑞−1 . We wish to consider the problem√︁of maximizing√︁𝑝 + 𝑞 subject to the√above constraint and 𝑝, 𝑞 ≥ 2. If we set 𝑥 = 𝑝 − 1 and 𝑦 = 𝑞 − 1, then 𝑥 + 𝑦 ≤ 𝑛 and 𝑥, 𝑦 ≥ 1 (noting that 𝑝, 𝑞 ≥ 2). Further, 𝑝 = 𝑥 2 + 1 and 𝑞 = 𝑦 2 + 1, which yields the following problem:

maximize 𝐹 (𝑥, 𝑦) = 𝑥 2 + 1 + 𝑦 2 + 1 √ subject to 𝑥 + 𝑦 ≤ 𝑛, and 𝑥, 𝑦 ≥ 1. In order to maximize the function 𝐹 (𝑥, 𝑦), we wish to make √ 𝑥 and 𝑦 as large as possible. Thus, to achieve the maximum we require 𝑥 + 𝑦 = 𝑛. This means that maximizing 𝐹 (𝑥, 𝑦) as a function of 𝑥 (with 𝑛 treated as fixed) is given by maximizing the function √ √
2 𝑓 (𝑥) = 𝑥 2 + 1 + 𝑛 − 𝑥 + 1 = 2𝑥 2 − 2𝑥 𝑛 + (𝑛 + 2). This function 𝑓 (𝑥) describes a parabola √ and therefore has an endpoint maximum. The two endpoints are 𝑥 = 1 and 𝑥 = 𝑛 − 1 (which to √𝑦 = 1). Both
√ correspond endpoints yield the same value, namely 𝑓 (1) = 𝑓 𝑛 − 1 = 𝑛 + 4 − 2 𝑛. Thus, the function 𝐹 (𝑥, 𝑦) is maximized at √ 𝐹 ★ (𝑥, 𝑦) = 𝑛 + 4 − 2 𝑛, √ √ and is attained for 𝑥 = 1 and √ 𝑦 = 𝑛 − 1 or 𝑥 = 𝑛 − 1 and 𝑦 = 1. Hence, 𝑝 + 𝑞 = 𝑖(𝐺) + 𝑖(𝐺) ≤ 𝑛 + 4 − 2 𝑛. To show that the bound in Theorem 16.54 is best possible, Goddard and Henning [350] provided the following construction. For positive integers 𝑟, 𝑠, and 𝑑 where 𝑠 ≥ 𝑑, let 𝐻 (𝑟, 𝑠, 𝑑) be a graph consisting of the disjoint union of a clique 𝑅 on 𝑟 vertices and an independent set 𝑆 on 𝑠 vertices with edges added in such a way that every vertex in 𝑅 is adjacent to 𝑑 vertices in 𝑆 and every vertex in 𝑆 is adjacent to either ⌊𝑟𝑑/𝑠⌋ or ⌈𝑟𝑑/𝑠⌉ vertices in 𝑅. By construction, the resulting graph 𝐺 = 𝐻 (𝑟, 𝑠, 𝑑) satisfies   𝑟𝑑 𝑖(𝐺) = 𝑠 − 𝑑 + 1 and 𝑖(𝐺) = + 1. (16.13) 𝑠

Section 16.4. Independent Domination Number

495

As shown in [350], for 𝑛 ≥ 4 if we let 
√ √ √ 𝐻𝑛 = 𝐻 v 𝑛w, 𝑛 − v 𝑛w, 𝑛 − 1 ,  √  then defining 𝐺 = 𝐻𝑛 , we have 𝑖(𝐺) = 𝑛 + 2 − 2 𝑛 and 𝑖(𝐺) = 2, and so  √  𝑖(𝐺) + 𝑖(𝐺) = 𝑛 + 4 − 2 𝑛 . The bound in Theorem 16.54 is therefore best possible (if one rounds down) for all 𝑛, where necessarily 𝑛 ≥ 4 (since 𝛿(𝐺) ≥ 1 and 𝛿(𝐺) ≥ 1). As an immediate corollary of Theorem 16.54, we have the upper bound for the independent domination number of a graph given in Theorem 6.82 of Chapter 6, which was first observed in 1988 by Favaron [274] (and also proved in 1995 by Gimbel and Vestergaard [336]). Corollary 16.55 ([274]) If 𝐺 is an isolate-free graph of order 𝑛, then √ 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛. Proof Since 𝐺 is an isolate-free graph, we have 𝛿(𝐺) ≥ 1 and so 𝑖(𝐺) ≥ 2. If 𝐺 has a dominating vertex, then 𝑖(𝐺) = 1 and the desired bound follows. If 𝐺 has no dominating then 𝛿(𝐺) ≥√ 1 and so by Theorem 16.54, we have
√ vertex, 𝑖(𝐺) ≤ 𝑛 + 4 − 2 𝑛 − 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛. We next turn our attention to determining a Nordhaus-Gaddum upper bound on the product of the independent domination numbers of a graph and its complement. In 1989 Cockayne and Mynhardt [198] constructed a family of graphs 𝐺 of order 𝑛 ≡ 0 (mod 4) with the property that 𝑖(𝐺) = 𝑖(𝐺) = 14 𝑛 + 1, yielding 1 𝑖(𝐺)𝑖(𝐺) = 16 (𝑛 + 4) 2 , thereby showing that the maximum value of the product 1 𝑖(𝐺)𝑖(𝐺) among all graphs 𝐺 of order 𝑛 ≡ 0 (mod 4) is at least 16 (𝑛 + 4) 2 . Subsequently, with Fricke [186] in 1995, they showed that this is best possible in the
1 2 sense that 𝑖(𝐺)𝑖(𝐺) ≤ 16 𝑛 + O 𝑛2 . In 2003 Goddard and Henning [350] completed the study and determined the maximum possible value for the product 𝑖(𝐺)𝑖(𝐺). They showed that the lower bound on the maximum product, due to Cockayne and Mynhardt [198], for order 𝑛 ≡ 0 (mod 4) is best possible. More generally, they proved the following upper bound for the product 𝑖(𝐺)𝑖(𝐺) using the key result given in Theorem 16.53. Theorem 16.56 ([350]) If 𝐺 is a graph of order 𝑛, then 𝑖(𝐺)𝑖(𝐺) ≤

1 16 (𝑛

+ 4) 2 .

Proof Let 𝑖(𝐺) = 𝑝 and 𝑖(𝐺) = 𝑞. Thus, every vertex of 𝐺 is in a 𝑝-clique and every vertex of 𝐺 is in an independent set of cardinality 𝑞. If 𝑝 = 1 or 𝑞 = 1, then the maximum product is clearly 𝑛, noting that 𝑖(𝐺) ≤ 𝑛 and 𝑖(𝐺) ≤ 𝑛. Hence, we may assume that 𝑝, 𝑞 ≥ 2. By Theorem 16.53, 𝑛≥

√︁

2 √︁ 𝑝−1+ 𝑞−1 .

Chapter 16. Nordhaus-Gaddum Bounds

496

𝑝𝑞 subject√to the above constraint We wish to consider the problem √︁ of maximizing √︁ and with 𝑝, 𝑞 ≥ 2. If we set 𝑥 = 𝑝 − 1, 𝑦 = 𝑞 − 1, and 𝑧 = 𝑛, then 𝑥 + 𝑦 ≤ 𝑧 and 𝑥, 𝑦 ≥ 1 (noting that 𝑝, 𝑞 ≥ 2). Further, 𝑝 = 𝑥 2 + 1 and 𝑞 = 𝑦 2 + 1, which yields the following problem:

maximize 𝐺 (𝑥, 𝑦) = 𝑥 2 + 1 𝑦 2 + 1 subject to

𝑥 + 𝑦 ≤ 𝑧, and 𝑥, 𝑦 ≥ 1.

In order to maximize the function 𝐺 (𝑥, 𝑦), we wish to make 𝑥 and 𝑦 as large as possible. Thus, to achieve the maximum we require 𝑥 + 𝑦 = 𝑧. This means that the maximum of the function 𝐺 (𝑥, 𝑦) as a function of 𝑥 (with 𝑧 treated as fixed) is given by maximizing the function

𝑔(𝑥) = 𝑥 2 + 1 (𝑧 − 𝑥) 2 + 1 . By calculus, the global maximum for the function 𝐺 (𝑥, 𝑦) is therefore given by 𝐺 ★ (𝑥, 𝑦) =

1 16 (𝑛

+ 4) 2 ,

√ attained for 𝑥 = 𝑦 = 12 𝑧 = 12 𝑛, that is, when 𝑝 = 𝑥 2 +1 = 14 𝑛+1 and 𝑞 = 𝑦 2 +1 = 14 𝑛+1. Hence,

1 𝑖(𝐺)𝑖(𝐺) = 𝑝𝑞 ≤ 14 𝑛 + 1 14 𝑛 + 1 = 16 (𝑛 + 4) 2 . The upper bound in Theorem 16.56 is achievable when the order 𝑛 is a multiple of 4. Consider, for example, the graphs 𝐻 (𝑟, 𝑠, 𝑑), where 𝑟, 𝑠, and 𝑑 are integers where 𝑠 ≥ 𝑑, whose construction was explained following the proof of Theorem 16.54. In this case, we let    
𝐺 𝑛 = 𝐻 v 𝑛2 w, 𝑛2 , 𝑛4 . If 𝑛 = 4𝑘 where 𝑘 ≥ 1, then 𝐺 𝑛 = 𝐻 (2𝑘, 2𝑘, 𝑘). Thus, in this case 𝐺 𝑛 is a graph consisting of the disjoint union of a clique 𝑅 on 2𝑘 vertices and an independent set 𝑆 on 2𝑘 vertices with edges added in such a way that every vertex in 𝑅 is adjacent to 𝑘 vertices in 𝑆 and every vertex in 𝑆 is adjacent to 𝑘 vertices in 𝑅. For example, one way to ensure an even distribution of the edges is as follows. Let 𝑅 = {𝑢 0 , 𝑢 1 , . . . , 𝑢 2𝑘−1 } and 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 2𝑘 }, and for 𝑖 ∈ [2𝑘] join 𝑣 𝑖 to the 𝑘 vertices 𝑢 𝑗 for 𝑘 (𝑖 − 1) ≤ 𝑗 ≤ 𝑘𝑖 − 1, where subscripts are taken modulo 2𝑘. We note that (or use Equation (16.13)) if 𝐺 = 𝐺 𝑛 , then 𝑖(𝐺) = 𝑖(𝐺) = 𝑘 + 1, and so 𝑖(𝐺)𝑖(𝐺) = (𝑘 + 1) 2 =

1 4𝑛

+1


2

=

1 16 (𝑛

+ 4) 2 .

1 Thus, the value 16 (𝑛 + 4) 2 is attained for the extremal graph 𝐺 𝑛 above when 𝑛 ≡ 0    𝑛+6  1 (mod 4). In this case when 𝑛 is a multiple of 4, we note that 16 (𝑛 + 4) 2 = 𝑛+4 4 4 . 1  2 We remark that when 𝑛 ≡ 2 (mod 4), then the value 16 (𝑛 + 4) is attained by   the graph 𝐺 𝑛 defined above. In this case, taking 𝐺 = 𝐺 𝑛 , we have 𝑖(𝐺) = 𝑛+4 and 4  𝑛+6  𝑖(𝐺) = 4 , and so

Section 16.4. Independent Domination Number 

𝑛+4 𝑖(𝐺)𝑖( 𝐺) = 4



497

   𝑛+6 1 2 = (𝑛 + 4) . 4 16

The authors in [350] actually gave a stronger result than that given in Theorem 16.56. However, the case when the order 𝑛 is odd is more complicated than the case when 𝑛 is even and requires a more intricate analysis to establish the exact maximum value, which we omit. In order to state this stronger result, we define S to be the set that contains the perfect squares of odd integers and the perfect squares minus 1 of even integers, that is,  S = 𝑛 : 𝑛 = 𝑥 2 for odd integers 𝑥 or 𝑛 = 𝑥 2 − 1 for even integers 𝑥 . Adopting the notation in [350], we define    𝑛+4 𝑛+6 𝑏(𝑛) = . 4 4 That is,  1 2  if 𝑛 ≡ 0 (mod 4)  16 (𝑛 + 4)       1 (𝑛 + 3) 2 if 𝑛 ≡ 1 (mod 4) 𝑏(𝑛) = 16 1    16 (𝑛 + 2) (𝑛 + 6) if 𝑛 ≡ 2 (mod 4)     1 (𝑛 + 1) (𝑛 + 5) if 𝑛 ≡ 3 (mod 4).  16 We are now in a position to state the result given in [350]. Theorem 16.57 ([350]) If 𝐺 is a graph of order 𝑛, then   𝑛 if 𝑛 ≤ 7     𝑖(𝐺)𝑖(𝐺) ≤ 𝑏(𝑛) + 1 if 𝑛 ∈ S     𝑏(𝑛) otherwise,  and this is best possible for all 𝑛. The extremal graphs are as follows. For 𝑛 ≤ 7, the graph 𝐺 = 𝐾𝑛 satisfies 𝑖(𝐺)𝑖(𝐺) is even, then the graph 𝐺 = 𝐺 𝑛 =  = 𝑛. 
As observed earlier, if 𝑛 ≥ 4𝑛+6 𝐻 v 𝑛2 w, 𝑛2 , 𝑛4 satisfies 𝑖(𝐺)𝑖(𝐺) = 𝑛+4 = 𝑏(𝑛). If 𝑛 = 𝑥 2 for 𝑥 ≥ 3 odd, 4 4 then 𝑛 ≡ 1 (mod 4) and the graph
𝐺 = 𝐻 𝑥(𝑥 + 1), 12 𝑥(𝑥 − 1), 14 (𝑥 2 − 1)    𝑛+6  satisfies 𝑖(𝐺) = 14 (𝑥 −1) 2 +1 and 𝑖(𝐺) = 14 (𝑥 +1) 2 +1, and 𝑖(𝐺)𝑖(𝐺) = 𝑛+4 = 4 4 𝑏(𝑛). If 𝑛 = (2𝑦) 2 − 1 for 𝑦 ≥ 2 an integer, then 𝑛 ≡ 3 (mod 4) and the graph
𝐺 = 𝐻 (𝑦 + 1) (2𝑦 − 1), 𝑦(2𝑦 − 1), 𝑦 2    𝑛+6  satisfies 𝑖(𝐺) = 𝑦 2 − 𝑦 + 1 and 𝑖(𝐺) = 𝑦 2 + 𝑦 + 1, and 𝑖(𝐺)𝑖(𝐺) = 𝑛+4 = 𝑏(𝑛). 4 4 All of the bounds in Theorem 16.57 are therefore tight.

Chapter 16. Nordhaus-Gaddum Bounds

498

16.5 Upper Domination Parameters In this section, we consider Nordhaus-Gaddum theorems for the upper domination number Γ(𝐺), the upper total domination number Γt (𝐺), and the independence number 𝛼(𝐺) of a graph 𝐺. These core upper domination parameters are discussed in Chapter 14.

16.5.1

Upper Domination and Independence Numbers

We repeat here the characterization of minimal dominating sets, which is first given in Chapter 2. Lemma 16.58 ([622]) A dominating set 𝑆 in a graph 𝐺 is a minimal dominating set of 𝐺 if and only if for every vertex 𝑣 ∈ 𝑆, ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. Nordhaus-Gaddum type results on the upper domination number were given by Cockayne and Mynhardt [198] in 1989. We remark that they proved a stronger result and showed that the result also holds for the upper irredundance number. Since our focus in this book is on the core domination parameters, we only present their result for the upper domination number; the proof for the upper irredundance number is identical. Theorem 16.59 ([198]) If 𝐺 is a graph of order 𝑛, then the following hold: (a) Γ(𝐺) + Γ( 𝐺) ≤  𝑛 + 1.  (b) Γ(𝐺)Γ(𝐺) ≤ 14 (𝑛 + 1) 2 . Proof Let 𝑋 be a Γ-set of 𝐺 and let 𝑌 be a Γ-set of 𝐺. Thus, Γ(𝐺) = |𝑋 | and Γ(𝐺) = |𝑌 |. Let |𝑋 | = 𝑥 and |𝑌 | = 𝑦. Let 𝑆 = 𝑋 ∩𝑌 and 𝑇 = 𝑉 (𝐺) \ (𝑋 ∪𝑌 ). Further, let |𝑆| = 𝑠 and |𝑇 | = 𝑡. We note that 𝑋 ∪ 𝑌 = 𝑉 (𝐺) \ 𝑇 and so 𝑥 + 𝑦 − 𝑠 = 𝑛 − 𝑡. Thus, 𝑥 + 𝑦 = 𝑛 + (𝑠 − 𝑡)

and 𝑥𝑦 = 𝑥(𝑛 − 𝑥 + 𝑠 − 𝑡).

(16.14)

We proceed further with the following claim. Claim 16.59.1

𝑠 ≤ 𝑡 + 1.

Proof Suppose, to the contrary, that 𝑠 ≥ 𝑡 + 2. Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑠 }. By supposition, 𝑠 ≥ 2. We note that at least one of 𝐺 [𝑆] and 𝐺 [𝑆] is an isolate-free graph. We may assume, without loss of generality, that 𝐺 [𝑆] is isolate-free. In particular, every vertex of 𝑆 ⊆ 𝑋 has at least one neighbor that belongs to the set 𝑋 in the graph 𝐺. Thus, ipn𝐺 [𝑣, 𝑋] = ∅ for every vertex 𝑣 ∈ 𝑆. Hence, by Lemma 16.58, we have epn𝐺 [𝑣, 𝑋] ≠ ∅ for every vertex 𝑣 ∈ 𝑆. For 𝑖 ∈ [𝑠], let 𝑤 𝑖 be an 𝑋-external private neighbor of 𝑣 𝑖 in the graph 𝐺 and so 𝑤 𝑖 ∈ 𝑉 (𝐺) \ 𝑋 and N𝐺 (𝑤 𝑖 ) ∩ 𝑋 = {𝑣 𝑖 }. We note that the vertices 𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑠 are distinct. Let 𝑊 = {𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑠 }. Since 𝑠 > 𝑡, at least one vertex in 𝑊 belongs to the set 𝑌 \ 𝑋. Renaming vertices if necessary, we may assume that 𝑤 𝑠 belongs to 𝑌 \ 𝑋. The vertex 𝑤 𝑠 is adjacent in 𝐺 to every vertex of 𝑋 \ {𝑣 𝑠 }. In particular, the vertices in 𝑆 \ {𝑣 𝑠 } are not isolated in the graph 𝐺 [𝑌 ]. By the minimality of the set 𝑌 , we therefore deduce by Lemma 16.58 that

Section 16.5. Upper Domination Parameters

499

epn𝐺 [𝑣, 𝑌 ] ≠ ∅ for every vertex 𝑣 ∈ 𝑆 \ {𝑣 𝑠 }. For 𝑖 ∈ [𝑠 − 1], let 𝑢 𝑖 be a 𝑌 -external private neighbor of 𝑣 𝑖 in the graph 𝐺 and so 𝑢 𝑖 ∈ 𝑉 (𝐺) \ 𝑌 and N𝐺 (𝑢 𝑖 ) ∩ 𝑌 = {𝑣 𝑖 }. We note that the vertices 𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑠−1 are distinct. Let 𝑈 = {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑠−1 }. If 𝑢 𝑖 belongs to the set 𝑋 \ 𝑌 for some 𝑖 ∈ [𝑠 − 1], then 𝑢 𝑖 would be adjacent to 𝑤 𝑠 in 𝐺, contradicting the fact that 𝑣 𝑖 is the only neighbor of 𝑢 𝑖 in 𝐺 that belongs to the set 𝑌 . Hence, 𝑈 ⊆ 𝑇 and so 𝑠 − 1 = |𝑈| ≤ |𝑇 | = 𝑡, contradicting our supposition that 𝑠 ≥ 𝑡 + 2. By Claim 16.59.1, we have 𝑠 ≤ 𝑡 + 1. By Equation (16.14), Γ(𝐺) + Γ(𝐺) = 𝑥 + 𝑦 = 𝑛 + (𝑠 − 𝑡) ≤ 𝑛 + 1. This proves (a). To prove (b), by Equation (16.14), we have 𝑥𝑦 = 𝑥(𝑛 − 𝑥 + 𝑠 − 𝑡) ≤ 𝑥(𝑛 − 𝑥 + 1). The function 𝑥(𝑛 − 𝑥 + 1) is maximized when 𝑥 = 12 (𝑛 + 1), yielding Γ(𝐺)Γ(𝐺) = 𝑥𝑦 ≤ 14 (𝑛 + 1) 2 .

(16.15)

Since Γ(𝐺) and Γ(𝐺) are integers, the upper bound in (b) now follows. Since every maximum independent set is a minimal dominating set, 𝛼(𝐺) ≤ Γ(𝐺) for every graph 𝐺. Hence, as an immediate consequence of Theorem 16.59, we have the following Nordhaus-Gaddum bounds for the independence numbers of a graph and its complement. Theorem 16.60 ([198]) If 𝐺 is a graph of order 𝑛, then the following hold: (a) 𝛼(𝐺) + 𝛼(𝐺) ≤  𝑛 + 1.  (b) 𝛼(𝐺)𝛼(𝐺) ≤ 14 (𝑛 + 1) 2 . The complete graph 𝐾𝑛 obtains the upper bound of 𝑛 + 1 on the sum in both Theorems 16.59 and 16.60. The upper bound 𝑛 + 1 on the sum is also tight for isolate-free graphs 𝐺 and 𝐺, as is the upper bound 14 (𝑛 + 1) 2 on the product for all 𝑛 ≥ 5. Such an extremal graph 𝐺 can be constructed as follows. Let 𝐺 be a graph  of order 𝑛 ≥ 5 with vertex set 𝑉 partitioned into 𝑉1 and 𝑉2 , where |𝑉1 | = 12 𝑛 and |𝑉2 | = 12 𝑛 . The edge set of 𝐺 is defined as follows. Let 𝑣 ∈ 𝑉2 and let 𝑉1 ∪ {𝑣} induce a complete graph in 𝐺. The only remaining edges of 𝐺 form a matching from 𝑉2 \ {𝑣} to vertices in 𝑉1 . When 𝑛 = 9, for example, the resulting graph 𝐺 is illustrated in Figure 16.4(a) and its complement 𝐺 in Figure 16.4(b). In the resulting  graph 𝐺, we have 𝛿(𝐺) ≥ 1 and 𝛿(𝐺)   ≥ 1. Moreover, 𝛼(𝐺) = Γ(𝐺) = |𝑉2 | = 12 𝑛 and 𝛼(𝐺) = Γ(𝐺) = |𝑉1 ∪ {𝑣}| = 12 𝑛 + 1. Hence, 𝛼(𝐺) + 𝛼(𝐺) = Γ(𝐺) + Γ(𝐺) = 𝑛 + 1 and 𝛼(𝐺)𝛼(𝐺) = Γ(𝐺)Γ(𝐺) =

1  1  
1 2 2𝑛 2 𝑛 + 1 = 4 (𝑛 + 1) .

Thus, the bounds in Theorems 16.59 and 16.60 are tight.

Chapter 16. Nordhaus-Gaddum Bounds

500 𝑉2 \ {𝑣}

𝑉1

𝑉2 \ {𝑣}

𝑉1

𝑣1

𝑢1

𝑣1

𝑢1

𝑣2

𝑢2

𝑣2

𝑢2

𝑣3

𝑢3

𝑣3

𝑢3

𝑣4

𝑢4

𝑣4

𝑢4

𝑣

𝑣

(a) 𝐺

(b) 𝐺

Figure 16.4 A graph 𝐺 and its complement 𝐺

16.5.2

Upper Total Domination Number

In 2022 Haynes and Henning [427] determined the following Nordhaus-Gaddum upper bounds on the sum Γt (𝐺) + Γt (𝐺) and the product Γt (𝐺)Γt (𝐺), where 𝐺 and 𝐺 are isolate-free. Theorem 16.61 ([427]) If 𝐺 and its complement 𝐺 are isolate-free graphs of order 𝑛, then the following inequalities hold: (a) Γt (𝐺) + Γt (𝐺) ≤  𝑛 + 2.  (b) Γt (𝐺)Γt (𝐺) ≤ 14 (𝑛 + 2) 2 . The bound in Theorem 16.61(a) on the sum of the upper total domination numbers of a graph and its complement is tight as can be seen by the graph 𝐺 = 𝑘𝐾2 , for which 𝐺 has order 𝑛 = 2𝑘, Γt (𝐺) = 2𝑘, and Γt (𝐺) = 2, and so Γt (𝐺) + Γt (𝐺) = 𝑛 + 2. To construct an extremal graph for the product of the upper total domination numbers of a graph and its complement that achieves the bound in Theorem 16.61(b), let 𝑘 ≥ 2 be an integer and let 𝐺 be a graph of order 𝑛 = 4𝑘 + 2 with 𝑉 (𝐺) partitioned into sets 𝑉1 and 𝑉2 , where |𝑉1 | = 2𝑘 + 2 and |𝑉2 | = 2𝑘. The edge set of 𝐺 is defined as follows. Let 𝐺 [𝑉1 ] = 𝐾2𝑘+2 − 𝑀1 , where 𝑀1 is a perfect matching in the complete graph 𝐾2𝑘+2 with vertex set 𝑉1 , and let 𝐺 [𝑉2 ] = 𝑘𝐾2 . We note that the vertices in 𝑉2 have degree 1 in 𝐺. Equivalently, in the complement 𝐺 of 𝐺, we have 𝐺 [𝑉1 ] = (𝑘 + 1)𝐾2 and 𝐺 [𝑉2 ] = 𝐾2𝑘 − 𝑀2 , where 𝑀2 is a perfect matching in the complete graph 𝐾2𝑘 with vertex set 𝑉2 . Further, all edges between the vertices 𝑉1 and the vertices 𝑉2 are present in 𝐺. The set 𝐷 2 = 𝑉2 ∪ {𝑢 1 , 𝑣 1 } is a Γt -set in 𝐺, where 𝑢 1 and 𝑣 1 are any two vertices in 𝑉1 that are adjacent in 𝐺. We note that 𝐺 [𝐷 2 ] = (𝑘 + 1)𝐾2 . Moreover, the set 𝑉1 is a Γt -set in 𝐺. Thus, Γt (𝐺) = |𝐷 2 | = 2(𝑘 + 1)

and

Γt (𝐺) = |𝑉1 | = 2(𝑘 + 1),

and so, in this example, Γt (𝐺)Γt (𝐺) = 4(𝑘 + 1) 2 = 14 (𝑛 + 2) 2 .

Section 16.6. Domatic Numbers of 𝐺 and 𝐺

16.6

501

Domatic Numbers of 𝑮 and 𝑮

In this section, we consider the domatic numbers of a graph 𝐺 and its complement 𝐺. We first repeat two useful results, which were given in Chapter 12. Observation 16.62 ([194]) If 𝐺 is a graph, then dom(𝐺) ≤ 𝛿(𝐺) + 1. Theorem 16.63 If 𝐺 is a graph of order 𝑛, then dom(𝐺) ≤

𝑛 𝛾 (𝐺) .

A set 𝑆 ⊂ 𝑉 is called loose if no vertex in 𝑉 \ 𝑆 is adjacent to every vertex in 𝑆. Notice, therefore, that if a set 𝑆 is loose in 𝐺, then every vertex in 𝑉 \ 𝑆 has at least one neighbor in 𝑆 in the complement 𝐺, that is, 𝑆 is a dominating set of 𝐺. A vertex partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } is called a loose partition if every set 𝑉𝑖 is a loose set for 𝑖 ∈ [𝑘]. Define ℓ(𝐺) to equal the maximum order of a loose partition of 𝐺. In 1977 Cockayne and Hedetniemi [194] determined the following. Theorem 16.64 ([194]) For every graph 𝐺, dom(𝐺) = ℓ(𝐺). The following Nordhaus-Gaddum bounds for the domatic number were also given in [194]. Theorem 16.65 ([194]) If 𝐺 is a graph of order 𝑛, then dom(𝐺) + dom(𝐺) ≤ 𝑛 + 1. Proof By Observation 16.62, we have dom(𝐺) ≤ 𝛿(𝐺)+1. Thus, dom(𝐺) ≤ 𝛿(𝐺)+ 1 ≤ Δ(𝐺) +1, and so dom(𝐺) +dom(𝐺) ≤ 𝛿(𝐺) +1+Δ(𝐺) +1 = (𝑛−1) +2 = 𝑛+1. Next we consider bounds on the sum of the domination number and domatic number of a graph 𝐺. The proofs of these results make use of the complement of 𝐺. The first bound follows directly from Theorem 16.65, since 𝛾(𝐺) ≤ dom(𝐺) by Corollary 16.4. However, we give a different proof to characterize when the bound is attained. Theorem 16.66 ([194]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) + dom(𝐺) ≤ 𝑛 + 1, with equality if and only if 𝐺 = 𝐾𝑛 or 𝐺 = 𝐾 𝑛 . Proof If 𝐺 = 𝐾𝑛 or 𝐺 = 𝐾 𝑛 , then trivially dom(𝐺) + 𝛾(𝐺) = 𝑛 + 1. Assume, therefore, that 𝐺 is neither a complete graph nor the complement of a complete graph. By Theorem 4.3 in Chapter 4, for any graph 𝐺 of order 𝑛, 𝛾(𝐺) ≤ 𝑛 − Δ(𝐺) ≤ 𝑛 − 𝛿(𝐺). Also, by Observation 16.62, dom(𝐺) ≤ 𝛿(𝐺) + 1. Next, we claim that at least one of the basic inequalities 𝛾(𝐺) ≤ 𝑛 − 𝛿(𝐺) or dom(𝐺) ≤ 𝛿(𝐺) + 1 is strict. Suppose, to the contrary, that 𝛾(𝐺) = 𝑛−𝛿(𝐺) and dom(𝐺) = 𝛿(𝐺)+1. Thus, 𝛾(𝐺) = 𝑛 − dom(𝐺) + 1. By Theorem 16.63, we have dom(𝐺) ≤ 𝑛/𝛾(𝐺), and therefore,

dom(𝐺) ≤ 𝑛/ 𝑛 − dom(𝐺) + 1 , or equivalently, dom(𝐺) 𝑛 − dom(𝐺) + 1 ≤ 𝑛, from which we conclude that

𝑛 − dom(𝐺) dom(𝐺) − 1 ≤ 0. (16.16) Since both of the factors on the left-hand side of Inequality (16.16) are nonnegative, at least one of them must equal 0, that is, dom(𝐺) = 𝑛 or dom(𝐺) = 1. Since 𝐺 ≠ 𝐾𝑛 ,

502

Chapter 16. Nordhaus-Gaddum Bounds

we have dom(𝐺) < 𝑛, implying that dom(𝐺) = 1. But then 𝐺 has an isolated vertex 𝑣 and so 𝛿(𝐺) = 0. Since 𝐺 ≠ 𝐾 𝑛 , we note that 𝐺 − 𝑣 ≠ 𝐾 𝑛−1 , implying that 𝛾(𝐺 − 𝑣) < 𝑛 − 1 and therefore that 𝛾(𝐺) = 𝛾(𝐺 − 𝑣) + 1 < 𝑛 = 𝑛 − 𝛿(𝐺), a contradiction. Thus, at least one of the two inequalities, 𝛾(𝐺) ≤ 𝑛 −
𝛿(𝐺) or
dom(𝐺) ≤ 𝛿(𝐺)+1 is strict. Hence, 𝛾(𝐺)+dom(𝐺) < 𝑛−𝛿(𝐺) + 𝛿(𝐺)+1 = 𝑛+1, as claimed. This theorem was considerably strengthened in 1996 by Harary and Haynes [383] as follows. Theorem 16.67   ([383]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) + dom(𝐺) ≤ 𝑛2 + 2. In 2000 Baogen et al. [57] tightened the bound of Theorem 16.67 as follows. Theorem 16.68 ([57])  If 𝐺 and its complement  𝐺 are isolate-free graphs   of order 𝑛, then 𝛾(𝐺) + dom(𝐺) = 𝑛2 + 2 if and only if 𝛾(𝐺), dom(𝐺) = 2, 𝑛2 , or 𝑛 = 9 and 𝛾(𝐺) = dom(𝐺) = 3. Proof Let 𝐺 and 𝐺 be isolate-free graphs of order 𝑛. Assume that 𝛾(𝐺) + dom(𝐺) = 𝑛 Since 𝐺 is isolate-free, 𝛾(𝐺) ≥ 2. Further, by Theorem 16.2, we have + 2. 2 1 𝛾(𝐺) ≤ 2 𝑛. Similarly, since 𝐺 is isolate-free, 𝛾(𝐺) ≥ 2 and 𝛾(𝐺) ≤ 12 𝑛. Further, by Corollary 12.3 from Chapter 12, dom(𝐺) ≥ 2 and dom(𝐺) ≥ 2. By Theo𝑛 rem 16.63, dom(𝐺) ≤ 𝛾 (𝐺) ≤ 12 𝑛. Thus, since 𝛾(𝐺) ≤ 12 𝑛 and dom(𝐺) ≤ 12 𝑛, if   𝛾(𝐺) = 2 or dom(𝐺) = 2, then 𝛾(𝐺) + dom(𝐺) ≤ 𝑛2 + 2, with equality if and    𝑛  only if 𝛾(𝐺), dom(𝐺) = 2, 2 . Hence, we may assume that 𝛾(𝐺) ≥ 3 and dom(𝐺) ≥ 3.  𝑛   𝑛  By Theorem 16.63, we have 𝛾(𝐺) + dom(𝐺) ≤ dom(𝐺) . If both 𝛾(𝐺) +  𝑛   𝑛  𝛾 (𝐺) and dom(𝐺) are at least 4, then 𝛾(𝐺) + dom(𝐺) ≤ 2 4 < 2 + 2, a contradiction. Hence,   𝛾(𝐺) = 3 or dom(𝐺) = 3, implying that 𝑛 ≥ 6 and 𝛾(𝐺) + dom(𝐺) ≤ 3 + 𝑛3 ≤ 2 + 𝑛2 . Equality can only occur for 𝑛 = 6, 7, 9. But if 𝑛 = 6 or 𝑛 = 7, then at least one of 𝛾(𝐺) and dom(𝐺) equals 2, which we have assumed is not the case. Thus, 𝑛 = 9 and equality implies 𝛾(𝐺) = dom(𝐺) = 3, as required. Next we consider an upper bound on the sum of the total domination number and the total domatic number, which was established by Cockayne et al. [182]. To aid in the proof, we repeat here a result from Chapter 12. Theorem 16.69 The following hold in an isolate-free graph 𝐺: (a) tdom(𝐺) ≤ 𝛿(𝐺). 𝑛 (b) tdom(𝐺) ≤ 𝛾t (𝐺) . 1  (c) tdom(𝐺) ≤ 2 𝑛 . Theorem 16.70 ([182]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾t (𝐺) + tdom(𝐺) ≤ 𝑛 + 1, with equality if and only if 𝐺 = 𝑚𝐾2 . Proof Every set of 𝑛 − 𝛿(𝐺) + 1 vertices is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1. By Theorem 16.69(a), tdom(𝐺) ≤ 𝛿(𝐺). Therefore, it is immediate

Section 16.6. Domatic Numbers of 𝐺 and 𝐺

503

that 𝛾t (𝐺) + tdom(𝐺) ≤ 𝑛 + 1. For equality, we must have 𝛾t (𝐺) = 𝑛 − Δ(𝐺) + 1 and tdom(𝐺) = 𝛿(𝐺) = Δ(𝐺), and so 𝐺 is regular.
By Theorem 16.69(b), tdom(𝐺) ≤ 𝑛 𝛾t (𝐺) , and so tdom(𝐺) ≤ 𝑛/ 𝑛 − tdom(𝐺) + 1 , from which we conclude that

𝑛 − tdom(𝐺) tdom(𝐺) − 1 ≤ 0.

(16.17)

By Theorem 16.69(c), the first factor in Inequality (16.17) is positive, implying that tdom(𝐺) = 1 = 𝛿(𝐺) = Δ(𝐺), and so 𝐺 = 𝑚𝐾2 . We conclude this section with Nordhaus-Gaddum type results for the total domatic number. Theorem 16.71 ([182]) If 𝐺 is an isolate-free graph of order 𝑛 with Δ(𝐺) < 𝑛 − 1, then tdom(𝐺) + tdom(𝐺) ≤ 𝑛 − 1, with equality if and only if 𝐺 or 𝐺 = 𝐶4 . Proof Let 𝐺 be an isolate-free graph of order 𝑛 with Δ(𝐺) < 𝑛 − 1. Then, 𝐺 is an isolate-free graph and by Theorem 16.69(a), tdom(𝐺) ≤ 𝛿(𝐺) and tdom(𝐺) ≤ 𝛿(𝐺) ≤ Δ(𝐺). Therefore, tdom(𝐺) + tdom(𝐺) ≤ 𝛿(𝐺) + Δ(𝐺) = 𝑛 − 1.

(16.18)

In order to prove equality in Inequality (16.18), note that if 𝐺 is not a regular graph, then 𝛿(𝐺) < Δ(𝐺) and the inequality is strict. Hence, for equality, 𝐺 and 𝐺 must both be regular graphs and tdom(𝐺) = 𝛿(𝐺) and tdom(𝐺) = 𝛿(𝐺), and without loss of generality, we can assume that 𝛿(𝐺) ≤ 𝛿(𝐺). If 𝛿(𝐺) < 12 (𝑛 − 1), then 𝛿(𝐺) + 𝛿(𝐺) = tdom(𝐺) + tdom(𝐺) < 𝑛 − 1. Since by Theorem 16.69(c), 𝛿(𝐺) = tdom(𝐺) ≤ 12 𝑛, it follows that 12 (𝑛 − 1) ≤ 𝛿(𝐺) ≤ 12 𝑛. Thus, there are two possibilities: (i) 𝑛 is even, 𝛿(𝐺) = 12 𝑛, and 𝛿(𝐺) = 12 𝑛 − 1, or (ii) 𝑛 is odd and 𝛿(𝐺) = 𝛿(𝐺) = 12 (𝑛 − 1). Assume that each set 𝑉𝑖 of a total domatic partition of 𝐺 has at least three vertices, that is, |𝑉𝑖 | ≥ 3. If 𝑛 is odd, then 32 (𝑛 − 1) ≤ 𝑛, that is, 𝑛 = 3, but no graph of order 𝑛 = 3 satisfies these hypotheses. If 𝑛 is even, a similar argument shows that 𝑛 ≤ 6. If 𝑛 = 6, then 𝛿(𝐺) = 2 and there are two TD-sets, say 𝑉1 and 𝑉2 , of cardinality 3 in the total domatic partition of 𝐺. Since 𝐺 [𝑉1 ] and 𝐺 [𝑉2 ] both have at least two edges, there exists a vertex 𝑥 ∈ 𝑉1 of degree 2 in 𝐺 [𝑉1 ]. Hence, 𝑥 is not adjacent to any vertex in 𝑉2 , a contradiction. If 𝑛 = 4, then 𝛿(𝐺) = 1 and 𝛿(𝐺) = 2. Hence, 𝐺 = 2𝐾2 and 𝐺 = 𝐶4 , giving equality above. In all remaining cases, some TD-set in the domatic partition of 𝐺 has two vertices, which must be
adjacent. For 𝑛 even, two adjacent vertices are adjacent in 𝐺 to at most 2 12 𝑛 − 2 = 𝑛 − 4 other vertices, and hence cannot dominate 𝐺. Likewise, for 𝑛
odd, two adjacent vertices dominate at most 12 (𝑛 − 1) − 1 = 𝑛 − 3 other vertices of 𝐺, a similar contradiction.

504

Chapter 16. Nordhaus-Gaddum Bounds

In 1998 Arumugam and Thuraiswamy [39] characterized the class of all regular graphs for which tdom(𝐺) + tdom( 𝐺) = 𝑛 − 2. Theorem 16.71 was significantly strengthened in 2000 by Baogen et al. [57] with the following theorem. Theorem 16.72 ([57]) If 𝐺 is an isolate-free graph of order 𝑛 with tdom(𝐺) ≥ 2, then   𝛾t (𝐺) + tdom(𝐺) ≤ 12 𝑛 + 2,     with equality for 𝑛 ≠ 9 if and only if 𝛾t (𝐺), tdom(𝐺) = 2, 12 𝑛 .

Chapter 17

Domination in Grids and Hypercubes 17.1 Introduction In this chapter, we study domination in certain classes of graphs that can be constructed from the Cartesian product operation □ of two graphs. In particular, we present results on domination in grid-like graphs and also in the class of graphs called hypercubes. Recall that for graphs 𝐺 and 𝐻, the Cartesian product 𝐺 □ 𝐻 is the graph with vertex set 𝑉 (𝐺) ×𝑉 (𝐻) where vertices (𝑢 1 , 𝑣 1 ) and (𝑢 2 , 𝑣 2 ) are adjacent if and only if either 𝑢 1 = 𝑢 2 and 𝑣 1 𝑣 2 ∈ 𝐸 (𝐻) or 𝑣 1 = 𝑣 2 and 𝑢 1 𝑢 2 ∈ 𝐸 (𝐺). Grid-like graphs arise in applications from computer networks and integrated circuit design to city street layouts. Hypercubes are also popular models for computer network design. Our main focus of this chapter is on domination, independent domination, and total domination in grids (Cartesian products of two paths). We also briefly mention results on domination numbers of cylinders (Cartesian products of a path and a cycle) and tori (the Cartesian products of two cycles). Thereafter, we present selected results on domination in hypercubes.

17.2

Domination in Grids

We first review some definitions.  An 𝑚 × 𝑛 grid graph 𝐺 𝑚,𝑛 has a vertex set 𝑉 = (𝑖, 𝑗) : 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛] with vertex (𝑖, 𝑗) adjacent to vertex (𝑘, ℓ) if 𝑖 = 𝑘 and | 𝑗 − ℓ| = 1 or 𝑗 = ℓ and |𝑖 − 𝑘 | = 1. For a fixed value of 𝑖, the set of vertices of the form (𝑖, 𝑗), 𝑗 ∈ [𝑛], is called the 𝑖 th row of 𝐺 𝑚,𝑛 , and for a fixed value of 𝑗, the set of vertices of the form (𝑖, 𝑗), 𝑖 ∈ [𝑚], is called the 𝑗 th column of 𝐺 𝑚,𝑛 . Hence, a grid graph, or just a grid, is a Cartesian product of the form 𝑃𝑚 □ 𝑃𝑛 . A cylinder is a Cartesian product of the form 𝑃𝑚 □ 𝐶𝑛 or 𝐶𝑚 □ 𝑃𝑛 , and a torus or toroidal graph is a Cartesian product of the form 𝐶𝑚 □ 𝐶𝑛 . We note that in a grid 𝑃𝑚 □ 𝑃𝑛 , the subgraph induced by the vertices in any given © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_17

505

Chapter 17. Domination in Grids and Hypercubes

506

row 𝑖 is a copy of the path 𝑃𝑛 , while the subgraph induced by the vertices in any given column 𝑗 is a copy of the path 𝑃𝑚 . For example, in Figure 17.1(a), the set of four red vertices in the grid 𝑃4 □ 𝑃4 is a minimum dominating set, a minimum independent dominating set, and an efficient dominating set of this grid, in that every vertex is dominated exactly once. The row of five red vertices in the cylinder 𝐶3 □ 𝑃5 in Figure 17.1(c) is a minimum total dominating set of this cylinder. The set of four red vertices in the cylinder 𝑃3 □ 𝐶5 in Figure 17.1(b) is a minimum dominating set, while the set of four red vertices in the torus 𝐶3 □ 𝐶5 in Figure 17.1(d) is a minimum total dominating set.

(a) 𝑃4 □ 𝑃4

(b) 𝑃3 □ 𝐶5

(c) 𝐶3 □ 𝑃5

(d) 𝐶3 □ 𝐶5

Figure 17.1 A grid 𝑃4 □ 𝑃4 , cylinders 𝑃3 □ 𝐶5 and 𝐶3 □ 𝑃5 , and a torus 𝐶3 □ 𝐶5 Figure 17.2 illustrates a minimum dominating set for the cylinder 𝑃6 □ 𝐶6 , where each column is a 𝑃6 and each row is a 𝐶6 . In fact, for this cylinder, 𝛾(𝑃6 □ 𝐶6 ) = 𝑖(𝑃6 □ 𝐶6 ) = 9. For certain types of graphs, there are simple formulas for their domination numbers; for   𝑛 ) = 1 for 𝑛 ≥ 1; for paths,  example, for complete graphs, 𝛾(𝐾 𝛾(𝑃𝑛 ) = 𝑛3 for 𝑛 ≥ 1; and for cycles, 𝛾(𝐶𝑛 ) = 𝑛3 for 𝑛 ≥ 3. However, as we will see in this chapter, no single formula exists for grids, cylinders, or tori. Instead, for any fixed value of 𝑚, a formula is often shown to exist, while for 𝑚 sufficiently large, one formula for all 𝑚 and 𝑛, for 𝑚 ≤ 𝑛 can exist. Thus, in this chapter the reader will see quite a few formulas for the domination, independent domination, and total domination numbers of grids.

17.2.1

Domination Numbers of Grids

The problem of determining domination numbers of grids has received much attention in the literature. Several papers, including [164, 165, 188, 368, 387, 389, 569, 676],

Section 17.2. Domination in Grids

507

Figure 17.2 Minimum dominating set in 𝑃6 □ 𝐶6 have developed techniques for either computing 𝛾(𝑃𝑚 □ 𝑃𝑛 ) exactly or establishing lower or upper bounds for 𝛾(𝑃𝑚 □ 𝑃𝑛 ). Beginning in 1984, Jacobson and Kinch [505] determined 𝛾(𝑃𝑚 □ 𝑃𝑛 ) for 𝑚 ∈ [4] and all 𝑛. In 1993 Chang and Clark [151] extended this to 𝑚 ∈ {5, 6} and all 𝑛. In 1989 Hare [387] settled specific cases for 7 ≤ 𝑚 ≤ 11. In 1998 Fisher [302] developed a method for calculating 𝛾(𝑃𝑚 □ 𝑃𝑛 ) that is described in the 2001 PhD thesis of Spalding [689], who gave the values of 𝛾(𝑃𝑚 □ 𝑃𝑛 ) for 𝑚 ≤ 19 and all 𝑛. In 2011 Alanko et al. [12] used dynamic programming to extend these results to 𝑚 ≤ 29 and all 𝑛. Of particular interest in their paper is the illustration of minimum dominating sets for all grids 𝑃𝑛 □ 𝑃𝑛 , for 𝑛 ∈ [29]. Different papers have provided equivalent but different looking formulas for the domination numbers of grid graphs for various values of 𝑚 and 𝑛. Chang [150], in his 1992 PhD thesis, conjectured that for all 𝑚, 𝑛 ≥ 16,   (𝑚 + 2) (𝑛 + 2) 𝛾(𝑃𝑚 □ 𝑃𝑛 ) = − 4. 5 In 2001 Mao et al. [582] proved Chang’s Conjecture for sufficiently large 𝑚 and 𝑛. The major breakthrough came in 2011 when Chang’s Conjecture was proved true by Goncalves et al. [362], effectively solving the problem for domination in grids. Theorem 17.1 ([362]) For 1 ≤ 𝑚 ≤ 𝑛, the value of 𝛾(𝑃𝑚 □ 𝑃𝑛 ) is given by the following closed formulas:     𝛾(𝑃1 □ 𝑃𝑛 ) = 𝛾(𝑃𝑛 ) = 𝑛3 = 𝑛+2 3  𝑛+2   𝑛+1  𝛾(𝑃2 □ 𝑃𝑛 ) = 2 = 2     𝛾(𝑃3 □ 𝑃𝑛 ) = 3𝑛+4 = 𝑛 − 𝑛−1 4 4 ( 𝑛 + 1 if 𝑛 ∈ {5, 6, 9} 𝛾(𝑃4 □ 𝑃𝑛 ) = 𝑛 otherwise

Chapter 17. Domination in Grids and Hypercubes

508 (  6𝑛+6  𝛾(𝑃5 □ 𝑃𝑛 ) =  5  6𝑛+8 5

if 𝑛 = 7 otherwise

(  10𝑛+10  𝛾(𝑃6 □ 𝑃𝑛 ) =  7  10𝑛+12 7

𝛾(𝑃7 □ 𝑃𝑛 ) =

 5𝑛+3 

𝛾(𝑃8 □ 𝑃𝑛 ) =

 15𝑛+14 

𝛾(𝑃9 □ 𝑃𝑛 ) =

 23𝑛+20 

8 11

𝛾(𝑃10 □ 𝑃𝑛 ) =  13  30𝑛+24 13

(  38𝑛+21  𝛾(𝑃11 □ 𝑃𝑛 ) =  15  38𝑛+36 15

if 𝑛 ∉ {13, 16} and 𝑛 (mod 13) ∈ {0, 3} otherwise if 𝑛 ∈ {11, 18, 20, 22, 33} otherwise

 80𝑛+66  29

(  98𝑛+111  𝛾(𝑃13 □ 𝑃𝑛 ) =  33  98𝑛+78 33

(  35𝑛+40  𝛾(𝑃14 □ 𝑃𝑛 ) =  11  35𝑛+29 11

(  44𝑛+27  𝛾(𝑃15 □ 𝑃𝑛 ) =  13  44𝑛+40 13

𝛾(𝑃𝑚 □ 𝑃𝑛 ) =

otherwise

3

(  30𝑛+37 

𝛾(𝑃12 □ 𝑃𝑛 ) =

if 𝑛 ≡ 1 (mod 7)

 (𝑚+2) (𝑛+2)  5

if 𝑛 (mod 33) ∉ {14, 15, 17, 20} otherwise if 𝑛 ≡ 18 (mod 22) otherwise if 𝑛 ≡ 5 (mod 26) otherwise − 4 for all 𝑚, 𝑛 ≥ 16.

Most of the literature on domination in grid graphs has focused on developing techniques for computing the value of 𝛾(𝑃𝑚 □ 𝑃𝑛 ), but relatively little attention has been given to methods for constructing 𝛾-sets of grid graphs. In 1986 Hare et al. [389] displayed many 𝛾-sets for 𝑚 ≤ 11, and in 1985 Cockayne et al. [188] gave a method for constructing 𝛾-sets for square grids 𝑃𝑛 □ 𝑃𝑛 . In 2015 Hutson et al. [502] extended the work of Cockayne et al. [188] and showed how to construct 𝛾-sets for most of the values of 𝛾(𝑃𝑚 □ 𝑃𝑛 ) cited in Theorem 17.1, that is, for all values of 𝑚, 𝑛 ≥ 16 and most of the values of 𝑚, 𝑛 < 16, except 𝑚 ∈ {12, 13}. In concluding this section, we give sample results for domination in cylinders and tori. More details can be found in [137, 255, 334, 526, 605, 606, 631, 657] among others. Most of the results on 𝛾(𝑃𝑚 □ 𝑃𝑛 ) are for small values of 𝑚. For instance, in 2011 Nandi et al. [605] determined the domination numbers of 𝑃𝑚 □ 𝐶𝑛 for 𝑛 ≥ 3 and 𝑚 ∈ {2, 3, 4}, as follows.

Section 17.2. Domination in Grids

509

Theorem 17.2 ([605]) For 𝑛 ≥ 3, ( 𝑛+1  𝛾(𝑃2 □ 𝐶𝑛 ) =  2 𝑛 2

𝛾(𝑃3 □ 𝐶𝑛 ) =

otherwise

 3𝑛  4

( 𝛾(𝑃4 □ 𝐶𝑛 ) =

if 𝑛 . 0 (mod 4)

𝑛 + 1 if 𝑛 ∈ {3, 5, 9} 𝑛

otherwise.

In 2014 Nandi et al. [606] gave the domination numbers of 𝑃5 □ 𝐶𝑛 for 𝑛 ≥ 3 and tight bounds on domination numbers of 𝑃6 □ 𝐶𝑛 for 𝑛 ≥ 3. We state their result for 𝑃5 □ 𝐶𝑛 , noting first that 𝛾(𝑃5 □ 𝐶3 ) = 4. Theorem 17.3 ([606]) For 𝑛 ≥ 4, 𝛾(𝑃5 □ 𝐶𝑛 ) = 𝑛 +

𝑛 5

+ 𝑝𝑛,

where   0 if 𝑛 ≡ 0 (mod 10)     𝑝 𝑛 = 2 if 𝑛 (mod 10) ∈ {3, 9}     1 otherwise.  In 1995 Klavžar and Seifter [526] presented the first formulas for the domination numbers of tori 𝐶𝑚 □ 𝐶𝑛 for small 𝑚. Theorem 17.4 ([526]) For tori 𝐶𝑚 □ 𝐶  𝑛, the following hold: (a) For all 𝑛 ≥ 3, 𝛾(𝐶3 □ 𝐶𝑛 ) = 𝑛 − 𝑛4 . (b) For all 𝑛 ≥ 4, 𝛾(𝐶4 □ 𝐶𝑛 ) = 𝑛. (c) For all 𝑛 ≥ 5, ( 𝑛 if 𝑛 ≡ 0 (mod 5) 𝛾(𝐶5 □ 𝐶𝑛 ) = 𝑛 + 1 if 𝑛 (mod 5) ∈ {1, 2, 4}. (d) 𝛾(𝐶5 □ 𝐶5𝑘+3 ) ≤ 5(𝑘 + 1). Figure 17.3 illustrates the formula in Theorem 17.4 for 𝛾(𝐶4 □ 𝐶𝑛 ) = 𝑛, where the 𝑛 = 7 red vertices form a 𝛾-set of 𝐶4 □ 𝐶7 . The case for 𝑚 = 5 and 𝑛 ≡ 3 (mod 5) was settled by El-Zahar and Shaheen [255] in 2007. Theorem 17.5 ([255]) If 𝑛 ≡ 3 (mod 5) and 𝑛 ≠ 3, then 𝛾(𝐶5 □ 𝐶𝑛 ) = 𝑛 + 2. El-Zahar and Shaheen [255] also solved the case for 𝑚 ∈ {6, 7}. We state their result for 𝑚 = 6 as an example. Theorem 17.6 ([255]) For 𝑛 ≥ 3,  4𝑛    if 𝑛 (mod 6) ∈ {0, 1, 4}     3  4𝑛 𝛾(𝐶6 □ 𝐶𝑛 ) = 3 + 1 if 𝑛 (mod 6) ∈ {2, 3, 5} but 𝑛 . 5 (mod 18)    4𝑛    if 𝑛 ≡ 5 (mod 18).  3

510

Chapter 17. Domination in Grids and Hypercubes

Figure 17.3 A minimum dominating set in the torus 𝐶4 □ 𝐶7

17.2.2

Independent Domination Numbers of Grids

The first results on independent domination in grid graphs appear to have gone unnoticed, largely because they do not seem to have been published, except in the Applied Mathematics Master’s Thesis of Cortés [202] at the University of Colorado at Denver in 1994, which was directed by Fisher. Cortés [202] determined the independent domination number of 𝑃𝑚 □ 𝑃𝑛 for small values of 𝑚 and all 𝑛. Noteworthy is that the following upper bound is given in this thesis:  𝑖(𝑃𝑚 □ 𝑃𝑛 ) ≤

 (𝑚 + 2) (𝑛 + 2) − 4. 5

(17.1)

In 2015 Crevals and Östergård [206] determined the independent domination number of 𝑃𝑚 □ 𝑃𝑛 for all values of 𝑚 and 𝑛 as stated in the two following theorems. Apparently unaware of the results in [202], they proved the upper bound of Inequality (17.1) to show that for 𝑚, 𝑛 ≥ 16, 𝛾(𝑃𝑚 □ 𝑃𝑛 ) = 𝑖(𝑃𝑚 □ 𝑃𝑛 ). Thus, with the exception of some of the small values of 𝑚, 𝛾(𝑃𝑚 □ 𝑃𝑛 ) = 𝑖(𝑃𝑚 □ 𝑃𝑛 ). In other words, most grids have an independent 𝛾-set. Theorem 17.7 ([206]) For all 1 ≤ 𝑚 ≤ 𝑛, the value of 𝑖(𝑃𝑚 □ 𝑃𝑛 ) is given by the following closed formulas: 𝛾(𝑃1 □ 𝑃𝑛 ) = 𝑖(𝑃1 □ 𝑃𝑛 ) = 𝛾(𝑃2 □ 𝑃𝑛 ) = 𝑖(𝑃2 □ 𝑃𝑛 ) =

𝑛

=  𝑛+2 

 𝑛+2 

3

= 2 (  3𝑛+8 

𝑖(𝑃3 □ 𝑃𝑛 ) =  4  3𝑛+4 4

( 𝛾(𝑃4 □ 𝑃𝑛 ) = 𝑖(𝑃4 □ 𝑃𝑛 ) =

2

if 𝑛 ≡ 2 (mod 4) otherwise

𝑛 + 1 if 𝑛 ∈ {5, 6, 9} otherwise

𝑛 𝑖(𝑃5 □ 𝑃𝑛 ) =

3

 𝑛+1 

 6𝑛+8  5

Section 17.2. Domination in Grids

511

(  10𝑛+17  𝑖(𝑃6 □ 𝑃𝑛 ) =  7  10𝑛+10 7

𝛾(𝑃7 □ 𝑃𝑛 ) = 𝑖(𝑃7 □ 𝑃𝑛 ) =

3

16 𝑖(𝑃8 □ 𝑃𝑛 ) =   15𝑛+16 8

(  21𝑛+28  𝑖(𝑃9 □ 𝑃𝑛 ) =  10  21𝑛+18 10

(  21𝑛+23  𝑖(𝑃10 □ 𝑃𝑛 ) =  9  21𝑛+14 9

𝑖(𝑃11 □ 𝑃𝑛 ) =

 28𝑛+26 

𝑖(𝑃12 □ 𝑃𝑛 ) =

 36𝑛+28 

𝑖(𝑃15 □ 𝑃𝑛 ) =

if 𝑛 = 8 otherwise if 𝑛 (mod 10) ∈ {0, 7, 9} otherwise if 𝑛 ∈ {12, 18, 21, 30} otherwise

11 13

(

𝑖(𝑃14 □ 𝑃𝑛 ) =

otherwise

 5𝑛+3  (

𝑖(𝑃13 □ 𝑃𝑛 ) =

if 𝑛 (mod 7) ∈ {0, 3} and 𝑛 ≠ 7

3𝑛 + 1 if 𝑛 (mod 12) ∈ {1, 4, 7, 10}

3𝑛 + 2 otherwise  16𝑛+12  5

 17𝑛+14 

.

5

Theorem 17.8 ([206]) For all 𝑚 and 𝑛, where 16 ≤ 𝑚 ≤ 𝑛,   (𝑚 + 2) (𝑛 + 2) 𝛾(𝑃𝑚 □ 𝑃𝑛 ) = 𝑖(𝑃𝑚 □ 𝑃𝑛 ) = − 4. 5 Proof Sketch The lower bound follows from Theorem 17.1, since for 𝑚, 𝑛 ≥ 16,   (𝑚 + 2) (𝑛 + 2) 𝑖(𝑃𝑚 □ 𝑃𝑛 ) ≥ 𝛾(𝑃𝑚 □ 𝑃𝑛 ) = − 4. 5   To show that 𝑖(𝑃𝑚 □ 𝑃𝑛 ) ≤ (𝑚+2)5(𝑛+2) − 4, Crevals and Östergård [206] gave a proof similar to the one given by Chang [150] for the domination number of sufficiently large grids, which we sketch as follows.  Given a grid 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 with vertex set 𝑈 = 𝑉 (𝐺 𝑚,𝑛 ) = (𝑖, 𝑗) : 𝑖 ∈ [𝑚] and 𝑗 ∈ [𝑛] , consider the grid 𝑃𝑚+2 □ 𝑃𝑛+2 formed from 𝐺 𝑚,𝑛 by adding  two rows and two columns, that is, let 𝑃𝑚+2 □ 𝑃𝑛+2 have vertex set (𝑖, 𝑗) : 𝑖 ∈ [𝑚 + 1] 0 and 𝑗 ∈ [𝑛 + 1] 0 . Thus, the original graph 𝐺 𝑚,𝑛 , which we shall call the inner grid, is the subgraph induced by 𝑈 in the grid 𝑃 𝑚+2 □ 𝑃𝑛+2 . For 𝑠 ∈ [4] 0 , let 𝑉𝑠 = (𝑖, 𝑗) : 2𝑖 + 𝑗 ≡ 𝑠 (mod 5) in 𝑃𝑚+2 □ 𝑃𝑛+2 . We note that each 𝑉𝑠 is a set of vertices placed on the grid according to the well-known 𝐾1,4 -star pattern. Since 𝜋 = {𝑉0 , 𝑉1 , 𝑉2 , 𝑉3 , 𝑉4 } is a partition of the vertices of 𝑃𝑚+2 □ 𝑃𝑛+2 ,

Chapter 17. Domination in Grids and Hypercubes

512

  there is at least one 𝑠 for which |𝑉𝑠 | ≤ (𝑚+2)5(𝑛+2) . For the given values of 𝑚 and 𝑛, choose an 𝑠 such that this inequality holds for 𝑉𝑠 . Note that the set 𝑉𝑠 dominates the inner grid. To complete the proof, an independent dominating set 𝑆 ⊂ 𝑈 of the inner grid with |𝑆| = |𝑉𝑠 | − 4 is constructed. In the construction, 𝑆 is obtained from the set 𝑉𝑠 by removing four select vertices (one per each corner of the inner grid) from 𝑉𝑠 \ 𝑈, replacing each of the remaining vertices in 𝑉𝑠 \𝑈 with a vertex in the inner grid, and if necessary swapping vertices in the resulting set with other vertices in 𝑈 in such a way to produce an independent dominating set   of the inner grid. Hence, 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) ≤ |𝑆| = |𝑉𝑠 | − 4 = (𝑚+2)5(𝑛+2) − 4.

17.2.3

Total Domination Numbers of Grids

In 1995 Garnick and Nieuwejaar [326] initiated the study of total domination in chessboards by kings, crosses, and knights. Since the cross graph is equivalent to a grid graph, their study determined the total domination number of 𝑃𝑚 □ 𝑃𝑛 for 𝑚 ∈ [2]. Theorem 17.9 ([326]) For 𝑚 ∈ [2] and 𝑛 ≥ 𝑚, 𝑛   if 𝑚 = 1 and 𝑛 ≡ 0 (mod 4)     2  𝑛 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) = + 1 if 𝑚 = 1, 𝑛 > 1, and 𝑛 . 0 (mod 4)  2    𝑛 2 if 𝑚 = 2.  3

The following bounds were also given in [326]. Theorem 17.10 ([326]) For 𝑚, 𝑛 > 2, 𝑚𝑛 𝑚𝑛 + 2𝑚 + 2𝑛 + 5 < 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) ≤ . 4 4 Using a backtracking search program, the authors in [326] obtained the exact values given in Table 17.1 and the upper bounds given in Table 17.2 for the total domination number of square grids, which improve on the bounds in Theorem 17.10.

𝑛

2

3

4

5

6

7

8

9

10

11

𝛾t

2

3

6

9

12

15

20

25

30

35

Table 17.1 Exact values of 𝛾t (𝑃𝑛 □ 𝑃𝑛 )

In 2002 Gravier [364] studied total domination in grids and gave formulas for 𝛾t (𝑃𝑚 □ 𝑃𝑛 ), for 𝑚 ∈ [4] and 𝑛 ≥ 𝑚. We present a proof for 𝑚 = 3.

Section 17.2. Domination in Grids

513

𝑛

12

13

14

15

16

17

18

19

20

𝛾t ≤

42

49

56

64

72

81

90

105

111

𝑛

21

22

23

24

25

30

40

50

60

70

𝛾t ≤

121

132

145

160

177

241

429

658

944 1284

Table 17.2 Upper bounds for 𝛾t (𝑃𝑛 □ 𝑃𝑛 )

Theorem 17.11 ([364]) For all 𝑛 ≥ 3, 𝛾t (𝑃3 □ 𝑃𝑛 ) = 𝑛. Proof Let 𝐺 = 𝑃3 □ 𝑃𝑛 . A TD-set of 𝐺 can be obtained simply by choosing all 𝑛 vertices in the middle of the three rows of 𝐺, and so 𝛾t (𝐺) ≤ 𝑛. We show next that this is optimal in the sense that such a set is a 𝛾t -set of 𝐺. For a set 𝑆 of vertices of 𝐺 and 𝑘 ∈ [3] 0 , a column of 𝐺 having exactly 𝑘 vertices in 𝑆 is called a 𝑘-column of 𝑆 and we say that 𝑆 has a 𝑘-column. Among all 𝛾t -sets of 𝐺, let 𝑆 be chosen so that (i) 𝑆 has the minimum number of 3-columns, and (ii) subject to (i), 𝑆 has the minimum number of 0-columns. We proceed with the following two claims. Claim 17.11.1 The set 𝑆 has no 3-columns. Proof Suppose, to the contrary, that 𝑆 has a 3-column 𝐶 𝑗 . Suppose firstly that 1 < 𝑗 < 𝑛. In this case, consider the set 
 𝑆 ′ = 𝑆 \ (1, 𝑗), (3, 𝑗) ∪ (2, 𝑗 − 1), (2, 𝑗 + 1) . For illustration, suppose 𝐶 𝑗 is the second column of a 3 × 3 subgrid of 𝑃3 □ 𝑃𝑛 as shown by the three red vertices in Figure 17.4(a). In this case, the set 𝑆 ′ replaces the top and bottom red vertices in column 𝐶 𝑗 by the two green vertices in the second row to create another TD-set of 𝐺, illustrated in Figure 17.4(b). Note that the vertices not highlighted in red may or may not be in 𝑆, and so |𝑆 ′ | ≤ |𝑆| = 𝛾t (𝐺).

(a)

(b)

Figure 17.4 A rotation in the first 3 × 3 grid of 𝑃3 □ 𝑃𝑛 Since 𝑆 ′ is a TD-set of 𝐺, it follows that |𝑆| = 𝛾t (𝐺) ≤ |𝑆 ′ |, and so |𝑆| = |𝑆 ′ | and 𝑆 ′ is a 𝛾t -set of 𝐺. Thus, our choice of 𝑆 implies that 𝑆 ′ has at least as many

Chapter 17. Domination in Grids and Hypercubes

514

3-columns as does 𝑆. Since 𝐶 𝑗 is not a 3-column of 𝑆 ′ , at least one of 𝐶 𝑗 −1 and 𝐶 𝑗+1 must be a 3-column of 𝑆 ′ but not a 3-column of 𝑆. If both 𝐶 𝑗 −1 and 𝐶 𝑗+1 are 3columns of 𝑆 ′ , then 𝑆 ′ \ (2, 𝑗) is a TD-set of 𝐺 having cardinality less than 𝛾t (𝐺), a contradiction. Hence, exactly one of 𝐶 𝑗 −1 and 𝐶 𝑗+1 is 3-column of 𝑆 ′ , and 𝑆 ′ has the same number of 3-columns as 𝑆 does. Without loss of generality, assume that 𝐶 𝑗+1 is a 3-column of 𝑆 ′ . Since creating 𝑆 ′ from 𝑆 does not add any 0-columns in 𝑆 ′ , the set 𝑆 ′ has at most the number of 0-columns as 𝑆 has. It follows by our choice of 𝑆 that 𝑆 ′ has the same number of 0-columns as 𝑆 has. This implies that neither 𝐶 𝑗 −1 nor 𝐶 𝑗+1 is a 0-column in 𝑆. Since |𝑆 ′ | = |𝑆|, it follows that (2, 𝑗− 1) ∉ 𝑆 and that at least one of (1, 𝑗 − 1) and (3, 𝑗 − 1) is in 𝑆. But then again 𝑆 ′ \ (2, 𝑗) is a TD-set of 𝐺 having cardinality less than 𝛾t (𝐺), a contradiction. Hence, 𝑗 = 1 or 𝑗 = 𝑛. If 𝑗 = 𝑛, then let 
 𝑆 ′ = 𝑆 \ (1, 𝑛), (3, 𝑛) ∪ (2, 𝑛 − 1) , and if 𝑗 = 1, then let 
 𝑆 ′ = 𝑆 \ (1, 1), (3, 1) ∪ (2, 2) . In both cases, 𝑆 ′ is a TD-set of 𝐺 having cardinality less than 𝛾t (𝐺), a contradiction. Claim 17.11.2

|𝑆| = 𝑛.

Proof Suppose, to the contrary, that |𝑆| ≠ 𝑛. As observed earlier, 𝛾t (𝐺) ≤ 𝑛, implying that |𝑆| < 𝑛. Hence, 𝑆 has at least one 0-column. Let 𝐶 𝑗 be a 0-column of 𝑆 with the smallest index. If 𝑗 = 1, then 𝐶2 must be a 3-column of 𝑆, contradicting Claim 17.11.1. Hence, 𝑗 ≥ 2. By our choice of 𝑗, every column 𝐶𝑖 has at least one vertex in 𝑆 for all 𝑖 ∈ [ 𝑗 − 1]. By Claim 17.11.1, every column has at most two vertices in 𝑆. Suppose that 𝐶𝑖 is a 2-column of 𝑆 for some 𝑖 ∈ [ 𝑗 − 1]. This implies that at least 𝑗 vertices of 𝑆 are located in the first 𝑗 − 1 columns. But then  replacing all vertices of 𝑆 that belong to the first 𝑗 − 1 columns with the vertices (2, 1), (2, 2), . . . , (2, 𝑗) , and keeping all vertices of 𝑆 in the columns from column 𝑗 + 1 to column 𝑛 unchanged, we produce a new TD-set 𝑆 ′ of 𝐺 such that (i) |𝑆 ′ | ≤ |𝑆|, (ii) 𝑆 ′ has no 3-column, and (iii) 𝑆 ′ has fewer 0-columns than the 𝛾t -set 𝑆. This contradicts our choice of the set 𝑆. Hence, every column 𝐶𝑖 is a 1-column of 𝑆 for all 𝑖 ∈ [ 𝑗 − 1]. Suppose that the vertex of 𝑆 in column 𝑗 −1 belongs to row 1, that is, (1, 𝑗 −1) ∈ 𝑆. Since 𝑆 is a TD-set, and neither (2, 𝑗 − 1) nor (1, 𝑗) is in 𝑆, vertex (1, 𝑗 − 2) must be in 𝑆 in order to totally dominate the vertex (1, 𝑗 − 1). But now the vertex (3, 𝑗 − 1) is not (totally) dominated by 𝑆 since none of its neighbors, namely (3, 𝑗 − 2), (2, 𝑗 − 1), and (3, 𝑗), belongs to the set 𝑆. This contradicts the fact that 𝑆 is a TD-set of 𝐺. Hence, the vertex of 𝑆 in column 𝑗 − 1 does not belong to row 1. Analogous arguments show that the vertex of 𝑆 in column 𝑗 − 1 does not belong to row 3. Therefore, (2, 𝑗 − 1) is the vertex of 𝑆 in column 𝑗 − 1. Since 𝐶 𝑗 is a 0-column of 𝑆, in order to totally dominate the vertices (1, 𝑗) and (3, 𝑗), the set 𝑆 must contain the vertices (1, 𝑗 + 1) and (3, 𝑗 + 1). Since 𝑆 has no

Section 17.2. Domination in Grids

515

3-columns, the vertex (2, 𝑗 + 1) does not belong to 𝑆, implying that in order to totally dominate the vertices (1, 𝑗 + 1) and (3, 𝑗 + 1), the set 𝑆 must contain the vertices (1, 𝑗 + 2) and (3, 𝑗 + 2). Let 𝑅 be the set of four vertices in columns 𝑗 + 1 and 𝑗 + 2 that belong to the set 𝑆, that is,  𝑅 = (1, 𝑗 + 1), (3, 𝑗 + 1), (1, 𝑗 + 2), (3, 𝑗 + 2) .  If 𝑛 = 𝑗 + 2, then the set (𝑆 \ 𝑅) ∪ (2, 𝑗), (2, 𝑗 + 1), (2, 𝑗 + 2) is a TD-set of 𝐺 of cardinality |𝑆| − 1, contradicting the minimality of 𝑆. Hence, 𝑛 ≥ 𝑗 + 3. If at least one of (1, 𝑗 + 3) and (3, 𝑗 + 3) is not in 𝑆, then consider the set  𝑆 ′ = (𝑆 \ 𝑅) ∪ (2, 𝑗), (2, 𝑗 + 1), (2, 𝑗 + 2), (2, 𝑗 + 3) . If (2, 𝑗 + 3) ∈ 𝑆, then 𝑆 ′ is a TD-set of 𝐺 such that |𝑆 ′ | < |𝑆| = 𝛾t (𝐺), a contradiction. If (2, 𝑗 + 3) ∉ 𝑆, then 𝑆 ′ is a TD-set of 𝐺 such that (i) |𝑆 ′ | = |𝑆|, (ii) 𝑆 ′ has no 3-column, and (iii) 𝑆 ′ has fewer 0-columns than 𝑆 has, contradicting our choice of 𝑆. Therefore, both (1, 𝑗 + 3) and (3, 𝑗 + 3) must belong to 𝑆. Since 𝑆 has no 3-columns, (2, 𝑗 + 3) ∉ 𝑆. In this case, the set 
 𝑆 ′′ = 𝑆 \ (1, 𝑗 + 1), (3, 𝑗 + 1) ∪ (2, 𝑗), (2, 𝑗 + 1) is a TD-set of 𝐺 such that (i) |𝑆 ′′ | = |𝑆|, (ii) 𝑆 ′′ has no 3-column, and (iii) 𝑆 ′′ has fewer 0-columns than 𝑆 has, contradicting our choice of 𝑆. We deduce, therefore, that |𝑆| = 𝑛. By Claim 17.11.2, we have 𝛾t (𝐺) = |𝑆| = 𝑛. Gravier [364] presented slightly refined bounds for 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) for 𝑚, 𝑛 ≥ 16. Theorem 17.12 ([364]) For all 𝑚, 𝑛 ≥ 16,   3𝑚𝑛 + 2(𝑚 + 𝑛) (𝑚 + 2) (𝑛 + 2) − 1 ≤ 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) ≤ − 4. 12 4 In concluding his paper, Gravier noted that because of the results of Klavžar and Žerovnik [527], there is a constant time algorithm for computing 𝛾t (𝑃𝑚 □ 𝑃𝑛 ), the constant only depending on the value of 𝑚. This raises the question of whether the following decision problem is NP-complete: GRIDTOTALDOMSET

Instance: A grid graph 𝑃𝑚 □ 𝑃𝑛 and a positive integer 𝑘. Question: Does 𝑃𝑚 □ 𝑃𝑛 have a total dominating set of cardinality at most 𝑘? In 2004 Klobučar [528] continued the development of the total domination numbers of grid graphs for the cases of 𝑚 ∈ {4, 5, 6}. The case for 𝑚 = 4 is attributed to Gravier, who stated the formula given in [364] without proof. We present a sketch of the proof given by Klobučar.

Chapter 17. Domination in Grids and Hypercubes

516

Recall than an open packing in a graph 𝐺 is a set 𝑆 of vertices whose open neighborhoods are pairwise disjoint. In order to totally dominate the vertices in an open packing 𝑆, every TD-set of 𝐺 contains at least one vertex from the open neighborhood of every vertex of 𝑆, implying that 𝛾t (𝐺) ≥ |𝑆|. Theorem 17.13 ([364]) For all 𝑛 ≥ 4,     6 𝑛5 + 2 if 𝑛 (mod 5) ∈ {0, 1}       𝛾t (𝑃4 □ 𝑃𝑛 ) = 6 𝑛5 + 4 if 𝑛 ≡ 2 (mod 5)      6 𝑛 if 𝑛 (mod 5) ∈ {3, 4}.  5 Proof Sketch For 𝑛 ≥ 4, let 𝐺 = 𝑃4 □ 𝑃𝑛 . Let 𝑆 be the set defined as follows: ⌊ 𝑛5 ⌋−1

𝑆=

Ø

(2, 1 + 5𝑘), (3, 1 + 5𝑘), (1, 3 + 5𝑘), (4, 3 + 5𝑘), (1, 4 + 5𝑘), (4, 4 + 5𝑘) .

𝑘=0

  We note that |𝑆| = 6 𝑛5 . In the special case when 𝑛 = 13, we have |𝑆| = 12 and the set 𝑆 is illustrated by the red vertices in Figure 17.5, and they occur in groups of six every consecutive five columns, starting from the first five columns. We note that the set 𝑆 (given by the red vertices in Figure 17.5) is an open packing in 𝐺, and so every TD-set of must contain at least one vertex from the open neighborhood of every vertex of 𝑆, implying that 𝛾t (𝐺) ≥ |𝑆|. However, additional vertices need to be added to the set 𝑆 in order to construct a TD-set of 𝐺.

Figure 17.5 The set 𝑆 of vertices in 𝑃4 □ 𝑃13 We now extend the set 𝑆 to a TD-set of 𝐺 as follows. Let (𝑃4 ) 𝑗 denote the four vertices in the 𝑗 th column where 𝑗 ∈ [𝑛], that is, we write (𝑃4 ) 𝑗 to mean (1, 𝑗), (2, 𝑗), (3, 𝑗), and (4, 𝑗).  (a) For 𝑛 ≡ 1 (mod 5), let 𝑆1 = 𝑆 ∪ (2, 𝑛), (3, 𝑛) . This set 𝑆1 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆1 |. When 𝑛 = 11, we illustrate the set 𝑆1 in Figure 17.6, where the vertices (2, 𝑛) and (3, 𝑛) are colored green. The set 𝑆1 (consisting of the vertices colored red and green) is an openpacking in 𝐺, and so 𝛾t (𝐺) ≥ |𝑆1 |.  Consequently, 𝛾t (𝐺) = |𝑆1 | = |𝑆| + 2 = 6 𝑛5 + 2.  (b) For 𝑛 ≡ 2 (mod 5), let 𝑆2 = 𝑆 ∪ (2, 𝑛−1), (3, 𝑛−1), (2, 𝑛), (3, 𝑛) . This set 𝑆2 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆2 | = |𝑆| + 4. When 𝑛 = 12, we illustrate the

Section 17.2. Domination in Grids

517

Figure 17.6 A TD-set in 𝑃4 □ 𝑃11

set 𝑆2 in Figure 17.7, where the vertices (2, 𝑛 − 1), (3, 𝑛 − 1), (2, 𝑛), and (3, 𝑛) are colored green. Every TD-set of 𝐺 contains at least one vertex from each of the open neighborhoods of vertices in the open packing 𝑆. Further, in order to totally dominate the ten vertices (2, 𝑛 − 2), (3, 𝑛 − 2), (𝑃4 ) 𝑛−1 , and (𝑃4 ) 𝑛 , we need at least four additional vertices, since 𝛾t (𝑃2 □ 𝑃4 )= 4, implying that 𝛾t (𝐺) ≥ |𝑆| + 4 = |𝑆2 |. Consequently, 𝛾t (𝐺) = |𝑆2 | = 6 𝑛5 + 4.

Figure 17.7 A TD-set in 𝑃4 □ 𝑃12  (c) For 𝑛 ≡ 3 (mod 5), let 𝑆3 = 𝑆 ∪ (2, 𝑛 − 2), (3, 𝑛 − 2), (2, 𝑛 − 1), (3, 𝑛 − 1), (2, 𝑛), (3, 𝑛) . This set 𝑆3 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆3 | = |𝑆| + 6. When 𝑛 = 13, we illustrate the set 𝑆3 in Figure 17.8, where the vertices (2, 𝑛 − 2), (3, 𝑛 − 2), (2, 𝑛 − 1), (3, 𝑛 − 1), (2, 𝑛) and (3, 𝑛) are colored green. Let 𝐷 be an arbitrary TD-set of 𝐺. The set 𝐷 contains at least one vertex from each of the open neighborhoods of vertices in the open packing 𝑆. Note that there is only one case when with only four vertices we can totally dominate all the vertices (𝑃4 ) 𝑛−2 , (𝑃4 ) 𝑛−1 , and (𝑃4 ) 𝑛 . This occurs if (1, 𝑛 − 1), (2, 𝑛 − 1), (3, 𝑛 − 1), and (4, 𝑛 − 1) all belong to the set 𝐷. But then vertices (2, 𝑛 − 3) and (3, 𝑛 − 3) are not totally dominated by 𝐷. To totally dominate them, we need at least two more vertices. With a more detailed case analysis, we can show that in order to totally dominate the 14 vertices (2, 𝑛 − 3), (3, 𝑛 − 3), (𝑃4 ) 𝑛−2 , (𝑃4 ) 𝑛−1 , and (𝑃4 ) 𝑛 , we need at least six additional vertices, implying     that 𝛾t (𝐺) ≥ |𝑆| + 6 = |𝑆3 |. Consequently, 𝛾t (𝐺) = |𝑆3 | = 6 𝑛5 + 6 = 6 𝑛5 .  (d) Let 𝑛 ≡ 4 (mod 5). Let 𝑆4 = 𝑆 ∪ (2, 𝑛 − 3), (3, 𝑛 − 3), (1, 𝑛 − 1), (1, 𝑛), (4, 𝑛 − 1), (4, 𝑛) . This set 𝑆4 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆4 |. When 𝑛 = 14, we illustrate the set 𝑆4 in Figure 17.9, where the vertices (2, 𝑛 − 3),

518

Chapter 17. Domination in Grids and Hypercubes

Figure 17.8 A TD-set in 𝑃4 □ 𝑃13

(3, 𝑛 − 3), (1, 𝑛 − 1), (1, 𝑛), (4, 𝑛 − 1), and (4, 𝑛) are colored green. The set 𝑆4 (consisting of the vertices colored red and green) is an openpacking in 𝐺,and  so 𝛾t (𝐺) ≥ |𝑆4 |. Consequently, 𝛾t (𝐺) = |𝑆4 | = |𝑆| + 6 = 6 𝑛5 + 6 = 6 𝑛5 .

Figure 17.9 A TD-set in 𝑃4 □ 𝑃14  (e) Let 𝑛 ≡ 0 (mod 5). Let 𝑆5 = 𝑆 ∪ (2, 𝑛), (3, 𝑛) . This set 𝑆5 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆5 |. When 𝑛 = 10, we illustrate the set 𝑆5 in Figure 17.10, where the vertices (2, 𝑛) and (3, 𝑛) are colored green. Every TD-set of 𝐺 contains at least one vertex from the open neighborhood of every vertex in 𝑆. With a more detailed case analysis, in order to totally dominate the two vertices (2, 𝑛) and (3, 𝑛), we need two additional vertices,   and so 𝛾t (𝐺) ≥ |𝑆| +2 = |𝑆5 |. Consequently, 𝛾t (𝐺) = |𝑆5 | = |𝑆| + 2 = 6 𝑛5 + 2.

Figure 17.10 A TD-set in 𝑃4 □ 𝑃10

This completes the proof sketch of Theorem 17.13.

Section 17.2. Domination in Grids

519

Unfortunately, the formulas given in [528] for 𝑚 ∈ {5, 6} are incorrect. In 2008 Goddard [344] determined correct results for 𝑚 ≤ 8. We state his results for 𝑚 ∈ {5, 6}. For any 𝑛 ≥ 𝑚 = 5, ( 3𝑛+2  + 1 if 𝑛 ≡ 0 (mod 4) 𝛾t (𝑃5 □ 𝑃𝑛 ) =  2  3𝑛+2 otherwise. 2

Theorem 17.14 ([344])

For example, in Figure 17.11, the red vertices form a 𝛾t -set of 𝑃5 □ 𝑃10 . We acknowledge Dr. Östergård for showing us this unusual and interesting example.

Figure 17.11 A minimum TD-set in 𝑃5 □ 𝑃10

Theorem 17.15 ([344]) For any 𝑛 ≥ 𝑚 = 6, (  6𝑛+6  2 − 2 if 𝑛 ≡ 4 (mod 7) 𝛾t (𝑃6 □ 𝑃𝑛 ) =  7  6𝑛+6 2 7 otherwise. For examples, a 𝛾t -set of 𝑃6 □ 𝑃6 is given by the 12 red vertices in Figure 17.12(a), and a 𝛾t -set of 𝑃6 □ 𝑃8 is given by the 16 red vertices in Figure 17.12(b). In 2017 Crevals and Östergård [207] considerably extended the known results by providing formulas for 𝛾t (𝑃𝑚 □ 𝑃𝑛 ) for all 𝑚 ∈ [28]; we only present formulas for 𝑚 ∈ {7, 8, 9, 10}. Theorem 17.16 ([207]) The value of 𝛾t (𝑃𝑚 □ 𝑃𝑛 ), for 7 ≤ 𝑚 ≤ 10 and 𝑛 ≥ 𝑚, is given by the following formulas: ( 2𝑛 + 2 if 𝑛 ≡ 0 (mod 2) or 𝑛 ∈ {9, 11, 15, 21} 𝛾t (𝑃7 □ 𝑃𝑛 ) = 2𝑛 + 1 otherwise  20𝑛+42    if 𝑛 (mod 9) ∈ {0, 7} and 𝑛 ∉ {9, 16}     9  20𝑛+33 𝛾t (𝑃8 □ 𝑃𝑛 ) = if 𝑛 (mod 9) ∈ {2, 3, 4, 5} 9    20𝑛+24    otherwise 9 

Chapter 17. Domination in Grids and Hypercubes

520

(a) 𝛾t -set of 𝑃6 □ 𝑃6

(b) 𝛾t -set of 𝑃6 □ 𝑃8

Figure 17.12 Minimum TD-sets in 𝑃6 □ 𝑃6 and 𝑃6 □ 𝑃8

(  10𝑛+15  𝛾t (𝑃9 □ 𝑃𝑛 ) =  4  10𝑛+11 4

 30𝑛+67     11         30𝑛+56 11 𝛾t (𝑃10 □ 𝑃𝑛 ) =   30𝑛+45    11       30𝑛+34  11

if 𝑛 ≡ 2 (mod 4) otherwise if 𝑛 ≡ 9 (mod 11) and 𝑛 ≠ 20 if 𝑛 (mod 11) ∈ {2, 5, 7} and 𝑛 ∉ {13, 18} if 𝑛 (mod 11) ∈ {0, 1, 3, 6} or 𝑛 = 20 otherwise.

Based on these formulas, Crevals and Östergård [207] formulated four conjectures about a general formula for 𝑛 ≡ 𝑖 (mod 4), for 𝑖 ∈ {0, 1, 2, 3}.

17.3

Domination in Hypercubes

Determining the domination number of the 𝑛-dimensional hypercube 𝑄 𝑛 is a fundamental problem not only in graph theory, but also in areas such as in coding theory and computer science. In the language of coding theory, the problem of determining 𝛾(𝑄 𝑛 ) is that of finding the size of a minimum covering code of length 𝑛 and covering radius 1. In computer science, hypercubes form a central model for interconnection networks, where each node (vertex) in the hypercube 𝑄 𝑛 is labeled with a unique string consisting of 𝑛 bits such that two vertices are adjacent if their labels differ in exactly one bit. This labeling allows for efficient message passing between nodes. Different distribution type problems on such networks can be modeled by domination invariants. Here we will view the hypercube 𝑄 𝑛 from a graph theoretic perspective and represent it as the 𝑛th power of 𝐾2 with respect to the Cartesian product operation □,

Section 17.3. Domination in Hypercubes

521

that is, 𝑄 1 = 𝐾2 and 𝑄 𝑛 = 𝑄 𝑛−1 □ 𝐾2 for 𝑛 ≥ 2. Thus, 𝑄 2 is the cycle 𝐶4 . The first three hypercubes are illustrated in Figure 17.13.

1

0

(a) 𝑄 1

10

11

010

011 110

111

00

01

000

001 100

101

(b) 𝑄 2

(c) 𝑄 3

Figure 17.13 Hypercubes 𝑄 𝑛 for 𝑛 ∈ [3]

In this section, we discuss selected results on domination in hypercubes. The determination of the value of 𝛾(𝑄 𝑛 ) turns out to be an intrinsically difficult problem. To date, exact values are known for two infinite families of hypercubes, namely when 𝑛 = 2 𝑘 or 𝑛 = 2 𝑘 − 1 for all integers 𝑘 ≥ 1. These values are given in Theorem 17.17. Theorem 17.17 ([386, 728]) If 𝑘 ≥ 1, then 𝛾(𝑄 2𝑘 −1 ) = 22 𝑘 22 −𝑘 .

𝑘 −𝑘−1

and 𝛾(𝑄 2𝑘 ) =

The first assertion of Theorem 17.17, due to Harary and Livingston [386] in 1993, is based on the fact that hypercubes 𝑄 2𝑘 −1 contain perfect codes. Since the domination number of a graph with a perfect code is equal to the size of such a code, the assertion follows. Knowing the existence of such codes, by the divisibility condition, one immediately infers that 𝑄 𝑛 contains a perfect code if and only if 𝑛 = 2 𝑘 − 1 for some 𝑘 ≥ 1. Lee [556] further proved that this is equivalent to the fact that 𝑄 𝑛 is a regular covering of the complete graph 𝐾𝑛+1 . The second assertion of Theorem 17.17 is due to Van Wee [728] in 1988. Related aspects of domination in hypercubes were investigated by Weichsel [749] in 1994. Other than the above two infinite families of hypercubes, to date exact values are only known for 𝑛 ≤ 9. These results are summarized in Table 17.3, where the values 𝛾(𝑄 7 ) = 16 and 𝛾(𝑄 8 ) = 32 are special cases of the more general result given in Theorem 17.17.

𝑛

1

2

3

4

5

6

7

8

9

𝛾(𝑄 𝑛 )

1

2

2

4

7

12

16

32

62

Table 17.3 Domination numbers of hypercubes up to dimension 9

The result 𝛾(𝑄 9 ) = 62 due to Östergård and Blass [625] presented a breakthrough back in 2001. The value of 𝛾(𝑄 10 ) is currently unknown. In 2004 Bertolo et al. [70] showed that 𝛾(𝑄 10 ) ≥ 107, and in 2005 Kéri and Östergård [522] proved that

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Chapter 17. Domination in Grids and Hypercubes

𝛾(𝑄 10 ) ≤ 120. These results are currently the best known lower and upper bounds for 𝛾(𝑄 10 ). In 2017 Azarija et al. [42] studied a relation between the domination number and the total domination number in hypercubes. For this purpose, they proved a much more general result. Before stating their result, we recall the notion of a prism of a graph. For a graph 𝐺, the prism of 𝐺 is the Cartesian product 𝐺 □ 𝐾2 of 𝐺 and 𝐾2 . Thus, the prism is obtained by taking two disjoint copies 𝐺 1 and 𝐺 2 of 𝐺, which we will call layers, and adding an edge between each pair of corresponding vertices. If 𝐺 is a bipartite graph, then the prism 𝐺 □ 𝐾2 is called a bipartite prism. For each vertex 𝑣 in 𝐺, we denote its equivalent in 𝐺 𝑖 by 𝑣 𝑖 for 𝑖 ∈ [2], and refer to the vertices 𝑣 1 and 𝑣 2 as partners. The projection 𝜋 : 𝑉 (𝐺 □ 𝐾2 ) → 𝑉 (𝐺) from 𝐺 □ 𝐾2 to 𝐺 defined by 𝜋(𝑣 𝑖 ) = 𝑣 for 𝑖 ∈ [2] is called the natural projection 𝜋 from the prism to the graph. It is immediate that the projection of a dominating set of the prism 𝐺 □ 𝐾2 onto either layer dominates 𝐺. Similarly, taking the same dominating set in each layer provides a TD-set of the prism. This yields the following observation. Observation 17.18 If 𝐺 is a graph, then 𝛾(𝐺) ≤ 𝛾t (𝐺 □ 𝐾2 ) ≤ 2𝛾(𝐺). We are now in a position to discuss the result presented in [42]. The authors proved that the total domination number of a prism 𝐺 □ 𝐾2 of a bipartite graph 𝐺 is equal to twice the domination number of 𝐺. The original proof of Theorem 17.19 given in [42] uses the interplay of transversals in hypergraphs and total domination in graphs. Subsequently, Goddard and Henning [353] presented a simple graph theoretic proof of this result, which we give below. Theorem 17.19 ([42]) If 𝐺 is a bipartite graph, then 𝛾t (𝐺 □ 𝐾2 ) = 2𝛾(𝐺). Proof Let 𝐺 be a bipartite graph, and let 𝐻 = 𝐺 □ 𝐾2 be the bipartite prism of 𝐺. Since 𝐺 is bipartite, the prism 𝐻 is also bipartite. Let 𝑋 and 𝑌 be the partite sets of 𝐻. Further, let 𝜋 be the natural projection from the prism 𝐻 to the graph 𝐺. Open neighborhoods in the prism 𝐻 project onto closed neighborhoods in the graph 𝐺. That is, for any vertex 𝑣 in 𝐻 it holds that N𝐺 [𝜋(𝑣)] = 𝜋(N 𝐻 (𝑣)). For every vertex 𝑥 ∈ 𝑋, all neighbors of 𝑥 in the (bipartite) prism 𝐻 belong to the set 𝑌 , implying that
N𝐺 [𝜋(𝑥)] = 𝜋 N 𝐻 (𝑥) ∩ 𝑌 . In particular, if 𝑇 is a TD-set of 𝐻, then N 𝐻 (𝑥) ∩𝑌 contains a vertex of 𝑇 ∩𝑌 , and so N𝐺 [𝜋(𝑥)] contains a vertex of 𝜋(𝑇 ∩ 𝑌 ). Since 𝜋(𝑋) is all of 𝑉 (𝐺), it follows that 𝜋(𝑇 ∩ 𝑌 ) is a dominating set of 𝐺, and so 𝛾(𝐺) ≤ |𝜋(𝑇 ∩ 𝑌 )|. Similarly, 𝜋(𝑇 ∩ 𝑋) is a dominating set of 𝐺, and so 𝛾(𝐺) ≤ |𝜋(𝑇 ∩ 𝑋)|. Since 𝑇 ∩ 𝑌 and 𝑇 ∩ 𝑋 are disjoint, it follows that 𝛾t (𝐻) = |𝑇 | = |𝑇 ∩ 𝑌 | + |𝑇 ∩ 𝑋 | = |𝜋(𝑇 ∩ 𝑌 )| + |𝜋(𝑇 ∩ 𝑋)| ≥ 2𝛾(𝐺). On the other hand, by Observation 17.18 we have 𝛾t (𝐻) ≤ 2𝛾(𝐺). Consequently, 𝛾t (𝐻) = 2𝛾(𝐺).

Section 17.3. Domination in Hypercubes

523

As shown in [42], the bipartite condition in the statement of Theorem 17.19 is essential. Indeed, they proved that for each integer 𝑘 ≥ 1, there exists a connected graph 𝐺 𝑘 satisfying 𝛾t (𝐺 𝑘 □ 𝐾2 ) − 2𝛾(𝐺 𝑘 ) = 𝑘. Since the hypercube 𝑄 𝑛 , 𝑛 ≥ 1, is a bipartite graph, as a special case of Theorem 17.19 we have the following result. Corollary 17.20 ([42]) For 𝑛 ≥ 1, 𝛾t (𝑄 𝑛+1 ) = 2𝛾(𝑄 𝑛 ). Combining Corollary 17.20 with Theorem 17.17, we deduce the following result. Corollary 17.21 ([42]) If 𝑘 ≥ 1, then 𝛾t (𝑄 2𝑘 +1 ) = 22

𝑘 −𝑘+1

and 𝛾t (𝑄 2𝑘 ) = 22

𝑘 −𝑘

.

The second assertion of Corollary 17.21 is due to Johnson [512] in 1972, see also [746]. As a consequence of Corollary 17.20 and the earlier results of the domination number 𝛾(𝑄 𝑛 ) given in Table 17.3, we also have the following known exact values of the total domination number 𝛾t (𝑄 𝑛 ) for 𝑛 ≤ 10 summarized in Table 17.4. 𝑛

1

2

3

4

5

6

7

8

9

10

𝛾t (𝑄 𝑛 )

2

2

4

4

8

14

24

32

64

124

Table 17.4 Total domination numbers of hypercubes up to dimension 10

In his 2014 Master’s thesis Verstraten [731] studied the total domination number 𝛾t (𝑄 𝑛 ) of the hypercube 𝑄 𝑛 from a coding theory perspective. In particular, the values 𝛾t (𝑄 𝑛 ) summarized in Table 17.4 for 𝑛 ≤ 10 were computed and some bounds established. Let 𝑛, 𝑘, and ℓ be positive integers with 𝑛 > 𝑘 > ℓ. In 2019 Badakhshian et al. [46] defined a bipartite graph 𝐺 𝑘,ℓ with partite sets consisting of all the 𝑘-element subsets and all the ℓ-element subsets of [𝑛], two vertices being adjacent if the corresponding ℓ-element set is a subset of the corresponding 𝑘-element subset. Formally, the graph 𝐺 𝑘,ℓ is a bipartite graph with   𝑉 (𝐺) = 𝐴 : 𝐴 ⊆ [𝑛] and | 𝐴| = 𝑘 ∪ 𝐵 : 𝐵 ⊆ [𝑛] and |𝐵| = ℓ , and the vertices 𝐴 and 𝐵 are adjacent if and only if 𝐴 ⊃ 𝐵. Thus, 𝐺 𝑘,ℓ is the graph defined by the 𝑘 th and ℓ th level of the 𝑛-cube. Badakhshian et al. [46] determined the domination number of 𝐺 𝑘,ℓ for ℓ = 1. Theorem 17.22 ([46]) For 𝑘 ≥ 2, 𝛾(𝐺 𝑘,1 ) = 𝑛 − 𝑘 + 1. Proof We adopt here the notation defined earlier. Let 𝑉 𝐴 and 𝑉𝐵 be the partite sets of 𝐺 𝑘,1 , where the vertices of 𝑉 𝐴 correspond to the 𝑘-element subsets of [𝑛] and the vertices of 𝑉𝐵 correspond to the 𝑛 singleton subsets of [𝑛]. For simplicity, we treat a vertex and its corresponding subset as the same. Consider an arbitrary vertex 𝐴 ⊂ 𝑉 𝐴, that is, 𝐴 is a 𝑘-element subset of [𝑛]. Let 𝑉𝐵′ be the subset of 𝑉𝐵 consisting of all singleton subsets of [𝑛] \ 𝐴. Every vertex in 𝑉𝐵 \ 𝑉𝐵′ is a subset of 𝐴 and

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Chapter 17. Domination in Grids and Hypercubes

is therefore adjacent to the vertex 𝐴 in the graph 𝐺 𝑘,1 . Moreover, every vertex in 𝑉 𝐴 \ {𝐴} is adjacent to at least one vertex in 𝑉𝐵′ in the graph 𝐺 𝑘,1 . Hence, 𝑉𝐵′ ∪ {𝐴} is a dominating set of 𝐺 𝑘,1 , implying that 𝛾(𝐺 𝑘,1 ) ≤ |𝑉𝐵′ | + 1 = 𝑛 − 𝑘 + 1. In order to prove the opposite direction, we show that every dominating set in 𝐺 𝑘,1 contains at least 𝑛 − 𝑘 + 1 vertices. Let 𝐷 be an arbitrary dominating set in 𝐺 𝑘,1 , and let 𝐷 𝐴 = 𝐷 ∩𝑉 𝐴 and 𝐷 𝐵 = 𝐷 ∩𝑉𝐵 . If |𝐷 𝐵 | ≥ 𝑛 − 𝑘 + 1, then |𝐷| ≥ |𝐷 𝐵 | ≥ 𝑛 − 𝑘 + 1, as desired. Hence, we may assume that |𝐷 𝐵 | ≤ 𝑛 − 𝑘. Thus, |𝐷 𝐵 | = 𝑛 − 𝑘 − 𝑡 where 𝑡 ≥ 0. We note that |𝑉𝐵 \ 𝐷 𝐵 | = 𝑘 + 𝑡 and every vertex in the set 𝑉𝐵 \ 𝐷 𝐵 is dominated by the set 𝐷 𝐴 in the graph 𝐺 𝑘,1 . In particular, all 𝑘-element subsets of the elements in the singleton sets of 𝑉𝐵 \ 𝐷 𝐵 belong to 𝐷 𝐴, implying that       1+𝑡 𝑘 +𝑡 𝑘 +𝑡 ≥ = = 𝑡 + 1. |𝐷 𝐴 | ≥ 𝑘 𝑡 𝑡 Therefore, |𝐷| = |𝐷 𝐴 | + |𝐷 𝐵 | ≥ (𝑡 + 1) + (𝑛 − 𝑘 − 𝑡) = 𝑛 − 𝑘 + 1, as desired. This is true for every dominating set 𝐷 of 𝐺 𝑘,1 , implying that 𝛾(𝐺 𝑘,1 ) ≥ 𝑛 − 𝑘 + 1. As observed earlier, 𝛾(𝐺 𝑘,1 ) ≤ 𝑛 − 𝑘 + 1. Consequently, 𝛾(𝐺 𝑘,1 ) = 𝑛 − 𝑘 + 1. Lower and upper bounds on the domination number 𝛾(𝐺 𝑘,2 ) are given in [46] and a conjecture on the asymptotic value of the domination number is given. This conjecture was subsequently proven in 2021 by Balogh et al. [53]. Theorem 17.23 ([53]) For every fixed 𝑘 ≥ 2,   𝑘 +3 𝛾(𝐺 𝑘,2 ) = 𝑛2 + O (𝑛2 ). 2(𝑘 − 1)(𝑘 + 1) As remarked in [53], the case of ℓ > 2 is more complex and the major difficulty in settling this case is that the Turán problem is still open for hypergraphs. An asymptotic lower bound for 𝛾(𝐺 𝑘,ℓ ) for every fixed 𝑘 > ℓ ≥ 2 is also proven in [53], and it is conjectured that the lower bound is asymptotically tight. Conjectures on the asymptotic value of 𝛾(𝐺 5,3 ) and 𝛾(𝐺 4,3 ) are also posed. We omit these very interesting results given in [53] on the domination number of the graph defined by the 𝑘 th and ℓ th level of the 𝑛-cube for selected values of 𝑘 and ℓ. One can easily prove that for any 𝑛 ≥ 4, 𝛾(𝐺 𝑛−1,2 ) = 3.

Chapter 18

Domination and Vizing’s Conjecture 18.1 Introduction In this chapter, we study Vizing’s Conjecture from 1968 which asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. The conjecture was first posed by Vizing as a question in 1963. Vizing’s Conjecture is considered by many to be the main open problem in the area of domination in graphs. We also present Vizing-like conjectures for the total domination number, the independent domination number, the independence number, the upper domination number, and the upper total domination number in Cartesian products of graphs. For a vertex 𝑔 of 𝐺, the subgraph of the Cartesian product 𝐺 □ 𝐻 of two  graphs 𝐺 and 𝐻 induced by the set (𝑔, ℎ) : ℎ ∈ 𝑉 (𝐻) is called an 𝐻-fiber and is denoted by 𝑔 𝐻. Similarly, for ℎ ∈ 𝑉 (𝐻), the 𝐺-fiber 𝐺 ℎ is the subgraph induced by  (𝑔, ℎ) : 𝑔 ∈ 𝑉 (𝐺) . We note that all 𝐺-fibers are isomorphic to 𝐺 and all 𝐻-fibers are isomorphic to 𝐻. The projection to 𝐺 is the map p𝐺 : 𝑉 (𝐺 □ 𝐻) → 𝑉 (𝐺) defined by p𝐺 (𝑔, ℎ) = 𝑔. Similarly, the projection to 𝐻 is the map p 𝐻 : 𝑉 (𝐺 □ 𝐻) → 𝑉 (𝐻) defined by p 𝐻 (𝑔, ℎ) = ℎ. We remark that our drawings of the Cartesian product 𝐺 □ 𝐻 in this chapter differ from the drawings in the previous chapters in that now the copies of the graph 𝐺 are horizontal and appear across the rows, and copies of the graph 𝐻 are vertical and appear in the columns. In other words, we draw the Cartesian product 𝐺 □ 𝐻 on the Cartesian plane with the vertex (𝑔, ℎ) placed on the (𝑔, ℎ) th coordinate. The reason for this convention is to be consistent with the drawings presented in the domination papers written to date on Vizing’s Conjecture.

18.2

Vizing’s Conjecture for the Domination Number

The following conjecture concerning the domination number of a Cartesian product © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_18

525

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was made by the Ukrainian (former Soviet) mathematician Vadim Georgievich Vizing in 1968, after being posed by him in 1963 as a question in [732]. Conjecture 18.1 (Vizing’s Conjecture [734]) For any two graphs 𝐺 and 𝐻, 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻).

(18.1)

Despite the numerous efforts to solve Vizing’s Conjecture over the past 60 years, neither a proof nor a counterexample has been found. Indeed, this simply stated conjecture remains the central open problem in domination theory. The literature on Vizing’s Conjecture was surveyed by Hartnell and Rall [396] in 1998 and by Brešar et al. [113] in 2012. In Vizing’s 1963 paper [732], he observed that a dominating set for 𝐺 □ 𝐻 can be constructed by using a copy of a 𝛾-set of 𝐻 in each 𝐻-fiber or a copy of a 𝛾-set of 𝐺 in each 𝐺-fiber. This establishes the following trivial upper bound on the domination number of 𝐺 □ 𝐻. Proposition 18.2 ([732]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≤ min 𝛾(𝐻)|𝑉 (𝐺)|, 𝛾(𝐺)|𝑉 (𝐻)| .  For two graphs 𝐺 and 𝐻, suppose that 𝛾(𝐺 □ 𝐻) < min |𝑉 (𝐺)|, |𝑉 (𝐻)| . Let 𝑆 be a 𝛾-set of 𝐺 □ 𝐻. In this case, there must exist a vertex 𝑔 ∈ 𝑉 (𝐺) and a vertex ℎ ∈ 𝑉 (𝐻) such that neither the 𝐺-fiber 𝐺 ℎ nor the 𝐻-fiber 𝑔 𝐻 contains a vertex of 𝑆. But this implies that the vertex (𝑔, ℎ) is not dominated by 𝑆 in 𝐺 □ 𝐻, a contradiction. This establishes the following lower bound, first observed in 1991 by El-Zahar and Pareek [254]. Proposition 18.3 ([254]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≥ min |𝑉 (𝐺)|, |𝑉 (𝐻)| . As a consequence of Proposition 18.3, we have the following result. Proposition 18.4 If 𝐺 and 𝐻 are graphs such that |𝑉 (𝐺)| ≥ 𝛾(𝐺) 𝛾(𝐻) and |𝑉 (𝐻)| ≥ 𝛾(𝐺) 𝛾(𝐻), then 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻). √ By Proposition 18.4, if 𝐺 and 𝐻 are graphs of order 𝑛 such that 𝛾(𝐺) ≤ 𝑛 and √ 𝛾(𝐻) ≤ 𝑛, then 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻). A graph 𝐺 satisfies Vizing’s Conjecture if Inequality (18.1) holds for every graph 𝐻. Trivially if 𝛾(𝐺) = 1, then since 𝐺 □ 𝐻 contains a copy of 𝐻, we observe that 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐻) = 𝛾(𝐺) 𝛾(𝐻). Thus, a graph 𝐺 with 𝛾(𝐺) = 1 satisfies Vizing’s Conjecture. Let 𝐺 be a graph that satisfies Vizing’s Conjecture. Suppose that 𝐺 ′ is a spanning subgraph of 𝐺 such that 𝛾(𝐺 ′ ) = 𝛾(𝐺). Since 𝐺 ′ □ 𝐻 is a spanning subgraph of 𝐺 □ 𝐻, we have 𝛾(𝐺 ′ ) 𝛾(𝐻) = 𝛾(𝐺) 𝛾(𝐻) ≤ 𝛾(𝐺 □ 𝐻) ≤ 𝛾(𝐺 ′ □ 𝐻).

Section 18.2. Vizing’s Conjecture for the Domination Number

527

This yields the following result. Proposition 18.5 If 𝐺 is a graph satisfying Vizing’s Conjecture and if 𝐺 ′ is a spanning subgraph of 𝐺 such that 𝛾(𝐺 ′ ) = 𝛾(𝐺), then 𝐺 ′ satisfies Vizing’s Conjecture. Recall that a clique is any complete subgraph of a graph. The clique cover number 𝜃 (𝐺) of a graph 𝐺 is the minimum number of cliques in 𝐺 whose union is 𝑉 (𝐺). One of the most pleasing partial results on Vizing’s Conjecture is about socalled decomposable graphs, a term coined in 1979 by Barcalkin and German [60]. A decomposable graph is a graph 𝐺 whose vertex set can be covered by 𝛾(𝐺) cliques, or equivalently, a graph for which 𝜃 (𝐺) = 𝛾(𝐺). Barcalkin and German [60] proved that every decomposable graph satisfies Vizing’s Conjecture. By Proposition 18.5, every spanning subgraph 𝐺 ′ of a decomposable graph 𝐺 such that 𝛾(𝐺 ′ ) = 𝛾(𝐺) satisfies Vizing’s Conjecture. Motivated by these observations, a graph 𝐺 ′ is called a BG-graph if 𝐺 ′ is a spanning subgraph of some graph 𝐺 such that 𝛾(𝐺 ′ ) = 𝜃 (𝐺) = 𝛾(𝐺). Classes of BG-graphs include trees, cycles, all graphs 𝐺 with 𝛾(𝐺) = 2, and all graphs with equal domination and packing numbers. As observed earlier, a graph 𝐺 with 𝛾(𝐺) = 1 satisfies Vizing’s Conjecture. The only other classes of graphs that are currently known to satisfy Vizing’s Conjecture are the following: (a) BG-graphs [60] (b) chordal graphs [8] (c) graphs with domination number two [60] (d) graphs with domination number three [111, 698] (e) Type X graphs [395] (f) graphs whose fair domination number and domination number are equal [116].

18.2.1 A Framework The two main approaches to settling Vizing’s Conjecture are the following. The first approach, pioneered in 1979 by Barcalkin and German [60], restricts one of the factors of the product to some (large) class of graphs (such as chordal graphs or BG-graphs or graphs with domination number 3) and proves that for such a class, Vizing’s Conjecture is satisfied. The second approach uses what is called the double-projection argument, pioneered in 2000 by Clark and Suen [181, 692] and proves a weaker inequality related to Vizing’s Conjecture that holds for all pairs of graphs. In this section, we present a framework to Vizing’s Conjecture that unifies the above two main approaches to the conjecture. This framework, given in 2021 by Brešar et al. [114], partitions the vertex set of the Cartesian product of two graphs into cells. Each cell in the partition is colored with one of four colors, with the coloring of the cell determined by properties of the vertices in that cell with respect to a given dominating set. Let 𝐺 be a graph with 𝛾(𝐺) = 𝑘 and let {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } be a 𝛾-set of 𝐺. Let 𝜋 = {𝜋1 , 𝜋2 , . . . , 𝜋 𝑘 } be a partition of 𝑉 (𝐺) such that 𝑢 𝑖 ∈ 𝜋𝑖 and 𝜋𝑖 ⊆ N[𝑢 𝑖 ] for each 𝑖 ∈ [𝑘]. Given a graph 𝐻, let Π𝑖𝐻 = 𝜋𝑖 × 𝑉 (𝐻). For a vertex ℎ of 𝐻, the set

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Chapter 18. Domination and Vizing’s Conjecture

of vertices 𝜋𝑖 × {ℎ} is called a cell and is denoted by 𝜋𝑖ℎ . We note that the cell 𝜋𝑖ℎ belongs to the fiber 𝐺 ℎ . We also say that the cell 𝜋𝑖ℎ belongs to Π𝑖𝐻 and that Π𝑖𝐻 is the union of the cells 𝜋𝑖ℎ , that is, Ø 𝜋𝑖ℎ . Π𝑖𝐻 = ℎ∈𝑉 (𝐻 )

We call Π𝑖𝐻 the block of 𝐺 □ 𝐻 associated with the partition 𝜋 for 𝑖 ∈ [𝑘]. We note that 𝑘 Ø 𝑉 (𝐺 □ 𝐻) = Π𝑖𝐻 . 𝑖=1

Let 𝐷 be a 𝛾-set of 𝐺 □ 𝐻. For 𝑖 ∈ [𝑘], let 𝐷 𝑖 be the set of vertices of 𝐷 that belong to the block Π𝑖𝐻 and so 𝐷 𝑖 = 𝐷 ∩ Π𝑖𝐻 . For each vertex ℎ ∈ 𝑉 (𝐻), let 𝐷 ℎ be the set of vertices of 𝐷 that belong to the 𝐺-fiber 𝐺 ℎ , that is,
𝐷ℎ = 𝐷 ∩ 𝑉 𝐺ℎ . Suppose that 𝜋𝑖ℎ ∩ 𝐷 ≠ ∅ and so the cell 𝜋𝑖ℎ contains a vertex of 𝐷. In this case, we color the cell either blue or yellow as follows. If 𝜋𝑖ℎ is dominated by 𝐷 ℎ , then we color the cell 𝜋𝑖ℎ blue. In this case, we also say that 𝜋𝑖ℎ is horizontally dominated by 𝐷. If 𝜋𝑖ℎ is not horizontally dominated by 𝐷 ℎ , then we color the cell 𝜋𝑖ℎ yellow. Suppose that 𝜋𝑖ℎ ∩ 𝐷 = ∅. If 𝜋𝑖ℎ is horizontally dominated by 𝐷 ℎ and no vertex of 𝜋𝑖ℎ is dominated by 𝐷 𝑖 , then we color the cell 𝜋𝑖ℎ red. If some vertex of 𝜋𝑖ℎ is dominated by 𝐷 𝑖 , then we color the cell 𝜋𝑖ℎ white. We note that every vertex of 𝐷 belongs to a cell colored blue or yellow. We color the vertices of 𝐷 that belong to blue cells (respectively, yellow cells) with the color blue (respectively, yellow). Thus, if 𝜋𝑖ℎ is a blue cell, then the vertices in 𝐷 ∩ 𝜋𝑖ℎ are colored blue; while if 𝜋𝑖ℎ is a yellow cell, then the vertices in 𝐷 ∩ 𝜋𝑖ℎ are colored yellow. This coloring of the vertices of 𝐷 is a partition of 𝐷 into subsets of blue vertices and yellow vertices. We note that within the Cartesian product 𝐺 □ 𝐻, the only vertices that have a color are the blue vertices (the vertices from 𝐷 that belong to a blue cell) and the yellow vertices (the vertices of 𝐷 that belong to a yellow cell). To illustrate this method of coloring cells consider the following example. Let 𝐺 and 𝐻 be the graphs shown in Figure 18.1, where 𝑉 (𝐺) = {𝑔1 , 𝑔2 , . . . , 𝑔8 } and 𝑉 (𝐻) = {ℎ1 , ℎ2 , ℎ3 , ℎ4 }. The Cartesian product 𝐺 □ 𝐻 is illustrated in Figure 18.2, where we omit the edges in order to simplify the drawing and focus on the cell coloring. We note that 𝛾(𝐺) = 3 and that the set {𝑔3 , 𝑔5 , 𝑔8 } is an example of a 𝛾-set of 𝐺. We can therefore use the partition 𝜋 = {𝜋1 , 𝜋2 , 𝜋3 }, where 𝜋1 = {𝑔1 , 𝑔2 , 𝑔3 }, 𝜋2 = {𝑔4 , 𝑔5 , 𝑔6 }, and 𝜋3 = {𝑔7 , 𝑔8 }. We note that 𝑔3 ∈ 𝜋1 ⊆ N[𝑔3 ], 𝑔5 ∈ 𝜋2 ⊆ N[𝑔5 ], and 𝑔8 ∈ 𝜋3 ⊆ N[𝑔8 ]. In this example, the cell 𝜋1ℎ1 = 𝜋1 × {ℎ1 } = (𝑔1 , ℎ1 ),  (𝑔2 , ℎ1 ), (𝑔3 , ℎ1 ) , 𝜋2ℎ3 = 𝜋2 × {ℎ3 } = (𝑔4 , ℎ3 ), (𝑔5 , ℎ3 ), (𝑔6 , ℎ3 ) , and so on.

Section 18.2. Vizing’s Conjecture for the Domination Number

529

𝑔1 ℎ4 𝑔4

𝑔2

𝑔3

𝑔5

ℎ3 ℎ2

𝑔8 𝑔6

𝑔7

ℎ1

(a) 𝐺

(b) 𝐻

Figure 18.1 Graphs 𝐺 and 𝐻

We note that 𝛾(𝐺 □ 𝐻) = 8 and that the set  𝐷 = (𝑔1 , ℎ1 ), (𝑔1 , ℎ2 ), (𝑔2 , ℎ1 ), (𝑔4 , ℎ3 ), (𝑔4 , ℎ4 ), (𝑔6 , ℎ1 ), (𝑔6 , ℎ2 ), (𝑔8 , ℎ3 ) is an example of a 𝛾-set of 𝐺 □ 𝐻 (of minimum cardinality 8), where the vertices of 𝐷 are circled in Figure 18.2.

ℎ4 ℎ3
퐻 ℎ2 ℎ1


푔1


푔2


푔3


푔4


푔5


푔6


푔7


푔8
퐺


휋 1


휋 2


휋 3

Figure 18.2 An example of a cell coloring in 𝐺 □ 𝐻 With the 𝛾-set 𝐷 of 𝐺 □ 𝐻 and the partition 𝜋 = {𝜋1 , 𝜋2 , 𝜋3 } defined earlier, and using the definitions of a colored cell, we find that there are two blue cells, five yellow cells, one red cell, and four white cells, as described in Table 18.1.

In the projection p 𝐻 Π𝑖𝐻 of the cells of Π𝑖𝐻 to 𝐻, the vertex ℎ = p 𝐻 𝜋𝑖ℎ receives the color of the cell 𝜋𝑖ℎ for each 𝑖 ∈ [𝑘]. Let 𝐵𝑖 , 𝑌𝑖 , and 𝑅𝑖 be the resulting set of vertices in 𝐻 colored blue, yellow, and red, respectively, for 𝑖 ∈ [𝑘]. We shall also need the following notation, introduced in [114], yielding information about the cardinality of some of the sets having certain colors. For a vertex ℎ ∈ 𝑉 (𝐻), let 𝑏 ′ℎ be the number of blue cells in the 𝐺-fiber 𝐺 ℎ, let 𝑏 𝑖′ the number of blue cells

Chapter 18. Domination and Vizing’s Conjecture

530

color of cell blue yellow red white

description of cell 𝜋1ℎ1 , 𝜋3ℎ3 𝜋1ℎ2 , 𝜋2ℎ1 , 𝜋2ℎ2 , 𝜋2ℎ3 , 𝜋2ℎ4 𝜋3ℎ1 𝜋1ℎ3 , 𝜋1ℎ4 , 𝜋3ℎ2 , 𝜋3ℎ4

Table 18.1 The coloring of the 12 cells of 𝐺 □ 𝐻

in Π𝑖𝐻 , and let 𝑏 ′ the total number of all blue cells in 𝐺 □ 𝐻. We define analogously 𝑦 ′ℎ , 𝑦 𝑖′ , and 𝑦 ′ associated with the yellow cells and 𝑟 ℎ′ , 𝑟 𝑖′ , and 𝑟 ′ associated with red cells. To illustrate these definitions, consider our earlier example of the Cartesian product 𝐺 □ 𝐻 illustrated in Figure 18.2. Here we have 𝑏 ′ℎ1 = 𝑦 ′ℎ1 = 𝑟 ℎ′ 1 = 1, noting that there is one blue, one yellow, and one red cell in the 𝐺-fiber 𝐺 ℎ1 . Since there is one blue, one yellow, and two white cells in the block Π1𝐻 , we have 𝑏 1′ = 𝑦 1′ = 1 and 𝑟 1′ = 0. The total number of blue, yellow, and red cells in 𝐺 □ 𝐻 is 2, 5, and 1, respectively (see Table 18.1), and so 𝑏 ′ = 2, 𝑦 ′ = 5, and 𝑟 ′ = 1. For a vertex ℎ ∈ 𝑉 (𝐻), let 𝑏 ℎ be the number of blue vertices in the 𝐺-fiber 𝐺 ℎ , and let 𝑏 𝑖 denote the number of blue vertices in the block Π𝑖𝐻 for 𝑖 ∈ [𝑘]. Further, let 𝑏 be the total number of blue vertices in 𝐺 □ 𝐻. In an analogous way, we define 𝑦 ℎ , 𝑦 𝑖 , and 𝑦, associated with the yellow vertices. As remarked earlier, the only colored vertices are vertices of 𝐷 that belong to cells colored blue or yellow. Thus, |𝐷 | = 𝑏 + 𝑦. Since each blue cell contains at least one blue vertex, the number of blue vertices in the 𝐺-fiber 𝐺 ℎ is at least the number of blue cells in 𝐺 ℎ , that is 𝑏 ℎ ≥ 𝑏 ′ℎ . Further, the number of blue vertices in the block Π𝑖𝐻 is at least the number of blue cells in Π𝑖𝐻 , that is 𝑏 𝑖 ≥ 𝑏 𝑖′ . Moreover, the number of blue vertices in 𝐺 □ 𝐻 is at least the number of blue cells in 𝐺 □ 𝐻, that is, 𝑏 ≥ 𝑏 ′ . Analogously, 𝑦 ℎ ≥ 𝑦 ′𝑦 , 𝑦 𝑖 ≥ 𝑦 𝑖′ , and 𝑦 ≥ 𝑦 ′ . To illustrate these definitions, consider again our example illustrated in Figure 18.2. Here the vertices (𝑔1 , ℎ1 ) ∈ 𝐷 and (𝑔2 , ℎ1 ) ∈ 𝐷 both belong to the blue cell 𝜋1ℎ1 and are therefore colored blue, while the vertex (𝑔6 , ℎ1 ) ∈ 𝐷 that belongs to the yellow cell 𝜋2ℎ1 is colored yellow. Hence, in the 𝐺-fiber 𝐺 ℎ1 , there are two blue vertices and one yellow vertex, that is, 𝑏 ℎ1 = 2 and 𝑦 ℎ1 = 1. The block Π1𝐻 contains two blue vertices, namely (𝑔1 , ℎ1 ) and (𝑔2 , ℎ1 ), and one yellow vertex, namely (𝑔1 , ℎ2 ), and so 𝑏 1 = 2 and 𝑦 1 = 1. The total number of blue and yellow vertices in 𝐺 □ 𝐻 is 3 and 5, respectively, and so 𝑏 = 3 and 𝑦 = 5. We next consider the Π𝑖𝐻 for each 𝑖 ∈ [𝑘]. Recall that the
red cells in the block 𝐻 𝐻 projection map p 𝐻 Π𝑖 maps the block Π𝑖 onto the graph
𝐻, and by this projection, we color the vertices of 𝐻 so that the vertex ℎ = p 𝐻 𝜋𝑖ℎ receives the color of the cell 𝜋𝑖ℎ . Let 𝑅𝑖 be the set of vertices in 𝐻 colored red (by the projection), and let 𝑠𝑖′ be the minimum number of vertices in 𝐻 needed to dominate the set 𝑅𝑖 . Let 𝑘 ∑︁ 𝑠′ = 𝑠𝑖′ . 𝑖=1

Section 18.2. Vizing’s Conjecture for the Domination Number

531

To illustrate this definition, consider the block Π3𝐻 from our earlier example illustrated in Figure 18.2. The block Π3𝐻 consists of four cells, namely 𝜋3ℎ1 , 𝜋3ℎ2 , 𝜋3ℎ3 , and 𝜋3ℎ4 which are colored red, white, blue, and white, respectively. Thus, in the projection map, the vertices ℎ1 , ℎ2 , ℎ3 , and ℎ4 are colored red, white, blue, and white, respectively. The set 𝑅3 = {ℎ1 } and since one vertex in 𝐻 is needed to dominate the set 𝑅1 , we have 𝑠3′ = 1. Since the blocks Π1𝐻 and Π2𝐻 contain no red cell, 𝑅1 = 𝑅2 = ∅ and 𝑠1′ = 𝑠2′ = 0. Hence, 𝑠′ = 𝑠1′ + 𝑠2′ + 𝑠3′ = 1.

18.2.2

Key Preliminary Lemmas

In this section, we present five key preliminary lemmas that will be used in the proofs of the main results in the next section. Throughout this section, we follow the notation introduced in Section 18.2.1. In particular, recall that 𝐺 and 𝐻 are graphs, 𝛾(𝐺) = 𝑘, and 𝐷 is a 𝛾-set of 𝐺 □ 𝐻. Further, we assume that the vertices of 𝐺 □ 𝐻 are assigned colors based on 𝐷 and the coloring method described in Section 18.2.1. Lemma 18.6 ([114])

𝑏 ′ + 𝑦 ′ + 𝑟 ′ ≥ 𝛾(𝐺) 𝛾(𝐻).

Proof Consider the cells in the block Π𝑖𝐻 for some 𝑖
∈ [𝑘]. Let 𝑆 be the set of vertices in 𝐻 that receive, by the projection map p 𝐻 Π𝑖𝐻 onto 𝐻, one of the colors blue, yellow, or red, and so the vertices in 𝑉 (𝐻) \ 𝑆 are all colored white (by the projection). We show that 𝑆 is a dominating set of 𝐻, that is, we show that every white vertex in the projection is adjacent to a vertex of 𝑆. Let ℎ be a white vertex of 𝐻 in the projection. Hence, the vertex ℎ is projected from a white cell 𝜋𝑖ℎ in 𝐺 □ 𝐻. By definition, the white cell 𝜋𝑖ℎ contains no vertex of 𝐷 and contains a vertex (𝑔, ℎ) that is dominated in 𝐺 □ 𝐻 by a vertex (𝑔, ℎ′ ) ∈ 𝐷 𝑖 . We note that the vertex (𝑔, ℎ′ ) is ′
′ ℎ colored blue or yellow, implying that the vertex ℎ = p 𝐻 𝜋𝑖 received color blue or yellow, and so ℎ′ ∈ 𝑆. Further, the vertex ℎ′ is adjacent to the vertex ℎ in 𝐻, implying that the set 𝑆 is a dominating set of 𝐻. Thus, 𝛾(𝐻) ≤ |𝑆| = 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ . This is true for every 𝑖 ∈ [𝑘], implying that 𝑏′ + 𝑦′ + 𝑟 ′ =

𝑘 ∑︁ 𝑖=1

𝑘
∑︁ 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ ≥ 𝛾(𝐻) = 𝛾(𝐺) 𝛾(𝐻). 𝑖=1

We show next that Lemma 18.6 can be improved as follows. Lemma 18.7 ([114])

𝑏 ′ + 𝑦 ′ + 𝑠′ ≥ 𝛾(𝐺) 𝛾(𝐻).

𝐻 Proof Consider the
cells in the block Π𝑖 for
some 𝑖 ∈ [𝑘] and, as before, in the 𝐻 ℎ receives the color of the cell 𝜋 ℎ . In projection p 𝐻 Π𝑖 to 𝐻, the vertex ℎ = p 𝜋 𝐻 𝑖 𝑖
the projection p 𝐻 Π𝑖𝐻 , let 𝐵𝑖 , 𝑌𝑖 , and 𝑅𝑖 be the set of vertices in 𝐻 colored blue, yellow, and red, respectively. We note that |𝐵𝑖 | = 𝑏 𝑖′ , |𝑌𝑖 | = 𝑦 𝑖′ , and |𝑅𝑖 | = 𝑟 𝑖′ . Let 𝑆𝑖 be a minimum set of vertices in 𝐻 that dominates the set 𝑅𝑖 , and so |𝑆𝑖 | = 𝑠𝑖′ . As shown in the proof of Lemma 18.6, every white vertex of 𝐻 in the projection is dominated by a blue or yellow vertex, while every red vertex of 𝐻 (that belongs to the set 𝑅𝑖 ) is dominated by the set 𝑆𝑖 . Hence, every white vertex in the projection is adjacent to a

Chapter 18. Domination and Vizing’s Conjecture

532

vertex of 𝐵𝑖 ∪ 𝑌𝑖 and every red vertex in the projection is adjacent to a vertex of 𝑆𝑖 , implying that the set 𝐵𝑖 ∪ 𝑌𝑖 ∪ 𝑆𝑖 is a dominating set of 𝐻. Thus, 𝛾(𝐻) ≤ 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑠𝑖′ . This is true for every 𝑖 ∈ [𝑘], implying that 𝑏 ′ + 𝑦 ′ + 𝑠′ =

𝑘 ∑︁

𝑘
∑︁ 𝛾(𝐻) = 𝛾(𝐺) 𝛾(𝐻). 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑠𝑖′ ≥ 𝑖=1

𝑖=1

Lemma 18.8 ([114]) 𝑟 ′ ≤ 𝑏 − 𝑏 ′ + 𝑦. Proof Let ℎ ∈ 𝑉 (𝐻) and consider the blue and yellow vertices in the 𝐺-fiber 𝐺 ℎ . We note that these 𝑏 ℎ + 𝑦 ℎ vertices
dominate all vertices in the red and blue cells in 𝐺 ℎ . In the projection p𝐺 𝐺 ℎ of the 𝐺-fiber 𝐺 ℎ onto 𝐺, the vertices projected from the blue and yellow vertices therefore dominate all vertices projected from red and blue cells in 𝐺 ℎ . To this set of projected vertices from the blue and yellow vertices, we add all vertices 𝑢 𝑖 from every projected white and yellow cell 𝜋𝑖ℎ . We note that each such added vertex 𝑢 𝑖 dominates all the vertices projected from the white and yellow cells 𝜋𝑖ℎ . Let 𝐴ℎ denote the resulting set of vertices in 𝐺, that is,
 𝐴ℎ = p𝐺 𝐷 ∩ 𝑉 (𝐺 ℎ ) ∪ 𝑢 𝑖 ∈ [𝑘] : 𝜋𝑖ℎ is a white or yellow cell . The set 𝐴ℎ is a dominating set of 𝐺. Since the number of white and yellow cells in 𝐺 ℎ is 𝑘 − 𝑏 ′ℎ − 𝑟 ℎ′ ,
𝑘 = 𝛾(𝐺) ≤ | 𝐴ℎ | = 𝑏 ℎ + 𝑦 ℎ + 𝑘 − 𝑏 ′ℎ − 𝑟 ℎ′ or, equivalently, 𝑟 ℎ′ ≤ 𝑏 ℎ − 𝑏 ′ℎ + 𝑦 ℎ . Summing over all vertices ℎ ∈ 𝑉 (𝐻), we have ∑︁ ∑︁
𝑟′ = 𝑟 ℎ′ ≤ 𝑏 ℎ − 𝑏 ′ℎ + 𝑦 ℎ = 𝑏 − 𝑏 ′ + 𝑦. ℎ∈𝑉 (𝐻 )

ℎ∈𝑉 (𝐻 )

Lemma 18.9 ([114]) 2𝑏 ′ + 𝑦 ′ + 𝑟 ′ ≥ 𝛾(𝐺) 𝛾t (𝐻). Proof Consider the cells in the block Π𝑖𝐻 for some 𝑖 ∈ [𝑘]. We adopt the notation in
the proof of Lemma 18.7. In particular, in the projection p 𝐻 Π𝑖𝐻 , let 𝐵𝑖 , 𝑌𝑖 , and 𝑅𝑖 be the set of vertices in 𝐻 colored blue, yellow, and red, respectively, and so |𝐵𝑖 | = 𝑏 𝑖′ , |𝑌𝑖 | = 𝑦 𝑖′ , and |𝑅𝑖 | = 𝑟 𝑖′ . For each vertex ℎ ∈ 𝐵𝑖 ∪ 𝑅𝑖 , let 𝑤 ℎ be a neighbor of ℎ in 𝐻, and let Ø 𝑊𝑖 = {𝑤 ℎ }. ℎ∈ 𝐵𝑖 ∪𝑅𝑖

We note that |𝑊𝑖 | ≤ |𝐵𝑖 | + |𝑅𝑖 | = 𝑏 𝑖′ + 𝑟 𝑖′ . Let 𝑋𝑖 = 𝐵𝑖 ∪ 𝑌𝑖 ∪ 𝑊𝑖 . We show that the set 𝑋𝑖 is a TD-set of 𝐻. Every white vertex in 𝐻 has a neighbor in 𝐵𝑖 ∪ 𝑌𝑖 and every blue and red vertex has a neighbor in 𝑊𝑖 . Hence, it suffices to show that every yellow vertex in 𝐻 has a neighbor in 𝑋𝑖 . Let ℎ be an arbitrary yellow vertex in 𝐻 and let 𝜋𝑖ℎ be the corresponding yellow cell in the block Π𝑖𝐻 . Since the cell 𝜋𝑖ℎ is not horizontally dominated, it contains a vertex (𝑔, ℎ) that is dominated in 𝐺 □ 𝐻 by a vertex (𝑔, ℎ′ ) ∈ 𝐷 𝑖 for some 𝑔 ∈ 𝑉 (𝐺) and some neighbor ℎ′ of ℎ in 𝐻. Since ′ the vertex (𝑔, ℎ′ ) is yellow or blue, the cell 𝜋𝑖ℎ is yellow or blue. Therefore, the

Section 18.2. Vizing’s Conjecture for the Domination Number

533


vertex ℎ′ received the color yellow or blue in the projection p 𝐻 Π𝑖𝐻 , implying that the vertex ℎ is totally dominated by a vertex of 𝐵𝑖 ∪ 𝑌𝑖 . Thus, every yellow vertex in 𝐻 has a neighbor in 𝐵𝑖 ∪ 𝑌𝑖 . These observations imply that the set 𝑋𝑖 is a TD-set of 𝐻. Thus, since |𝐵𝑖 | = 𝑏 𝑖′ , |𝑌𝑖 | = 𝑦 𝑖′ , and |𝑊𝑖 | ≤ 𝑏 𝑖′ + 𝑟 𝑖′ ,
𝛾t (𝐻) ≤ |𝑋𝑖 | ≤ |𝐵𝑖 | + |𝑌𝑖 | + |𝑊𝑖 | ≤ 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑏 𝑖′ + 𝑟 𝑖′ = 2𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ . This is true for every 𝑖 ∈ [𝑘], implying that 2𝑏 ′ + 𝑦 ′ + 𝑟 ′ =

𝑘 ∑︁ 𝑖=1

𝑘
∑︁ 𝛾t (𝐻) = 𝑘𝛾t (𝐻) = 𝛾(𝐺) 𝛾t (𝐻). 2𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ ≥ 𝑖=1

Lemma 18.10 ([114]) Renaming the graphs 𝐺 and 𝐻 if necessary, 𝑏 ≥ 𝛾(𝐺). Proof Suppose that the projection p𝐺 (𝐷) of 𝐷 onto 𝐺 is not the entire set 𝑉 (𝐺). In this case, there is a vertex 𝑔 ∈ 𝑉 (𝐺) such that the 𝐻-fiber 𝑔 𝐻 contains no vertex of 𝐷. Let ℎ be an arbitrary vertex of 𝐻. If the 𝐺-fiber 𝐺 ℎ contains no vertex of 𝐷, then the vertex (𝑔, ℎ) is not dominated by 𝐷 in 𝐺 □ 𝐻, contradicting the fact that 𝐷 is a dominating set of 𝐺 □ 𝐻. Hence, the 𝐺-fiber 𝐺 ℎ contains at least one vertex of 𝐷 for every vertex ℎ ∈ 𝑉 (𝐻), implying that the projection p 𝐻 (𝐷) of 𝐷 onto 𝐻 is the entire set 𝑉 (𝐻). Renaming the graphs 𝐺 and 𝐻 if necessary, we may therefore assume that the projection p𝐺 (𝐷) of 𝐷 onto 𝐺 is the entire set 𝑉 (𝐺). Hence, each vertex of 𝑔 has at least one vertex of 𝐷 mapped onto it by the projection map p𝐺 (𝐷). In particular, for each vertex 𝑢 𝑖 belonging to the 𝛾-set {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } of 𝐺 that formed our partition 𝜋 = {𝜋1 , 𝜋2 , . . . , 𝜋 𝑘 }, there is a vertex (𝑢 𝑖 , ℎ) that belongs to the set 𝐷 for some ℎ ∈ 𝑉 (𝐻). Therefore, the cell 𝜋𝑖ℎ contains a vertex of 𝐷 that dominates all the vertices in that cell, implying that such a cell is colored blue. Hence, for each vertex of the 𝛾-set {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } of 𝐺, we can associate at least one blue cell, implying that there are at least 𝑘 = 𝛾(𝐺) blue cells. Hence, 𝑏 ≥ 𝑏 ′ ≥ 𝛾(𝐺).

18.2.3

Classical Results Related to Vizing’s Conjecture

We are now in a position to show that the framework presented in Section 18.2.1 yields transparent proofs of classical results related to Vizing’s Conjecture, including the 1979 result due to Barcalkin and German [60], the important 2000 result due to Clark and Suen [181], the 2012 result due to Suen and Tarr [692], and the 2019 result due to Zerbib [785]. The new framework therefore unifies the classical results on the conjecture and combines them into a more transparent form. The proofs we present in this section are from Brešar et al. [114]. We continue to use the notation introduced in Section 18.2.1. In particular, 𝐺 and 𝐻 are graphs, 𝛾(𝐺) = 𝑘, and 𝐷 is a 𝛾-set of 𝐺 □ 𝐻. Further, the vertices of 𝐺 □ 𝐻 are assigned colors based on 𝐷 and the coloring method described in Section 18.2.1. As an immediate consequence of Lemmas 18.6 and 18.8, we have the following classical 2000 result due to Clark and Suen [181].

534

Chapter 18. Domination and Vizing’s Conjecture

Theorem 18.11 ([181]) For any two graphs 𝐺 and 𝐻, 𝛾(𝐺 □ 𝐻) ≥ 21 𝛾(𝐺) 𝛾(𝐻). Proof Let 𝐷 be a 𝛾-set of 𝐺 □ 𝐻. Adopting the notation introduced in Section 18.2.1, we note that |𝐷| = 𝑏 + 𝑦 and 𝑦 ≥ 𝑦 ′ . Thus, by Lemmas 18.6 and 18.8, 𝛾(𝐺) 𝛾(𝐻) ≤ 𝑏 ′ + 𝑦 ′ + 𝑟 ′
≤ 𝑏′ + 𝑦′ + 𝑏 − 𝑏′ + 𝑦 = 𝑏 + 𝑦 + 𝑦′ ≤ 2(𝑏 + 𝑦) = 2|𝐷| = 2𝛾(𝐺 □ 𝐻), or equivalently, 𝛾(𝐺 □ 𝐻) ≥ 12 𝛾(𝐺) 𝛾(𝐻). We show next that Vizing’s Conjecture holds if there are no yellow cells in 𝐺 □ 𝐻. Theorem 18.12 ([114]) If 𝐷 is a 𝛾-set of 𝐺 □ 𝐻 such that there is no yellow cell in 𝐺 □ 𝐻, then 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻). Proof Let 𝐷 be a 𝛾-set of 𝐺 □ 𝐻. Suppose there is no yellow cell in 𝐺 □ 𝐻, and so 𝑦 = 𝑦 ′ = 0. Thus, |𝐷| = 𝑏 + 𝑦 = 𝑏. By Lemma 18.8, 𝑏 ≥ 𝑏 ′ + 𝑟 ′ − 𝑦 = 𝑏 ′ + 𝑟 ′ , and by Lemma 18.6, 𝑏 ′ + 𝑟 ′ = 𝑏 ′ + 𝑦 ′ + 𝑟 ′ ≥ 𝛾(𝐺) 𝛾(𝐻). Hence, 𝛾(𝐺 □ 𝐻) = |𝐷 | = 𝑏 ≥ 𝑏 ′ + 𝑟 ′ ≥ 𝛾(𝐺) 𝛾(𝐻). As a consequence of Theorem 18.12, we have the classical 1979 result of Barcalkin and German [60]. Theorem 18.13 ([60]) Vizing’s Conjecture is true for decomposable graphs. Proof Suppose that 𝐺 is a decomposable graph. Thus, 𝑉 (𝐺) can be partitioned into 𝛾(𝐺) subsets each of which induces a clique in 𝐺. We can therefore choose our partition 𝜋 = {𝜋1 , 𝜋2 , . . . , 𝜋 𝑘 } of 𝑉 (𝐺), where 𝑘 = 𝛾(𝐺), so that the sets in 𝜋 induce cliques in 𝐺. This implies that every cell 𝜋𝑖ℎ of 𝐺 □ 𝐻 that contains a vertex of 𝐷 for some 𝑖 ∈ [𝑘] and ℎ ∈ 𝑉 (𝐻) is such that the cell 𝜋𝑖ℎ is dominated by that vertex of 𝐷, and is therefore colored blue. In particular, there are no yellow cells in 𝐺 □ 𝐻, and so 𝑦 = 0. Hence, by Theorem 18.12, we have 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻). We show next that the coloring framework can be used to prove the important 2012 result on Vizing’s Conjecture by Suen and Tarr [692] that was historically the first improvement of the Clark-Suen Theorem 18.11. Theorem 18.14 (Suen-Tarr Theorem [692]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≥ 12 𝛾(𝐺) 𝛾(𝐻) + 12 min 𝛾(𝐺), 𝛾(𝐻) .

Section 18.2. Vizing’s Conjecture for the Domination Number

535

Proof Let 𝐷 be a 𝛾-set of 𝐺 □ 𝐻. By Lemma 18.10, we may assume that 𝑏 ≥ 𝛾(𝐺). Recall that |𝐷| = 𝑏 + 𝑦 and 𝑦 ≥ 𝑦 ′ . As shown in the proof of Theorem 18.11, we have 𝛾(𝐺) 𝛾(𝐻) ≤ 𝑏 + 𝑦 + 𝑦 ′ . By assumption, 𝑏 ≥ 𝛾(𝐺). Hence,
𝛾(𝐺) + 𝛾(𝐺) 𝛾(𝐻) ≤ 𝑏 + 𝑏 + 𝑦 + 𝑦 ′ = 2𝑏 + 𝑦 + 𝑦 ′ ≤ 2(𝑏 + 𝑦) = 2|𝐷 | = 2𝛾(𝐺 □ 𝐻).  Since 𝛾(𝐺) ≥ min 𝛾(𝐺), 𝛾(𝐻) , the desired lower bound in the statement of the theorem follows. In 2019 Zerbib [785] showed that the Suen-Tarr Theorem 18.14 can be improved. The proof we present of Zerbib’s theorem again uses the coloring framework of [114], except that the 𝛾-set of 𝐺 is chosen differently than the way it was chosen before. However, the coloring, projecting, and counting proceeds exactly as in the underlying framework, noting that the color of the resulting cells determines the color of the set of vertices in the dominating set 𝐷 of the Cartesian product 𝐺 □ 𝐻. The proof of Theorem 18.15 follows and is given in [114]. Theorem 18.15 ([785]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≥ 12 𝛾(𝐺) 𝛾(𝐻) + 12 max 𝛾(𝐺), 𝛾(𝐻) . Proof Renaming the graphs if necessary, we may assume that 𝛾(𝐺) ≥ 𝛾(𝐻). In what follows, we adopt precisely the notation defined in Section 18.2.1 for our initial framework, except that here 𝑘 ≥ 𝛾(𝐺) (recalling that previously, 𝑘 = 𝛾(𝐺)) and the dominating set {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } of 𝐺 is defined differently. Let 𝐷 be a dominating set of 𝐺 □ 𝐻 and let 𝐷 𝐺 be the projection p𝐺 (𝐷) of the set 𝐷 to the graph 𝐺. Since 𝐷 is a 𝛾-set of 𝐺 □ 𝐻, the set 𝐷 𝐺 is a dominating set of 𝐺. Among all subsets of vertices of 𝐷 𝐺 that form a dominating set of 𝐺, let 𝐴 = {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } be one of minimum cardinality, where possibly, 𝐴 = 𝐷 𝐺 . Since 𝐴 is a dominating set of 𝐺, we note that 𝛾(𝐺) ≤ | 𝐴| = 𝑘. Proceeding exactly as in the proof of Lemma 18.6, we have that 𝛾(𝐻) ≤ 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ holds for every 𝑖 ∈ [𝑘], implying that 𝑏′ + 𝑦′ + 𝑟 ′ =

𝑘 ∑︁

𝑘
∑︁ 𝑏 𝑖′ + 𝑦 𝑖′ + 𝑟 𝑖′ ≥ 𝛾(𝐻) = 𝑘𝛾(𝐻).

𝑖=1

𝑖=1

(18.2)

For each vertex ℎ ∈ 𝑉 (𝐻), let 𝐴ℎ be the set defined exactly as in the proof of Lemma 18.8. Since the set 𝐴ℎ is a dominating set of 𝐺 and since 𝐴ℎ ⊆ 𝐷 𝐺 , by the minimality of
the set 𝐴, we have | 𝐴| ≤ | 𝐴ℎ |. Thus, 𝑘 = | 𝐴| ≤ | 𝐴ℎ | = 𝑏 ℎ + 𝑦 ℎ + 𝑘 − 𝑏 ′ℎ − 𝑟 ℎ′ or equivalently, 𝑟 ℎ′ ≤ 𝑏 ℎ − 𝑏 ′ℎ + 𝑦 ℎ . Summing over all vertices ℎ ∈ 𝑉 (𝐻), ∑︁ ∑︁
𝑟′ = 𝑟 ℎ′ ≤ 𝑏 ℎ − 𝑏 ′ℎ + 𝑦 ℎ = 𝑏 − 𝑏 ′ + 𝑦. (18.3) ℎ∈𝑉 (𝐻 )

ℎ∈𝑉 (𝐻 )

536

Chapter 18. Domination and Vizing’s Conjecture

For each vertex 𝑢 𝑖 ∈ 𝐴, let (𝑢 𝑖 , ℎ) be a vertex of 𝐷 that belongs to the 𝐻-fiber 𝑢𝑖 𝐻. We note that the vertex (𝑢 𝑖 , ℎ) ∈ 𝐷 dominates the cell 𝜋𝑖ℎ , implying that 𝜋𝑖ℎ is a blue cell. Hence, there are at least 𝑘 blue cells, and therefore 𝑏 ′ ≥ 𝑘. Since 𝑏 ≥ 𝑏 ′ , this yields 𝑏 ≥ 𝑘. (18.4) By Inequalities (18.2), (18.3), and (18.4), 𝛾(𝐺) + 𝛾(𝐺) 𝛾(𝐻) ≤ 𝑘 + 𝑘𝛾(𝐻) ≤ 𝑏 + 𝑘𝛾(𝐻)
≤ 𝑏 + 𝑏′ + 𝑦′ + 𝑟 ′

≤ 𝑏 + 𝑏′ + 𝑦′ + 𝑏 − 𝑏′ + 𝑦 = 2𝑏 + 𝑦 + 𝑦 ′ ≤ 2(𝑏 + 𝑦) = 2|𝐷| = 2𝛾(𝐺 □ 𝐻),  implying that 𝛾(𝐺 □ 𝐻) = |𝐷| ≥ 21 𝛾(𝐺) 𝛾(𝐻) + 12 max 𝛾(𝐺), 𝛾(𝐻) . We present next the following improvement of the Clark-Suen Theorem 18.11. Theorem 18.16 ([114]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≥ 12 max 𝛾(𝐺) 𝛾t (𝐻), 𝛾t (𝐺) 𝛾(𝐻) . Proof Let 𝐷 be a 𝛾-set of 𝐺 □ 𝐻. By Lemmas 18.8 and 18.9, 𝛾(𝐺) 𝛾t (𝐻) ≤ 2𝑏 ′ + 𝑦 ′ + 𝑟 ′
≤ 2𝑏 ′ + 𝑦 ′ + 𝑏 − 𝑏 ′ + 𝑦

= 𝑏 + 𝑏′ + 𝑦 + 𝑦′ ≤ 2(𝑏 + 𝑦) = 2|𝐷| = 2𝛾(𝐺 □ 𝐻), implying that 𝛾(𝐺 □ 𝐻) = |𝐷| ≥ 12 𝛾(𝐺) 𝛾t (𝐻). Analogously, interchanging the roles of 𝐺 and 𝐻, we have 𝛾(𝐺 □ 𝐻) ≥ 12 𝛾t (𝐺) 𝛾(𝐻). We close this section with the following 2017 result due to Brešar [112]. We omit the proof, which as shown in [114], can be established using the coloring framework approach of Section 18.2.1. Recall that 𝜌(𝐺) denotes the packing number of 𝐺. Theorem 18.17 ([112]) For any two graphs 𝐺 and 𝐻, 

𝛾(𝐺 □ 𝐻) ≥ 13 max 2𝛾(𝐺) − 𝜌(𝐺) 𝛾(𝐻), 𝛾(𝐺) 2𝛾(𝐻) − 𝜌(𝐻) .

Section 18.3. Total Domination Number

18.3

537

Total Domination Number

In this section, we present a Vizing-like bound for the total domination number. In 2005 Henning and Rall [475] conjectured that the total domination number of the Cartesian product of two graphs without isolated vertices is at least one-half the product of the total domination numbers of the two graphs. This conjecture was proved in 2008 by Ho [498]. We present here a simple proof of this result that uses a similar coloring framework to that presented in Section 18.2.1. Let 𝐺 and 𝐻 be isolate-free graphs. Let 𝛾t (𝐺) = 𝑘 and let {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } be a 𝛾t -set of 𝐺. Let 𝜋 = {𝜋1 , 𝜋2 , . . . , 𝜋 𝑘 } be a partition of 𝑉 (𝐺) chosen so that 𝜋𝑖 ⊆ N(𝑢 𝑖 ) for each 𝑖 ∈ [𝑘]. We define the blocks Π𝑖𝐻 = 𝜋𝑖 × 𝑉 (𝐻) for 𝑖 ∈ [𝑘] exactly as before. Further, for a vertex ℎ of 𝐻, we define a cell 𝜋𝑖ℎ exactly as before, namely the set of vertices 𝜋𝑖ℎ = 𝜋𝑖 × {ℎ}. Let 𝐷 be a 𝛾t -set of 𝐺 □ 𝐻. As before, let 𝐷 𝑖 be the set of vertices of 𝐷 that belong to the block Π𝑖𝐻 for 𝑖 ∈ [𝑘], and let 𝐷 ℎ be the set of vertices of 𝐷 that belong to the 𝐺-fiber 𝐺 ℎ for ℎ ∈ 𝑉 (𝐻). Thus, for 𝑖 ∈ [𝑘] and for ℎ ∈ 𝑉 (𝐻), 𝐷 𝑖 = 𝐷 ∩ Π𝑖𝐻

𝐷ℎ = 𝐷 ∩ 𝐺ℎ.

and

If a vertex in 𝐺 □ 𝐻 is totally dominated by a vertex of 𝐷 in some 𝐺-fiber 𝐺 ℎ for some vertex ℎ ∈ 𝑉 (𝐻), then we say that 𝑣 is horizontally totally dominated by 𝐷, abbreviated ht-dominated. Otherwise, we say that 𝑣 is vertically totally dominated by 𝐷, abbreviated vt-dominated. For 𝑖 ∈ [𝑘] and ℎ ∈ 𝑉 (𝐻), if the cell 𝜋𝑖ℎ is totally dominated by the set 𝐷 ℎ (in the 𝐺-fiber 𝐺 ℎ ), then we say that 𝜋𝑖ℎ is horizontally totally dominated by 𝐷, abbreviated ht-dominated. For 𝑖 ∈ [𝑘], let 𝑐 dom,𝑖 be the number of cells in the block Π𝑖𝐻 that are ht-dominated and let 𝑐 ndom,𝑖 be the number of cells in the block Π𝑖𝐻 that are not ht-dominated. For ℎ ∈ 𝑉 (𝐻), let 𝑑dom,ℎ be the number of cells in the 𝐺-fiber 𝐺 ℎ that are ht-dominated and let 𝑑ndom,ℎ be the number of cells in 𝐺 ℎ that are not ht-dominated. Further, let 𝑐 dom be the number of ht-dominated cells in 𝐺 □ 𝐻, and so 𝑐 dom =

𝑘 ∑︁

𝑐 dom,𝑖 =

𝑖=1

∑︁

𝑑dom,ℎ .

ℎ∈𝑉 (𝐻 )

We shall need the following lemma. Lemma 18.18 If 𝐷 is a 𝛾t -set of 𝐺 □ 𝐻, then 𝑐 dom ≤ |𝐷|. Proof We adopt our earlier notation. In particular, 𝐷 is a 𝛾t -set of 𝐺 □ 𝐻 and so |𝐷| = 𝛾t (𝐺 □ 𝐻). For each ℎ ∈ 𝑉 (𝐻), we consider the projection map p𝐺 (𝐷 ℎ ) of 𝐷 ℎ

Chapter 18. Domination and Vizing’s Conjecture

538

onto the graph 𝐺. For each cell 𝜋𝑖ℎ in the 𝐺-fiber 𝐺 ℎ that is not ht-dominated, we add the vertex 𝑢 𝑖 to the set p𝐺 (𝐷 ℎ ). Let 𝑆 ℎ denote the resulting set. Every vertex of 𝐺 that
belongs to a cell in 𝐺 ℎ that is ht-dominated is totally dominated by the set p𝐺 𝐷 ℎ , while every vertex of 𝐺 that belongs to a cell in 𝐺 ℎ that is not ht-dominated is vt-dominated, and is therefore totally dominated by the set 𝑆 ℎ \ p𝐺 (𝐷 ℎ ). Hence, 𝑆 ℎ is a TD-set of 𝐺. Since there are 𝑑ndom,ℎ = 𝑘 − 𝑑dom,ℎ cells that are not ht-dominated in 𝐺 ℎ , we therefore have that 𝑘 = 𝛾t (𝐺) ≤ |𝑆 ℎ | = |p𝐺 (𝐷 ℎ )| + 𝑘 − 𝑑dom,ℎ ,
or equivalently, 𝑑dom,ℎ ≤ p𝐺 𝐷 ℎ . Summing over all vertices ℎ ∈ 𝑉 (𝐻), ∑︁ ∑︁ 𝑐 dom = |p𝐺 (𝐷 ℎ )| = |𝐷|. 𝑑dom,ℎ ≤ ℎ∈𝑉 (𝐻 )

ℎ∈𝑉 (𝐻 )

We are now in a position to prove Ho’s Vizing-like bound for the total domination number. Theorem 18.19 ([498]) If 𝐺 and 𝐻 are isolate-free graphs, then 𝛾t (𝐺 □ 𝐻) ≥ 21 𝛾t (𝐺) 𝛾t (𝐻). Proof We adopt our earlier notation. In particular, 𝐷 is a 𝛾t -set of 𝐺 □ 𝐻. For each 𝑖 ∈ [𝑘], we consider the projection map p 𝐻 (𝐷 𝑖 ) of 𝐷 𝑖 onto the graph 𝐻. For each cell 𝜋𝑖ℎ in the block Π𝑖𝐻 that is ht-dominated, we add a neighbor ℎ′ of ℎ in 𝐻 to the set p 𝐻 (𝐷 𝑖 ). Let 𝑆𝑖 denote the resulting set. Every vertex of 𝐻 that belongs to a cell in Π𝑖𝐻 that is not ht-dominated is totally dominated by the set p 𝐻 (𝐷 𝑖 ), while every vertex of 𝐻 that belongs to a cell in Π𝑖𝐻 that is ht-dominated is totally dominated by the set 𝑆𝑖 \ p 𝐻 (𝐷 𝑖 ). Hence, 𝑆𝑖 is a TD-set of 𝐻, implying that 𝛾t (𝐻) ≤ |𝑆𝑖 | = |p 𝐻 (𝐷 𝑖 )| + 𝑐 dom,𝑖 ≤ |𝐷 𝑖 | + 𝑐 dom,𝑖 . Summing over all 𝑖 ∈ [𝑘], this yields by Lemma 18.18 that 𝛾t (𝐺) 𝛾t (𝐻) =

𝑘 ∑︁

𝛾t (𝐻)

𝑖=1

≤

𝑘 ∑︁

|𝐷 𝑖 | +

𝑖=1

𝑘 ∑︁

𝑐 dom,𝑖

𝑖=1

= |𝐷| + 𝑐 dom ≤ 2|𝐷| = 2𝛾(𝐺 □ 𝐻) or equivalently, 𝛾t (𝐺 □ 𝐻) ≥ 21 𝛾t (𝐺) 𝛾t (𝐻). The bound of Theorem 18.19 is tight. In the case when at least one of 𝐺 or 𝐻 is a tree of order at least 3, the graphs 𝐺 and 𝐻 for which 𝛾t (𝐺 □ 𝐻) = 12 𝛾t (𝐺) 𝛾t (𝐻)

Section 18.4. Independent Domination Number

539

are characterized in [475]. We remark that a constructive characterization of trees 𝐺 satisfying 𝛾t (𝐺) = 2𝛾(𝐺) is given in [454]. Theorem 18.20 ([475]) If 𝐺 is a tree of order at least 3 and 𝐻 is an arbitrary isolate-free graph, then 𝛾t (𝐺 □ 𝐻) ≥ 21 𝛾t (𝐺) 𝛾t (𝐻), with equality if and only if 𝛾t (𝐺) = 2𝛾(𝐺) and 𝐻 consists of disjoint copies of 𝐾2 . It remains, however, an open problem to characterize the isolate-free graphs 𝐺 and 𝐻 that achieve equality in the bound of Theorem 18.19.

18.4

Independent Domination Number

The equivalent of Vizing’s Conjecture for the independent domination number is not true. To see this, let 𝐺 be the graph of order 11 obtained from 𝐾3 by adding two pendant edges to one vertex of the 𝐾3 and adding three pendant edges to each of the other two vertices. The graph 𝐺 is illustrated in Figure 18.3. Let 𝐻 = 𝐺 be the complement of the graph 𝐺. We observe that 𝑖(𝐺) = 6 and 𝛾(𝐻) = 𝑖(𝐻) = 2. However, as observed in [113] we have 𝑖(𝐺 □ 𝐻) = 11.

Figure 18.3 A graph 𝐺

Proposition 18.21 ([113]) There exist nontrivial connected graphs 𝐺 and 𝐻 such that the following inequalities hold: (a) 𝛾(𝐺 □ 𝐻) < 𝑖(𝐺) 𝑖(𝐻), (b) 𝑖(𝐺 □ 𝐻) < 𝑖(𝐺) 𝑖(𝐻), and (c) 𝑖(𝐺 □ 𝐻) < 𝑖(𝐺) 𝛾(𝐻). The following conjecture was posed by Brešar et al. [113] in their 2012 survey paper on Vizing’s Conjecture. Conjecture 18.22 ([113]) For any two graphs 𝐺 and 𝐻,  𝛾(𝐺 □ 𝐻) ≥ min 𝑖(𝐺) 𝛾(𝐻), 𝛾(𝐺) 𝑖(𝐻) . The truth of Conjecture 18.22 would clearly imply Vizing’s Conjecture. Conjecture 18.22 was subsequently disproved by Pilipczuk et al. [636]. Proposition 18.23 ([636]) There exists an infinite family of graphs 𝐺 𝑘 , 𝑘 ≥ 2, such that 𝛾(𝐺 𝑘 ) = 𝑘 + 2, 𝑖(𝐺 𝑘 ) = 𝑘 2 + 𝑘 + 1, and 𝛾(𝐺 𝑘 □ 𝐺 𝑘 ) ≤ 10𝑘 2 + 8𝑘 + 4. Proof For
𝑘 ≥ 2, let 𝐺 𝑘 be obtained from the disjoint union of a double star 𝑆 𝑘 2 , 𝑘 2 and 𝑘 copies of 𝐾2 . The resulting graph 𝐺 𝑘 has order 𝑛(𝐺 𝑘 ) = 2𝑘 2 + 2𝑘 + 2.

540

Chapter 18. Domination and Vizing’s Conjecture

The two central vertices of the double star, together with one vertex from every added copy of 𝐾2 , is a 𝛾-set of 𝐺 𝑘 , while one central vertex of the double star and the leaf-neighbors of the other central vertex, together with one vertex from every added copy of 𝐾2 , is an 𝑖-set of 𝐺 𝑘 . Therefore, 𝛾(𝐺 𝑘 ) = 𝑘 + 2 and 𝑖(𝐺 𝑘 ) = 𝑘 2 + 𝑘 + 1. In the graph 𝐺 𝑘 , let 𝐴 be the set of 2𝑘 vertices that belong to a 𝐾2 -component, let 𝐵 be the set of 2𝑘 2 leaves in the double star, and let 𝐶 be the set of two central vertices of the double star. Thus, {𝐴, 𝐵, 𝐶} is a partition of 𝑉 (𝐺 𝑘 ). Let 𝐺 and 𝐻 be two vertex-disjoint copies of 𝐺 𝑘 . For 𝑋 ∈ {𝐺, 𝐻}, let 𝐴 𝑋 , 𝐵 𝑋 , and 𝐶𝑋 be the sets of vertices in the graph 𝑋 that correspond to the sets 𝐴, 𝐵, and 𝐶 in 𝐺 𝑘 . We consider the Cartesian product 𝐺 □ 𝐻 of the two graphs 𝐺 and 𝐻. The subgraph of 𝐺 □ 𝐻 induced by the set 𝐴𝐺 × 𝐴 𝐻 consists of 𝑘 2 vertex-disjoint copies of 𝐶4 . Each such copy of 𝐶4 is a component in 𝐺 □ 𝐻. Let 𝐷 1 be the set consisting of two vertices from each such 𝐶4 -component of 𝐺 □ 𝐻, and so 𝐷 1 dominates all vertices from these 𝑘 2 components and |𝐷 1 | = 2𝑘 2 . Let 𝐷 2 = 𝐶𝐺 × 𝑉 (𝐻) and let 𝐷 3 = 𝑉 (𝐺) × 𝐶 𝐻 . Let 𝐷 = 𝐷 1 ∪ 𝐷 2 ∪ 𝐷 3 . Let (𝑔, ℎ) be an arbitrary vertex in 𝑉 (𝐺 □ 𝐻) \ 𝐷. By construction of the set 𝐷, the vertex 𝑔 ∈ 𝐵𝐺 or the vertex ℎ ∈ 𝐵 𝐻 . Renaming the graphs 𝐺 and 𝐻 if necessary, we may assume without loss of generality that 𝑔 ∈ 𝐵𝐺 . Let 𝑔1 be the neighbor of the vertex 𝑔 in the 𝐺-fiber 𝐺 ℎ , and so 𝑔1 is the central vertex of the double star in this copy of 𝐺 with 𝑔 as one of its leaf neighbors. Thus, (𝑔1 , ℎ) dominates the vertex (𝑔, ℎ) in 𝐺 □ 𝐻. Since 𝑔1 ∈ 𝐶𝐺 , we note that (𝑔1 , ℎ) ∈ 𝐷 2 ⊂ 𝐷, implying that the vertex (𝑔, ℎ) is dominated by 𝐷 in 𝐺 □ 𝐻. We infer that the set 𝐷 is
a 2 2 dominating set of 𝐺 □ 𝐻. Since |𝐷
1 | = 2𝑘 , |𝐷 2 | = 2𝑛(𝐻) = 2 2𝑘 + 2𝑘 + 2 , 2 |𝐷 3 | = 2𝑛(𝐺) = 2 2𝑘 + 2𝑘 + 2 , and
|𝐷 2 ∩ 𝐷 3 | = 4, we therefore have that 𝛾(𝐺 □ 𝐻) ≤ |𝐷| = 2𝑘 2 + 4 2𝑘 2 + 2𝑘 + 2 − 4 = 10𝑘 2 + 8𝑘 + 4. By Proposition 18.23, there exists an infinite family of graphs 𝐺 𝑘 , 𝑘 ≥ 2, such that 𝛾(𝐺 𝑘 ) 𝑖(𝐺 𝑘 ) = Θ(𝑛3 ) and 𝛾(𝐺 𝑘 □ 𝐺 𝑘 ) = Θ(𝑛2 ). Thus, Proposition 18.23 disproved Conjecture 18.22. As remarked in [636], the counterexample in Proposition 18.23 can be made connected by creating a graph 𝐺 ′𝑘 from 𝐺 𝑘 by adding a new vertex 𝑣 and joining it to one central vertex of the double star and to one vertex from each 𝐾2 -component. The resulting graph satisfies 𝛾(𝐺 ′𝑘 ) = 𝛾(𝐺 𝑘 ), 𝑖(𝐺 ′𝑘 ) = 𝑖(𝐺 𝑘 ), and
𝛾(𝐺 ′𝑘 □ 𝐺 ′𝑘 ) ≤ 𝛾(𝐺 𝑘 □ 𝐺 𝑘 ) + 2 2𝑘 2 + 2𝑘 + 2 − 1 ≤ 14𝑘 2 + 12𝑘 + 7, as we can add to the set 𝐷 constructed in the proof of Proposition 18.23 the set

of vertices with one coordinate equal to 𝑣, that is, {𝑣} × 𝑉 (𝐺 ′𝑘 ) ∪ 𝑉 (𝐺 ′𝑘 ) × {𝑣} .

18.5

Independence Number

In this section, we present a Vizing-like bound for the independence number. For completeness, we present the following result and proof. Theorem 18.24 For any two nonempty graphs 𝐺 and 𝐻, 𝛼(𝐺 □ 𝐻) ≥ 𝛼(𝐺) 𝛼(𝐻) + 1. Proof Let 𝐼𝐺 and 𝐼 𝐻 be maximum independent sets of 𝐺 and 𝐻, respectively. Thus, |𝐼𝐺 | = 𝛼(𝐺) and |𝐼 𝐻 | = 𝛼(𝐻). Let 𝐼𝐺𝐻 be a maximum independent set in the

Section 18.6. Upper Domination Number

541



subgraph of 𝐺 □ 𝐻 induced by the set 𝑉 (𝐺) \ 𝐼𝐺 × 𝑉 (𝐻) \ 𝐼 𝐻 . Consider the set 𝐼 = (𝐼𝐺 × 𝐼 𝐻 ) ∪ 𝐼𝐺𝐻 . We show that 𝐼 is a maximal independent set in 𝐺 □ 𝐻. By construction, the set 𝐼 is an independent set. Hence, it suffices to show that every vertex in 𝐺 □ 𝐻 that does not belong to the set 𝐼 has a neighbor in 𝐼. Let (𝑔, ℎ) ∈ 𝑉 (𝐺 □ 𝐻) \ 𝐼. Assume that 𝑔 ∈ 𝐼𝐺 . Since 𝐼𝐺 × 𝐼 𝐻 ⊂ 𝐼, we note that ℎ ∈ 𝑉 (𝐻) \ 𝐼 𝐻 . By the maximality of the independent set 𝐼 𝐻 , there is a vertex ℎ′ ∈ 𝐼 𝐻 that is adjacent to ℎ in the graph 𝐻, implying that in the graph 𝐺 □ 𝐻 the vertex (𝑔, ℎ) is adjacent to the vertex (𝑔, ℎ′ ) ∈ 𝐼. Assume next that 𝑔 ∉ 𝐼𝐺 . If ℎ ∈ 𝐼 𝐻 , then analogously as in the previous case (with the roles of 𝐺 and 𝐻 switched) in the graph 𝐺 □ 𝐻 the vertex (𝑔, ℎ) has a neighbor in 𝐼. Hence, we may assume that ℎ ∉
𝐼 𝐻 . Thus, (𝑔,
ℎ) is a vertex in the subgraph of 𝐺 □ 𝐻 induced by the set 𝑉 (𝐺) \ 𝐼𝐺 × 𝑉 (𝐻) \ 𝐼 𝐻 . By the maximality of the independent set 𝐼𝐺𝐻 in this subgraph, the vertex (𝑔, ℎ) has a neighbor in 𝐼. Hence in both cases, the vertex (𝑔, ℎ) is adjacent to a vertex in 𝐼. Therefore, the set 𝐼 is a maximal independent set in 𝐺 □ 𝐻. Since 𝐺 and 𝐻 are nonempty graphs, 𝑉 (𝐺) \ 𝐼𝐺 ≠ ∅ and 𝑉 (𝐻) \ 𝐼 𝐻 ≠ ∅, and so certainly the set 𝐼𝐺𝐻 is nonempty. Thus, |𝐼𝐺𝐻 | ≥ 1. Thus, 𝛼(𝐺 □ 𝐻) ≥ |𝐼 | = |𝐼𝐺 | · |𝐼 𝐻 | + |𝐼𝐺𝐻 | = 𝛼(𝐺) 𝛼(𝐻) + |𝐼𝐺𝐻 | ≥ 𝛼(𝐺) 𝛼(𝐻) + 1. The bound in Theorem 18.24 is best possible in the sense that there exist graphs 𝐺 and 𝐻 of small orders for which equality is achieved. The simplest example is when 𝐺 = 𝑃2 and 𝐻 = 𝑃2 , in which case 𝐺 □ 𝐻 = 𝑃2 □ 𝑃2 = 𝐶4 . In this example, 𝛼(𝐺) = 𝛼(𝐻) = 1 and 𝛼(𝐺 □ 𝐻) = 2 = 𝛼(𝐺) 𝛼(𝐻) + 1. As a further example, take 𝐺 = 𝑃3 and 𝐻 = 𝑃3 . In this case, the graph 𝐺 □ 𝐻 = 𝑃3 □ 𝑃3 is illustrated in Figure 18.4, where the highlighted vertices form a maximum independent set of 𝐺 □ 𝐻 of cardinality 5. Thus, 𝛼(𝐺) = 𝛼(𝐻) = 2, while 𝛼(𝐺 □ 𝐻) = 5 = 𝛼(𝐺) 𝛼(𝐻) + 1.

Figure 18.4 The graph 𝑃3 □ 𝑃3

18.6

Upper Domination Number

In 1996 Nowakowski and Rall [616] posed the following Vizing-like conjecture for the upper domination number of Cartesian products of graphs.

542

Chapter 18. Domination and Vizing’s Conjecture

Conjecture 18.25 ([616]) For any two graphs 𝐺 and 𝐻, Γ(𝐺 □ 𝐻) ≥ Γ(𝐺) Γ(𝐻). In 2005 Brešar [110] proved the Nowakowski-Rall Conjecture. In fact, if both graphs 𝐺 and 𝐻 are nontrivial (that is, have at least two vertices), then he proved a slightly stronger bound. Before we formally state this result, we shall adopt the following notation in this section. Let 𝐺 be a nontrivial graph. Let 𝑆 be a Γ-set of 𝐺. Since 𝑆 is a minimal dominating set, by Lemma 2.72, ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅ for every vertex 𝑣 ∈ 𝑆. Let  𝐴 = 𝑣 ∈ 𝑆 : epn[𝑣, 𝑆] = ∅ . Thus, ipn[𝑣, 𝑆] ≠ ∅ for every vertex 𝑣 ∈ 𝐴 and so every vertex in 𝐴 is isolated in 𝐺 [𝑆]. Let 𝐵 = 𝑆 \ 𝐴 and so  𝐵 = 𝑣 ∈ 𝑆 : epn[𝑣, 𝑆] ≠ ∅ .

Let

Possibly, some vertices in 𝐵 may be isolated in 𝐺 [𝑆], in which case ipn[𝑣, 𝑆] ≠ ∅. Ø epn[𝑣, 𝑆] 𝐶= 𝑣∈𝐵

and so 𝐶 consists of all vertices in 𝑉 \ 𝑆 that have exactly one neighbor in 𝑆 (and by definition such a neighbor belongs to the set 𝐵). Let 𝑁 = 𝜕 (𝐵) \ 𝐶 and so 𝑁 consists of all vertices in the boundary of 𝐵 that do not belong to the set 𝐶. Thus, each vertex in 𝑁 belongs to 𝑉 \ 𝑆, has at least one neighbor in 𝐵, and has at least two neighbors in 𝑆. Let 𝑅 be the set of remaining vertices in 𝐺, that is, 𝑅 = 𝜕 (𝑆) \ (𝐶 ∪ 𝑁). We note that no vertex in 𝑅 is adjacent to a vertex in 𝐵. Thus, since 𝑆 is a minimal dominating set of 𝐺 and since no vertex in 𝐴 has an 𝑆-external private neighbor, the set 𝑅 is defined by 
𝑅 = 𝑣 ∈ 𝑉 \ 𝑆 : N(𝑣) ∩ 𝑆 ⊆ 𝐴 and deg 𝐴 (𝑣) ≥ 2 . We note that 𝑉 = 𝐴 ∪ 𝐵 ∪ 𝐶 ∪ 𝑁 ∪ 𝑅, where 𝑆 = 𝐴 ∪ 𝐵 and 𝑉 \ 𝑆 = 𝐶 ∪ 𝑁 ∪ 𝑅. Further, some of these sets may possibly be empty. We next define two operations. Operation 1. For each subset 𝐼 of 𝑅, we let 𝐵(𝐼) be a subset of vertices in 𝐵 such that the set 𝐵(𝐼) ∪ 𝐼 dominates the set 𝐶 ∪ 𝑁 and is minimal in the sense that for every vertex 𝑣 ∈ 𝐵(𝐼), there exists a vertex in 𝐶 ∪ 𝑁 that is dominated by 𝑣 but by no other vertex in 𝐵(𝐼) ∪ 𝐼. As remarked in [110], such a set 𝐵(𝐼) always exists, noting that the set 𝐵 itself dominates 𝐶 ∪ 𝑁 and thus the minimality condition can always be met by sequentially removing from 𝐵 vertices that are not needed to dominate the set 𝐶 ∪ 𝑁.

Section 18.6. Upper Domination Number

543

Operation 2. For each subset 𝐽 of 𝐴 ∪ 𝑅, we let 𝐵(𝐽) be a subset of vertices in 𝐵 such that the set 𝐵(𝐽) ∪ 𝐽 dominates the set 𝐶 ∪ 𝑁, and is minimal in the sense that each vertex 𝑣 ∈ 𝐵(𝐽) uniquely dominates a vertex in 𝐶 ∪ 𝑁 that is not adjacent to any other vertex of 𝐵(𝐽) ∪ 𝐽. To specify the graph 𝐺, we write 𝑆𝐺 , 𝐴𝐺 , 𝐵𝐺 , 𝐶𝐺 , 𝑁𝐺 , 𝑅𝐺 instead of 𝑆, 𝐴, 𝐵, 𝐶, 𝑁, 𝑅, respectively. However, if the graph 𝐺 is clear from the context, we omit the subscript. We are now in a position to state and prove the 2005 theorem due to Brešar [110]. Theorem 18.26 ([110]) For any two nontrivial graphs 𝐺 and 𝐻, Γ(𝐺 □ 𝐻) ≥ Γ(𝐺) Γ(𝐻) + 1. Proof Let 𝑆𝐺 and 𝑆 𝐻 be Γ-sets of 𝐺 and 𝐻, respectively. Thus, |𝑆𝐺 | = Γ(𝐺) and |𝑆 𝐻 | = Γ(𝐻). Suppose that in one of the factors (say 𝐺) the set 𝐴 is empty. In this case, the set 𝐷 = 𝐵𝐺 × 𝑉 (𝐻) is a minimal dominating set of 𝐺 □ 𝐻, every vertex in private neighbor. Hence, Γ(𝐺 □ 𝐻) ≥ |𝐷| = Γ(𝐺)|𝑉 (𝐻)| ≥ which has a 𝐷-external
Γ(𝐺) Γ(𝐻) + 1 ≥ Γ(𝐺) Γ(𝐻) + 1. Hence, we may assume that in both factors the set 𝐴 is nonempty, that is, 𝐴𝐺 ≠ ∅ and 𝐴 𝐻 ≠ ∅. Suppose that in both factors the set 𝐵 is empty. In this case, Γ(𝐺) = 𝛼(𝐺) and Γ(𝐻) = 𝛼(𝐻), and so by Theorem 18.24, we have Γ(𝐺 □ 𝐻) ≥ 𝛼(𝐺 □ 𝐻) ≥ 𝛼(𝐺) 𝛼(𝐻) + 1 = Γ(𝐺) Γ(𝐻) + 1. Hence, we may assume that the set 𝐵 is nonempty in at least one of the factors. Renaming the factors if necessary, we may assume that 𝐵 𝐻 ≠ ∅. We now construct a minimal dominating set 𝐷 in 𝐺 □ 𝐻 as a union of six pairwise disjoint sets, some of which may be empty. Let 𝐷 1 = 𝐴𝐺 × 𝑆 𝐻 . We note that |𝐷 1 | = | 𝐴𝐺 | · Γ(𝐻).

(18.5)

Let 𝐷 2 be a maximum independent set 𝐼 of the subgraph of 𝐺 □ 𝐻 induced by 𝑅𝐺 × 𝑅 𝐻 . For every vertex 𝑔 ∈ 𝑅𝐺 , let 𝐼𝑔 be the set of vertices in 𝐼 that belong to the 𝐻-fiber 𝑔 𝐻, that is,  𝐼𝑔 = 𝐼 ∩ (𝑔, ℎ) : ℎ ∈ 𝑉 (𝐻) .
We now consider the subset 𝐵 𝐻 p 𝐻 (𝐼𝑔 ) of 𝐵 𝐻 obtained by Operation 1, and we subset of 𝐺 □ 𝐻 in the 𝐻-fiber 𝑔 𝐻, that is, the set  consider the corresponding
(𝑔, ℎ) : ℎ ∈ 𝐵 𝐻 p 𝐻 (𝐼𝑔 ) . Let 𝐷 3 be the union of all such sets, that is, Ø
𝐷3 = (𝑔, ℎ) : ℎ ∈ 𝐵 𝐻 p 𝐻 (𝐼𝑔 ) . 𝑔∈𝑅𝐺

We note that 𝐷 3 is a subset of 𝑅𝐺 × 𝐵 𝐻 . We define the set 𝐷 4 similarly, but with the roles of 𝐺 and 𝐻 reversed. Thus, for every vertex ℎ ∈ 𝑅 𝐻 , we let 𝐼 ℎ be the set of vertices in 𝐼 that belong to the 𝐺-fiber 𝐺 ℎ , that is,

Chapter 18. Domination and Vizing’s Conjecture

544

 𝐼 ℎ = 𝐼 ∩ (𝑔, ℎ) : 𝑔 ∈ 𝑉 (𝐺) , and let 𝐷4 =

Ø


(𝑔, ℎ) : 𝑔 ∈ 𝐵𝐺 p𝐺 (𝐼 ℎ ) .

ℎ∈𝑅 𝐻

We note that 𝐷 4 is a subset of 𝐵𝐺 × 𝑅 𝐻 . For each vertex ℎ ∈ 𝐵 𝐻 , let 𝐽 𝑦 be the set of vertices from 𝑉 (𝐺) × {ℎ} that already belong to the set 𝐷, that is,
𝐽ℎ = (𝐷 1 ∪ 𝐷 3 ) ∩ 𝑉 (𝐺) × {ℎ} . We now project the set 𝐽ℎ in the graph 𝐺 □ 𝐻 onto the graph 𝐺 by the projection
map, yielding the set p𝐺 (𝐽ℎ ) ⊆ 𝐵𝐺 . Let 𝐷 5 be the union of the sets 𝐵𝐺 p𝐺 (𝐽ℎ ) ×{ℎ} taken over all vertices ℎ ∈ 𝐵 𝐻 obtained by Operation 2, that is, Ø
𝐷5 = 𝐵𝐺 p𝐺 (𝐽ℎ ) × {ℎ}. ℎ∈ 𝐵 𝐻

We note that 𝐷 5 ⊆ 𝐵𝐺 × 𝐵 𝐻 . Finally, let
𝐷 6 = 𝐵𝐺 × 𝑉 (𝐻) \ (𝐵 𝐻 ∪ 𝑅 𝐻 ) . We note that 𝐴 𝐻 ∪ 𝐶 𝐻 ⊆ 𝑉 (𝐻) \ (𝐵 𝐻 ∪ 𝑅 𝐻 ), and so |𝑉 (𝐻) \ (𝐵 𝐻 ∪ 𝑅 𝐻 )| ≥ | 𝐴 𝐻 | + |𝐶 𝐻 | ≥ | 𝐴 𝐻 | + |𝐵 𝐻 | = |𝑆 𝐻 |. Therefore, |𝐷 6 | ≥ |𝐵𝐺 | · |𝑆 𝐻 | = |𝐵𝐺 | · Γ(𝐻).

(18.6)

We now let 𝐷=

6 Ø

𝐷𝑖 .

𝑖=1

By our earlier observations, the six sets 𝐷 1 , 𝐷 2 , . . . , 𝐷 6 are vertex-disjoint. We proceed further by proving three claims. Claim 18.26.1

|𝐷| ≥ Γ(𝐺) Γ(𝐻) + 1.

Proof By Inequalities (18.5) and (18.6), |𝐷 1 | + |𝐷 6 | ≥ | 𝐴𝐺 | · Γ(𝐻) + |𝐵𝐺 | · Γ(𝐻) = |𝑆𝐺 | · Γ(𝐻) = Γ(𝐺) Γ(𝐻).

(18.7)

By assumption, 𝐴𝐺 ≠ ∅ and 𝐺 is a nontrivial graph, implying that the set 𝑅𝐺 ∪ 𝑁𝐺 is nonempty. Analogously, since 𝐴 𝐻 ≠ ∅ and 𝐻 is a nontrivial graph, the set 𝑅 𝐻 ∪ 𝑁 𝐻 is nonempty. If 𝑅𝐺 = ∅, then 𝐷 5 must be nonempty. If 𝑅𝐺 ≠ ∅ and 𝑅 𝐻 = ∅, then 𝐷 3 must be nonempty. If 𝑅𝐺 ≠ ∅ and 𝑅 𝐻 ≠ ∅, then 𝐷 2 must be nonempty. These observations imply that |𝐷 2 | + |𝐷 3 | + |𝐷 5 | ≥ 1.

(18.8)

Section 18.6. Upper Domination Number

545

By Inequalities (18.7) and (18.8), 6 ∑︁

|𝐷 𝑖 | ≥ |𝐷 1 | + |𝐷 6 | + |𝐷 2 | + |𝐷 3 | + |𝐷 5 | ≥ Γ(𝐺) Γ(𝐻) + 1. |𝐷| = 𝑖=1

Claim 18.26.2 The set 𝐷 is a dominating set of 𝐺 □ 𝐻. Proof Since 𝑆 𝐻 is a dominating set of 𝐻, all vertices in 𝐴𝐺 × 𝑉 (𝐻) are dominated by the set 𝐷 1 = 𝐴𝐺 × 𝑆 𝐻 . We next consider vertices in the set 𝑅𝐺 × 𝑉 (𝐻). Since 𝐼 = 𝐷 2 is a maximum independent set of the subgraph of 𝐺 □ 𝐻 induced by 𝑅𝐺 × 𝑅 𝐻 , the set 𝐷 2 dominates the set 𝑅𝐺 × 𝑅 𝐻 . Since the set 𝐴𝐺 dominates the set 𝑅𝐺 , the set 𝐷 1 dominates the set 𝑅𝐺 × 𝑆 𝐻 . The set 𝐷 2 ∪ 𝐷 3 dominates the set 𝑅𝐺 × (𝐶 𝐻 ∪ 𝑁 𝐻 ). Hence, all vertices in 𝐴𝐺 × 𝑉 (𝐻) are dominated by the set 𝐷 1 ∪ 𝐷 2 ∪ 𝐷 3 . Since 𝐴 𝐻 ∪𝐶 𝐻 ∪ 𝑁 𝐻 is a dominating set of 𝐻, the set 𝐷 6 = 𝐵𝐺 × ( 𝐴 𝐻 ∪𝐶 𝐻 ∪ 𝑁 𝐻 ) dominates the set 𝐵𝐺 × 𝑉 (𝐻). We prove next that 𝐷 dominates the set (𝐶𝐺 ∪ 𝑁𝐺 ) × 𝑉 (𝐻). Let ℎ ∈ 𝑉 (𝐻). If ℎ ∈ 𝑅 𝐻 , then (𝐶𝐺 ∪ 𝑁𝐺 ) × 𝑉 (𝐻) is dominated by the set 𝐷 2 ∪ 𝐷 4 . If ℎ ∈ 𝐵 𝐻 , then (𝐶𝐺 ∪ 𝑁𝐺 ) × 𝑉 (𝐻) is dominated by the set 𝐷 3 ∪ 𝐷 5 . If ℎ ∈ 𝐴 𝐻 ∪ 𝐶 𝐻 ∪ 𝑁 𝐻 , then (𝐶𝐺 ∪ 𝑁𝐺 ) × 𝑉 (𝐻) is dominated by the set 𝐷 6 , noting that the set 𝐵𝐺 dominates the set 𝐶𝐺 ∪ 𝑁𝐺 in 𝐺. Therefore, the set 𝐷 is a dominating set of 𝐺 □ 𝐻. Claim 18.26.3 The set 𝐷 is a minimal dominating set of 𝐺 □ 𝐻. Proof By Claim 18.26.2, the set 𝐷 is a dominating set of 𝐺 □ 𝐻. We show that in the graph 𝐺 □ 𝐻, we have ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 = (𝑔, ℎ) ∈ 𝐷, implying that the set 𝐷 is a minimal dominating set of 𝐺 □ 𝐻. Recall that 𝐷 = 𝐷 1 ∪ 𝐷 2 ∪ · · · ∪ 𝐷 6 . Case 1. 𝑣 = (𝑔, ℎ) ∈ 𝐷 1 . If ℎ ∉ 𝐵 𝐻 , then the vertex 𝑣 is isolated in (𝐺 □ 𝐻) [𝐷], and so ipn[𝑣, 𝐷] ≠ ∅ in 𝐺 □ 𝐻. If ℎ ∈ 𝐵 𝐻 , then in the graph 𝐻, there is a vertex ℎ′ ∈ 𝐶 𝐻 , where ℎ′ ∈ epn[ℎ, 𝑆 𝐻 ], implying that (𝑔, ℎ′ ) ∈ epn[𝑣, 𝐷] in the graph 𝐺 □ 𝐻. Case 2. 𝑣 = (𝑔, ℎ) ∈ 𝐷 2 . Recall that the set 𝐷 2 is a maximum independent set of the subgraph of 𝐺 □ 𝐻 induced by 𝑅𝐺 × 𝑅 𝐻 . In particular, 𝑔 ∈ 𝑅𝐺 and ℎ ∈ 𝑅 𝐻 . The only possible vertices adjacent to 𝑣 are vertices of the form (𝑔 ′ , ℎ) where 𝑔 ′ ∈ 𝐵𝐺 or (𝑔, ℎ′ ) where ℎ′ ∈ 𝐵 𝐻 . However, no vertex in 𝐵𝐺 is adjacent to a vertex in 𝑅𝐺 , and no vertex in 𝐵 𝐻 is adjacent to a vertex in 𝑅 𝐻 . Hence, the vertex 𝑣 is isolated in (𝐺 □ 𝐻) [𝐷], and so ipn[𝑣, 𝐷] ≠ ∅ in 𝐺 □ 𝐻.
Case 3. 𝑣 = (𝑔, ℎ) ∈ 𝐷 3 . In this case, ℎ ∈ 𝐵 𝐻 p 𝐻 (𝐼𝑔 ) , implying by Operation 1 ′ that the vertex
ℎ uniquely dominates a vertex ℎ ∈ 𝐶 𝐻 ∪ 𝑁 𝐻 in the projection 𝐵 𝐻 p 𝐻 (𝐼𝑔 ) in the graph 𝐻. This in turn implies that in the graph 𝐺 □ 𝐻, we have (𝑔, ℎ′ ) ∈ epn[𝑣, 𝐷]. Case 4. 𝑣 = (𝑔, ℎ) ∈ 𝐷 4 . As in the previous case (interchanging the roles of 𝐺 and 𝐻), the
vertex 𝑔 uniquely dominates a vertex 𝑔 ′ ∈ 𝐶𝐺 ∪ 𝑁𝐺 in the projection 𝐵𝐺 p𝐺 (𝐼 ℎ ) in the graph 𝐺, implying that (𝑔 ′ , ℎ) ∈ epn[𝑣,
𝐷] in 𝐺 □ 𝐻. Case 5. 𝑣 = (𝑔, ℎ) ∈ 𝐷 5 . In this case, 𝑔 ∈ 𝐵𝐺 p𝐺 (𝐽ℎ ) , implying by Operation 2 ′ that the vertex
𝑔 uniquely dominates a vertex 𝑔 ∈ 𝐶′𝐺 ∪ 𝑁𝐺 in the projection 𝐵𝐺 p𝐺 (𝐽ℎ ) in the graph 𝐻. This in turn implies that (𝑔 , ℎ) ∈ epn[𝑣, 𝐷] in 𝐺 □ 𝐻.

546

Chapter 18. Domination and Vizing’s Conjecture

Case 6. 𝑣 = (𝑔, ℎ) ∈ 𝐷 6 . In this case, 𝑔 ∈ 𝐵𝐺 and ℎ ∈ 𝐴 𝐻 ∪ 𝐶 𝐻 ∪ 𝑁 𝐻 . Since 𝑔 ∈ 𝐵𝐺 , there is a vertex 𝑔 ′ ∈ 𝐶𝐺 such that in the graph 𝐺, we have 𝑔 ′ ∈ epn[𝑔, 𝑆𝐺 ], implying that (𝑔 ′ , ℎ) ∈ epn[𝑣, 𝐷] in 𝐺 □ 𝐻. Therefore, for every vertex 𝑣 ∈ 𝐷 in the graph 𝐺 □ 𝐻, we have ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅, implying by Lemma 2.72 that the set 𝐷 is a minimal dominating set in 𝐺 □ 𝐻. By Claim 18.26.3, the set 𝐷 is a minimal dominating set in 𝐺 □ 𝐻. Hence, by Claim 18.26.1, we have Γ(𝐺 □ 𝐻) ≥ |𝐷| ≥ Γ(𝐺) Γ(𝐻) + 1. As remarked in [110], the bound in Theorem 18.26 is achievable for graphs 𝐺 and 𝐻 of small orders, as may be seen by taking 𝐺 = 𝑃3 and 𝐻 = 𝑃3 . In this case, the graph 𝐺 □𝐻 = 𝑃3 □𝑃3 is illustrated in Figure 18.4, where the highlighted vertices form a Γ-set of 𝐺 □ 𝐻. Thus, Γ(𝐺) = Γ(𝐻) = 2, while Γ(𝐺 □ 𝐻) = 5 = Γ(𝐺) Γ(𝐻) + 1.

18.7

Upper Total Domination Number

In this section, we present a Vizing-like bound for the upper total domination number established in 2008 by Dorbec et al. [237]. We first consider the case when both factors of the Cartesian product are connected graphs of order at least 3. Recall that for a graph 𝐺, a set 𝑆 ⊆ 𝑉, and a vertex 𝑣 ∈ 𝑆, the sets epn[𝑣, 𝑆] and epn(𝑣, 𝑆) are equal, that is, epn[𝑣, 𝑆] = epn(𝑣, 𝑆) and are called the 𝑆-external private neighborhoods of 𝑣. Theorem 18.27 ([237]) If 𝐺 and 𝐻 are connected graphs of order at least 3, then  Γt (𝐺 □ 𝐻) ≥ 12 Γt (𝐺) Γt (𝐻) + 12 max Γt (𝐺), Γt (𝐻) . Proof Let 𝐺 and 𝐻 be connected graphs of order at least 3. By Theorem 15.52, there exists a minimal dominating set 𝑆 of 𝐺 such that the following hold: (a) |𝑆| ≥ 12 Γt (𝐺), and (b) |epn(𝑣, 𝑆)| ≥ 1 for every vertex 𝑣 ∈ 𝑆. We now consider the set 𝐷 = 𝑆 × 𝑉 (𝐻). Since the set 𝑆 is a dominating set of 𝐺, the set 𝐷 dominates 𝐺 □ 𝐻. Further, for every vertex 𝑔 ∈ 𝑆, the vertices in {𝑔} × 𝑉 (𝐻) are totally dominated by their neighbors in the 𝐻-fiber 𝑔 𝐻, implying that 𝐷 is a TD-set of 𝐺 □ 𝐻. We show that 𝐷 is in fact a minimal TD-set of 𝐺 □ 𝐻. Let 𝑣 ∈ 𝐷 and so 𝑣 = (𝑔, ℎ) for some vertex 𝑔 ∈ 𝑆 and some vertex ℎ ∈ 𝑉 (𝐻). By property (b) of the minimal dominating set 𝑆, we have that |epn(𝑔, 𝑆)| ≥ 1 in the graph 𝐺. Let 𝑔 ′ ∈ epn(𝑔, 𝑆) and consider the vertex 𝑣 ′ = (𝑔 ′ , ℎ). In the graph 𝐺 □ 𝐻, it follows that 𝑣 ′ ∈ epn(𝑣, 𝐷). Hence, since 𝑣 is an arbitrary vertex in 𝐷 in the graph 𝐺 □ 𝐻, |epn(𝑣, 𝐷)| ≥ 1 for every 𝑣 ∈ 𝐷. Thus, by Lemma 15.51 in Chapter 15, the set 𝐷 is a minimal TD-set of 𝐺 □ 𝐻 and so Γt (𝐺 □ 𝐻) ≥ |𝐷| = |𝑆| × |𝑉 (𝐻)|. By property (b) of the minimal dominating set 𝑆, we have |𝑆| ≥ 12 Γt (𝐺). Since 𝐻 is a connected graph of order at least 3, Δ(𝐻) ≥ 2 and therefore by Observation 14.3 in Chapter 14, |𝑉 (𝐻)| ≥ Γt (𝐻) + 1. Hence,
Γt (𝐺 □ 𝐻) ≥ 12 Γt (𝐺) Γt (𝐻) + 1 . (18.9)

Section 18.7. Upper Total Domination Number

547

Interchanging the roles of 𝐺 and 𝐻, an analogous argument shows that
Γt (𝐺 □ 𝐻) ≥ 21 Γt (𝐻) Γt (𝐺) + 1 . (18.10) The desired lower bound in the statement of the theorem now follows from Inequalities (18.9) and (18.10). We show next that the bound in Theorem 18.27 is tight, that is, there exist connected graphs 𝐺 and 𝐻 of arbitrarily large orders that achieve equality in the bound. For 𝑘 ≥ 2, a (connected) graph can be constructed from 𝑘 ≥ 2 disjoint copies of 𝐾3 by identifying a set of 𝑘 vertices, one from each 𝐾3 , into one vertex. We call the resulting graph a daisy with 𝑘 petals. A daisy with 𝑘 = 4 petals is shown in Figure 18.5. If both 𝐺 and 𝐻 are daisies with the same number of petals, then equality is achieved in the bound in Theorem 18.27.

Figure 18.5 A daisy with four petals

Lemma 18.28 ([237]) For 𝑘 ≥ 2, if 𝐺 and 𝐻 are both daisies with 𝑘 petals, then  Γt (𝐺 □ 𝐻) = 12 Γt (𝐺) Γt (𝐻) + 12 max Γt (𝐺), Γt (𝐻) . We prove next the following Vizing-like bound for the upper total domination number given by Dorbec et al. [237]. Theorem 18.29 ([237]) If 𝐺 and 𝐻 are nontrivial connected graphs, then Γt (𝐺 □ 𝐻) ≥ 12 Γt (𝐺) Γt (𝐻), with equality if and only if 𝐺 = 𝐾2 and 𝐻 = 𝐾2 . Proof Let 𝐺 and 𝐻 be nontrivial connected graphs. If both 𝐺 and 𝐻 have order at least 3, then by Theorem 18.27,  Γt (𝐺□)𝐾2 ≥ 12 Γt (𝐺) Γt (𝐻) + 12 max Γt (𝐺), Γt (𝐻) > 12 Γt (𝐺) Γt (𝐻) + 1, noting that Γt (𝐺) ≥ 2 and Γt (𝐻) ≥ 2. Hence, we may assume that at least one of 𝐺 and 𝐻 has order 2. Renaming 𝐺 and 𝐻 if necessary, we may assume that 𝐻 = 𝐾2 and so Γt (𝐻) = 2. Let 𝑉 (𝐻) = {𝑢, 𝑣}. Consider the set  𝑆 = 𝑉 (𝐺) × {𝑣} = (𝑔, 𝑣) : 𝑔 ∈ 𝑉 (𝐺) . Let (𝑔1 , ℎ1 ) be an arbitrary vertex in 𝐺 □ 𝐻 and let 𝑔2 be a neighbor of 𝑔1 in the graph 𝐺. If ℎ1 = 𝑣, then (𝑔1 , ℎ1 ) is totally dominated by the vertex (𝑔2 , ℎ1 ) in

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𝐺 □ 𝐻. If ℎ1 = 𝑢, then (𝑔1 , ℎ1 ) is totally dominated by the vertex (𝑔1 , 𝑣) in 𝐺 □ 𝐻. Thus, the set 𝑆 is a TD-set of 𝐺 □ 𝐻. Further, for each vertex (𝑔, 𝑣) ∈ 𝑆, we have (𝑔, 𝑢) ∈ epn((𝑔, 𝑣), 𝑆) in the graph 𝐺 □ 𝐻, implying that the TD-set 𝑆 is a minimal TD-set of 𝐺 □ 𝐻. Hence, Γt (𝐺 □ 𝐻) ≥ |𝑆| = |𝑉 (𝐺)| ≥ Γt (𝐺) = 21 Γt (𝐺) Γt (𝐻).

(18.11)

Further, if Γt (𝐺 □ 𝐻) = 12 Γt (𝐺) Γt (𝐻), then the two inequalities in Inequality (18.11) are equalities. In particular, Γt (𝐺) = |𝑉 (𝐺)|. If Δ(𝐺) ≥ 2, then by Observation 14.3, Γt (𝐺) ≤ |𝑉 (𝐺)| − 1, a contradiction. Hence, Δ(𝐺) = 1, implying by the connectivity of 𝐺 that 𝐺 = 𝐾2 . Conversely, if 𝐺 = 𝐾2 and 𝐻 = 𝐾2 , then 𝐺 □ 𝐻 = 𝐶4 and Γt (𝐺 □ 𝐻) = 2 = 12 (2 × 2) = 12 Γt (𝐺) Γt (𝐻). Let 𝐺 and 𝐻 be isolate-free graphs. If 𝐺 has components 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 and 𝐻 has components 𝐻1 , 𝐻2 , . . . , 𝐻ℓ , then 𝐺 □ 𝐻 has 𝑘ℓ components, namely the components 𝐺 𝑖 □ 𝐻 𝑗 for 𝑖 ∈ [𝑘] and 𝑗 ∈ [ℓ]. Since the upper domination number of a graph is the sum of the upper domination numbers of its components, as an immediate consequence of Theorem 18.29, we have the following result. Theorem 18.30 ([237]) If 𝐺 and 𝐻 are isolate-free graphs, then Γt (𝐺 □ 𝐻) ≥ 12 Γt (𝐺) Γt (𝐻), with equality if and only if both 𝐺 and 𝐻 are a disjoint union of copies of 𝐾2 .

Epilogue In this book, we started In the Beginning ... with the mathematical roots of domination in graphs including four famous chessboard problems, and we concluded with the famous unsolved Vizing’s Conjecture. In the intermediate 16 chapters we provided a sampling of the most significant results and open problems involving the core concepts of domination, independent domination, and total domination. In this epilogue, we provide a brief recap of some of the salient open problems presented in the previous 18 chapters and offer some thoughts about the future directions of research in domination in graphs. We know that we have only scratched the surface of this vast field and the highlights included here are in no way comprehensive. The earliest mathematical roots of domination in graphs originated in the game of chess. The following four chess problems, all raised in the period from 1850 to 1900, are about placing queens on 𝑛-by-𝑛 chessboards, as given by Rouse Ball [51] in 1892: 𝒏 Queens Problem. In how many ways, denoted by Q (𝑛), can 𝑛 queens be placed on the squares of an 𝑛 × 𝑛 chessboard so that no two queens can attack each other? This is effectively the first occurrence of the concept of maximum independent sets in a graph. The problem of counting the number of maximum independent sets of queens, originally stated by a German chess player named Max Bezzel in 1848, was effectively solved by Simkin [675] in 2021, as follows: Theorem 19.1 ([675]) There exists a constant 𝛼, where 1.94 < 𝛼 < 1.9449, such that Q (𝑛) 1/𝑛 lim = 𝑒− 𝛼. 𝑛→∞ 𝑛 Queens Domination Problem. What is the minimum number 𝛾(Q𝑛 ) of queens that can be placed on the squares of an 𝑛 × 𝑛 chessboard so that every square is either occupied by a queen or is attacked by a queen? This is effectively the first occurrence of the concept of the domination number 𝛾(𝐺) of a graph 𝐺. The determination of the value of 𝛾(Q𝑛 ) is the oldest © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

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unsolved problem in domination theory. As shown by Östergård and Weakley [626] in 2001 and by Finozhenok and Weakley [299] in 2007, the value of 𝛾(Q𝑛 ) is either known exactly or known to be one of two consecutive values for all 𝑛 ≤ 122. Equally noteworthy along with the Queens Domination Problem is the following remarkably simple conjecture by Fricke et al. [312] given in 1995. Conjecture 19.2 ([312]) For every integer 𝑛 ≥ 1, 𝛾(Q𝑛 ) ≤ 𝛾(Q𝑛+1 ). Queens Independent Domination Problem. What is the minimum number 𝑖(Q𝑛 ) of queens that can be placed on the squares of an 𝑛 × 𝑛 chessboard so that every square is either occupied by a queen or is attacked by a queen and no two queens attack each other? This is effectively the first occurrence of the concept of the independent domination number 𝑖(𝐺) of a graph 𝐺. The determination of the value of 𝑖(Q𝑛 ) is also one of the oldest unsolved problems in domination theory. The following is a noteworthy conjecture relating queens domination to independent queens domination, in [312]. Conjecture 19.3 ([312]) There exists a positive integer 𝑛★ such that for all 𝑛 ≥ 𝑛★, 𝛾(Q𝑛 ) = 𝑖(Q𝑛 ). Queens Total Domination Problem. What is the minimum number 𝛾t (Q𝑛 ) of queens that can be placed on the squares of an 𝑛 × 𝑛 chessboard so that every square is either occupied by a queen or is attacked by a queen and every queen is attacked by at least one other queen? This is effectively the first occurrence of the concept of the total domination number 𝛾t (𝐺) of a graph 𝐺. Turning our attention from chessboards to general graphs, it is surprising that for the relatively simple class of connected cubic graphs, we do not yet know the supremum of the ratio of the domination number to the order of a graph, for graphs of large order. Indeed, the problem of determining a tight upper bound on the domination number of a connected cubic graph of sufficiently large order, in terms of its order, remains open. Among many conjectures on the domination number, it would be interesting to answer fundamental questions such as a best possible upper bound on the domination number of a graph with minimum degree 𝑘, for some given integer 𝑘 ≥ 4. The following conjectures were posed by Henning [460] in 2022. Conjecture 19.4 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤ 31 𝑛. Conjecture 19.5 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤ 14 𝑛. To date, these two conjectures have yet to be settled. Among many conjectures on the total domination number, it would be good to have a deeper understanding of graphs with large total domination and large minimum degree. Standout conjectures

Epilogue

551

are the conjectures due to Thomassé and Yeo [709] in 2007 and Henning and Yeo [493] in 2021. Conjecture 19.6 ([709]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾𝑡 (𝐺) ≤ 4 11 𝑛. Conjecture 19.7 ([493]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤ 4 13 𝑛. It would be pleasing to settle the following two conjectures of Goddard and Henning [352] in 2013. Conjecture 19.8 ([352]) If 𝐺 is a connected cubic graph of order 𝑛 > 10, then 𝑖(𝐺) ≤ 38 𝑛. Conjecture 19.9 ([352]) If 𝐺 ≠ 𝐾3,3 is a connected cubic, bipartite graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛. It would be interesting to explore in more depth the relationship between the size, order, and (total) domination number of a graph. Of special interest would be for each integer Δ ≥ 3, find the smallest value 𝑐 Δ , such that for every isolate-free graph of order 𝑛, size 𝑚, domination number 𝛾, and maximum degree Δ(𝐺) ≤ Δ, the inequality     Δ + 𝑐Δ Δ + 𝑐Δ + 2 𝑚≤ 𝑛− 𝛾 2 2 holds. The analogous problem for total domination which asks for each integer Δ ≥ 3, find the smallest value 𝑟 Δ , such that for every connected graph 𝐺 of order 𝑛 ≥ 3, size 𝑚, total domination number 𝛾t , and maximum degree Δ(𝐺) ≤ Δ, the inequality 𝑚 ≤ 12 (Δ + 𝑟 Δ ) (𝑛 − 𝛾t ) holds. As remarked by Yeo [764], √ it remains an open problem to determine if 𝑟 Δ grows proportionally with ln(Δ) or Δ, or some completely different function. A most intriguing conjecture is due to Verstraëte [730] in 2010. Conjecture 19.10 ([730]) If 𝐺 is a cubic graph of order 𝑛 and girth 𝑔(𝐺) ≥ 6, then 𝛾(𝐺) ≤ 13 𝑛. Equally appealing is a conjecture inspired by Kostochka in 2009. The conjecture was formally posed several years later by Henning [459], but was first posed as a question by Kostochka [536]. Conjecture 19.11 ([459]) If 𝐺 is a cubic bipartite graph of order 𝑛, then 𝛾(𝐺) ≤ 1 3 𝑛. Many conjectures on domination in planar graphs have yet to be settled. Of particular interest is the following conjecture due to Matheson and Tarjan [585] in 1996.

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Conjecture 19.12 ([585]) If 𝐺 is a planar triangulation of sufficiently large order 𝑛, then 𝛾(𝐺) ≤ 41 𝑛. The conjecture due to Murty [602] on diameter-2-critical graphs is a noteworthy conjecture that remains open in the general case. Conjecture 19.13([602])  If 𝐺 is a diameter-2-critical graph with 𝑛 vertices and 𝑚 edges, then 𝑚 ≤ 𝑛2 /4 , with equality if and only if 𝐺 is the complete bipartite graph 𝐾 ⌊ 𝑛2 ⌋, ⌈ 𝑛2 ⌉ . As observed by Hanson and Wang [379], this conjecture can be equated to a conjecture involving total domination in graphs. Conjecture 19.14 ([379]) If 𝐺 has order 𝑛, size 𝑚, and 𝛾t (𝐺) = 3, and the addition   of any edge decreases the total domination number, then 𝑚 > 𝑛(𝑛−2) . 4 Thus, armed with a toolbox of proof techniques in total domination theory, the Murty Conjecture may well be solvable. While not covered in this book, enumeration problems similar to the 𝑛-Queens Problem constitute a rich area for future research. In 2018 and in 2021, Mohr and Rautenbach [596, 597] investigated the problem of determining the number of maximum independent sets (the 𝑛-Queens Problem) in graphs and connected graphs. Results have also been obtained on counting the maximum number of other types of dominating sets. In 2019 Alvarado et al. [20] determined the maximum number of minimum dominating sets in forests and Henning et al. [472] considered the same problem for total dominating sets. It would be interesting to explore these enumeration problems in greater depth (and for other classes of connected graphs). Many open problems and conjectures in domination complexity theory and domination algorithms remain to be solved by those in domination theory who are attracted to complexity and algorithmic problems. Examples of some of these problems and conjectures can be found in the four chapters in [414] on: (i) algorithms and complexity of alliances in graphs, (ii) algorithms and complexity of power domination in graphs, (iii) algorithms and complexity of signed, minus and majority domination, and (iv) self-stabilizing domination algorithms. The subject of domination in graphs is a vibrant and growing area of research, and many problems wait to be solved. Each reader will be attracted to a unique subset of open problems and conjectures in this book, depending on his or her interests and expertise. Therefore, it is not possible for us to rank these problems ranging from the most attractive to the most challenging. However, there is a consensus that Vizing’s Conjecture [734] from 1968 remains the central open problem in domination theory. Conjecture 19.15 (Vizing’s Conjecture [734]) For any two graphs 𝐺 and 𝐻, 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺) 𝛾(𝐻).

(19.1)

Our hope is that the broad spectrum of open problems in domination theory presented and discussed in this book will attract researchers from a wide variety of disciplines to this diverse field of study in graph theory, and serve as a launching pad

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for future research. We are aware that this brief overview is just a small sample of challenging unsolved problems involving the core domination parameters. We have not even attempted to discuss the other rich areas of domination theory, including power domination, connected domination, paired domination, Roman domination, and game domination, to name a few. Entire chapters on each of these other major types of domination can be found in the two companion volumes to this book, [413] and [414]. These companion volumes also include many open problems and ideas for future research in domination theory.

Appendix A

Glossary A.1 Introduction This glossary contains an annotated listing of the most common terms and parameters of graphs which appear in this book. We include these in this appendix so that they need not be repeatedly defined in the chapters. Since graph theory terminology and notation sometimes varies, in this glossary we clarify the terminology that we adopted for this book. In Section A.2, we present basic graph theory definitions. We first define basic numbers that are used in defining the graph theory parameters in Section A.2.1 and then discuss common types of graphs in Section A.2.2. Some fundamental graph constructions are given in Section A.2.3. In Section A.3.1 and Section A.3.2, we present parameters related to connectivity and distance in graphs, respectively. The covering, packing, independence, and matching numbers are defined in Section A.3.3. In Section A.3.4, we define selected domination-type parameters that occur throughout the book. In Section A.4, we present the hypergraph terminology that we use. For more details and terminology, the reader is referred to the two books Fundamentals of Domination in Graphs, [417] and Domination in Graphs: Advanced Topics [416] written and edited by Haynes et al., and the book Total Domination in Graphs by Henning and Yeo [490]. For an integer 𝑘 ≥ 1, let [𝑘] = {1, 2, . . . , 𝑘 } and [𝑘] 0 = [𝑘] ∪ {0} = {0, 1, . . . , 𝑘 }. We denote the set of positive integers by N, that is, N = {1, 2, . . .}.

A.2

Basic Graph Theory Definitions

A graph 𝐺 = (𝑉, 𝐸) consists of a finite nonempty set 𝑉 (𝐺) of objects called vertices together with a possibly empty set 𝐸 (𝐺) of unordered pairs {𝑢, 𝑣} of distinct vertices 𝑢, 𝑣 ∈ 𝑉 (𝐺) called edges, which are denoted 𝑢𝑣 for simplicity. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

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Appendix A. Glossary

The order of a graph 𝐺 is 𝑛(𝐺) = |𝑉 (𝐺)| and the size of 𝐺 is 𝑚(𝐺) = |𝐸 (𝐺)|. If the graph 𝐺 is clear from the context, we omit it in the above expressions and write 𝑉, 𝐸, 𝑛, and 𝑚 rather than 𝑉 (𝐺), 𝐸 (𝐺), 𝑛(𝐺), and 𝑚(𝐺), respectively. If an edge 𝑢𝑣 ∈ 𝐸, then vertices 𝑢 and 𝑣 are said to be adjacent to each other, and vertex 𝑢 (respectively, 𝑣) and edge 𝑢𝑣 are said to be incident to each other, and edge 𝑢𝑣 is said to be between or connect vertices 𝑢 and 𝑣. Similarly, two distinct edges, say 𝑢𝑣 and 𝑣𝑤, are adjacent if they are incident to a common vertex, say 𝑣. A neighbor of a vertex 𝑣 in 𝐺 is any vertex 𝑢 that is adjacent to 𝑣. The open neighborhood of a vertex 𝑣 in 𝐺 is the set of neighbors of 𝑣, denoted N𝐺 (𝑣). Thus, N𝐺 (𝑣) = 𝑢 ∈ 𝑉 : 𝑢𝑣 ∈ 𝐸 (𝐺) . The closed neighborhood of 𝑣 is the set N𝐺 [𝑣] = {𝑣} ∪ N𝐺 (𝑣). For a set of vertices 𝑆 ⊆ 𝑉, the open neighborhood of 𝑆 is the Ð set N𝐺 (𝑆) = 𝑣 ∈𝑆 N𝐺 (𝑣) and its closed neighborhood is the set N𝐺 [𝑆] = N𝐺 (𝑆) ∪𝑆. If the graph 𝐺 is clear from the context, we often omit 𝐺 in the above expressions. For example, we write N(𝑣), N[𝑣], N(𝑆), and N[𝑆] rather than N𝐺 (𝑣), N𝐺 [𝑣], N𝐺 (𝑆), and N𝐺 [𝑆], respectively. For a set 𝑆⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the 𝑆-private neighborhood of 𝑣 is defined by pn[𝑣, 𝑆] = 𝑤 ∈ 𝑉 : N𝐺 [𝑤] ∩ 𝑆 = {𝑣} , while its open 𝑆-private neighborhood  is defined by pn(𝑣, 𝑆) = 𝑤 ∈ 𝑉 : N𝐺 (𝑤) ∩ 𝑆 = {𝑣} . The sets pn[𝑣, 𝑆] \ 𝑆 and pn(𝑣, 𝑆)\𝑆 are equivalent and we define the 𝑆-external private neighborhood of 𝑣 to be this set, abbreviated epn[𝑣, 𝑆] or epn(𝑣, 𝑆). The 𝑆-internal private neighborhood of 𝑣 is defined by ipn[𝑣, 𝑆] = pn[𝑣, 𝑆] ∩ 𝑆 and its open 𝑆-internal private neighborhood is defined by ipn(𝑣, 𝑆) = pn(𝑣, 𝑆) ∩ 𝑆. We define an 𝑆-external private neighbor of 𝑣 to be a vertex in epn(𝑣, 𝑆) and an 𝑆-internal private neighbor of 𝑣 to be a vertex in ipn(𝑣, 𝑆). We denote the degree of a vertex 𝑣 in 𝐺 by deg𝐺 (𝑣) = |N𝐺 (𝑣)|, or simply by deg(𝑣) if the graph 𝐺 is clear from context. For a subset of vertices 𝑆 ⊆ 𝑉, the degree of 𝑣 in 𝑆, denoted deg𝑆 (𝑣), is the number of vertices in 𝑆 that are adjacent to 𝑣; that is, deg𝑆 (𝑣) = |N𝐺 (𝑣) ∩ 𝑆|. In particular, if 𝑆 = 𝑉, then deg𝑆 (𝑣) = deg𝐺 (𝑣). An isolated vertex is a vertex of degree 0 in 𝐺. A graph is isolate-free if it does not contain an isolated vertex. The minimum degree among the vertices of 𝐺 is denoted by 𝛿(𝐺) and the maximum degree by Δ(𝐺). A leaf is a vertex of degree 1, while its only neighbor is called a support vertex. A strong support vertex is a (support) vertex with at least two leaf neighbors. A 𝑘-regular graph is a graph in which every vertex has degree 𝑘. A regular graph is a graph that is 𝑘-regular for some 𝑘 ≥ 0. A 3-regular graph is also called a cubic graph. For disjoint subsets 𝑋 and 𝑌 of vertices of 𝐺, we denote the set of edges that are incident to a vertex of 𝑋 and a vertex of 𝑌 by 𝐺 [𝑋, 𝑌 ], or simply by [𝑋, 𝑌 ] if 𝐺 is clear from context. We also say that [𝑋, 𝑌 ] denotes the set of edges between 𝑋 and 𝑌 in 𝐺. The contraction of an edge 𝑒 = 𝑥𝑦 in a graph 𝐺 is the graph obtained from 𝐺 by removing the vertices 𝑥 and 𝑦 (along with their incident edges), adding a new
vertex 𝑧, and adding edges to make 𝑧 adjacent to the vertices in N𝐺 (𝑥) ∪ N𝐺 (𝑦) \ {𝑥, 𝑦}. The union 𝐺 = 𝐺 1 ∪ 𝐺 2 of two graphs 𝐺 1 and 𝐺 2 has vertex set 𝑉 (𝐺) = 𝑉 (𝐺 1 ) ∪ 𝑉 (𝐺 2 ) and edge set 𝐸 (𝐺 1 ) ∪ 𝐸 (𝐺 2 ). If 𝐺 is a union of 𝑘 pairwise-disjoint copies of a graph 𝐹, we write 𝐺 = 𝑘 𝐹.

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557

Two graphs 𝐺 and 𝐻 are isomorphic, denoted 𝐺  𝐻, if there exists a bijection 𝜙 : 𝑉 (𝐺) → 𝑉 (𝐻) such that two vertices 𝑢 and 𝑣 are adjacent in 𝐺 if and only if the two vertices 𝜙(𝑢) and 𝜙(𝑣) are adjacent in 𝐻. A parameter of a graph 𝐺 is a numerical value (usually a non-negative integer) that can be associated with a graph such that whenever two graphs are isomorphic, they have the same associated parameter value. By a partition of the vertex set 𝑉 of a graph 𝐺 we mean a family 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of nonempty pairwise disjoint sets whose union equals 𝑉, that is, for all 𝑖 and 𝑗 with 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, 𝑉𝑖 ∩ 𝑉 𝑗 = ∅ and the union of the sets 𝑉𝑖 over all 𝑖 ∈ [𝑘] is the vertex set 𝑉, that is, 𝑉 = 𝑉1 ∪ 𝑉2 ∪ · · · ∪ 𝑉𝑘 . For such a partition 𝜋 we will say that 𝜋 has order 𝑘. A walk in a graph 𝐺 from a vertex 𝑢 to a vertex 𝑣, called a (𝑢, 𝑣)-walk, is a finite alternating sequence of vertices and edges, starting with vertex 𝑢 and ending with vertex 𝑣, in which each edge of the sequence joins the vertex that precedes it in the sequence to the vertex that follows it in the sequence. A trail is a walk containing no repeated edges. A path is a walk containing no repeated vertices. A path of order 𝑛 (and size 𝑛 − 1) is denoted by 𝑃𝑛 . A path from a vertex 𝑢 to a vertex 𝑣 is called a (𝑢, 𝑣)-path. A cycle is a trail in which the first and last vertices are the same and all other vertices are distinct. A cycle of order 𝑛 (and size 𝑛) is denoted by 𝐶𝑛 . A chord is an edge between two nonconsecutive vertices of a cycle. The length of a walk, path, trail, or cycle equals the number of edges it contains. A graph 𝐺 = (𝑉, 𝐸) is connected if between any two vertices 𝑢, 𝑣 ∈ 𝑉, there is a (𝑢, 𝑣)-path. A component of a graph is a maximal connected subgraph. An odd (even) component is a component of odd (even) order. Let oc(𝐺) equal the number of odd components of 𝐺. The distance 𝑑 (𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 equals the minimum length of a (𝑢, 𝑣)-path. A graph 𝐺 ′ = (𝑉 ′ , 𝐸 ′ ) is a subgraph of a graph 𝐺 = (𝑉, 𝐸) if 𝑉 ′ ⊆ 𝑉 and ′ 𝐸 ⊆ 𝐸. A subgraph 𝐺 ′ of a graph 𝐺 is a spanning subgraph if 𝑉 ′ = 𝑉. If 𝐺 = (𝑉, 𝐸) and 𝑆 ⊆ 𝑉, then the subgraph of 𝐺 induced by 𝑆 is the graph 𝐺 [𝑆] whose vertex set is 𝑆 and whose edges are all the edges in 𝐸 both of whose vertices are in 𝑆. Let 𝐹 be an arbitrary graph. A graph 𝐺 is said to be 𝐹-free if 𝐺 does not contain 𝐹 as an induced subgraph. If 𝐺 = (𝑉, 𝐸) and 𝑆 ⊆ 𝑉, the subgraph obtained from 𝐺 by deleting all vertices in 𝑆 and all edges incident with vertices in 𝑆 is denoted by 𝐺 − 𝑆; that is, 𝐺 − 𝑆 = 𝐺 [𝑉 \ 𝑆]. If 𝑆 = {𝑣}, we simply denote 𝐺 − {𝑣} by 𝐺 − 𝑣. For a vertex 𝑣 in a graph 𝐺, the graph 𝐺 − 𝑣 is obtained from 𝐺 by removing vertex 𝑣 and all edges incident to 𝑣. A vertex 𝑣 ∈ 𝑉 is a cut-vertex if the graph 𝐺 − 𝑣 has more components than 𝐺. An edge 𝑒 = 𝑢𝑣 is a bridge if the graph 𝐺 − 𝑒 obtained by deleting 𝑒 has more components than 𝐺.

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A graph 𝐺 of order 𝑛 is called 𝑘-vertex-connected (or simply, 𝑘-connected) if 𝑛 ≥ 𝑘 + 1 and the deletion of any 𝑘 − 1 or fewer vertices leaves a connected graph. A complete graph 𝐾𝑛 is a graph of order 𝑛 in which every two vertices are adjacent. A clique is a complete subgraph. A triangle is a subgraph isomorphic to 𝐾3 . A simplicial vertex is a vertex 𝑣 whose open neighborhood N(𝑣) induces a complete graph. A set 𝑆 ⊆ 𝑉 is called independent if no two vertices in 𝑆 are adjacent, that is, the induced subgraph 𝐺 [𝑆] has no edges.

A.2.1 Basic Numbers The following are the most basic numbers that are used in defining the parameters in this book: (a) order 𝑛(𝐺) = |𝑉 (𝐺)|, number of vertices in 𝐺. (b) size 𝑚(𝐺) = |𝐸 (𝐺)|, number of  edges in 𝐺. (c) minimum degree 𝛿(𝐺) = min deg(𝑢) : 𝑢 ∈ 𝑉 (𝐺) , minimum degree of a vertex in 𝐺.  (d) maximum degree Δ(𝐺) = max deg(𝑢) : 𝑢 ∈ 𝑉 (𝐺) , maximum degree of a vertex in 𝐺. (e) degree sequence of a graph 𝐺 with vertex set 𝑉 (𝐺) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } is the sequence 𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 where 𝑑𝑖 = deg𝐺 (𝑣 𝑖 ) for 𝑖 ∈ [𝑛]. Often the degree sequence 𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 is written in non-increasing order and so 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 . If the graph 𝐺 is clear from the context, we omit it in the above expressions and write 𝑛, 𝑚, 𝛿, and Δ rather than 𝑛(𝐺), 𝑚(𝐺), 𝛿(𝐺), and Δ(𝐺), respectively.

A.2.2

Common Types of Graphs

A tree is a connected acyclic graph, that is, a connected graph having no cycles. Equivalently, a tree is a connected graph having size one less than its order. Thus, if 𝑇 is a tree of order 𝑛 and size 𝑚, then 𝑇 is connected and 𝑚 = 𝑛 − 1. A graph 𝐺 is bipartite if its vertex set 𝑉 (𝐺) can be partitioned into two independent sets 𝑋 and 𝑌 , in which case every edge in 𝐺 lies between 𝑋 and 𝑌 . The sets 𝑋 and 𝑌 are called the partite sets of 𝐺. A complete bipartite graph, denoted 𝐾𝑟 ,𝑠 , is a bipartite graph with partite sets 𝑋 and 𝑌 , where |𝑋 | = 𝑟, |𝑌 | = 𝑠, and every vertex in 𝑋 is adjacent to every vertex in 𝑌 . The graph 𝐾𝑟 ,𝑠 has order 𝑟 +𝑠, size 𝑟 𝑠, 𝛿(𝐾𝑟 ,𝑠 ) = min{𝑟, 𝑠}, and Δ(𝐾𝑟 ,𝑠 ) = max{𝑟, 𝑠}. A star is a tree with at most one vertex that is not a leaf. Thus, a star is either the trivial graph 𝐾1 or the complete bipartite graph 𝐾1,𝑘 for some 𝑘 ≥ 1. Equivalently, a star is a tree with diameter at most 2. A claw is an induced copy of the graph 𝐾1,3 . Thus, a claw-free graph is a 𝐾1,3 -free graph. A galaxy is a union of stars. For 𝑟, 𝑠 ≥ 1, a double star 𝑆(𝑟, 𝑠) is a tree with exactly two (adjacent) vertices that are not leaves, one of which has 𝑟 leaf neighbors and the other 𝑠 leaf neighbors. Equivalently, a double star is a tree having diameter equal to 3.

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A diamond is an induced copy of the graph 𝐾4 − 𝑒, which is obtained from a copy of the complete graph of order 4 by deleting one edge 𝑒. A forest is an acyclic, or cycle-free, graph. Thus, a forest is a union of trees. A linear forest is a forest in which every component is a path. A graph 𝐺 = (𝑉, 𝐸) can be embedded on a surface 𝑆 if its vertices 𝑉 can be placed on 𝑆 and all of the edges in 𝐸 can be drawn on the surface 𝑆 in such a way that no two edges intersect. A graph 𝐺 is planar if it can be embedded in the plane; a plane graph is a graph that has been embedded in the plane. Every plane graph divides the plane into regions, called faces. A rooted tree 𝑇𝑟 is a tree having a distinguished vertex labeled 𝑟, called the root. Let 𝑇𝑟 be a rooted tree with root 𝑟. For each vertex 𝑣 ≠ 𝑟 of 𝑇𝑟 , the parent of 𝑣 is the neighbor of 𝑣 on the unique (𝑟, 𝑣)-path between 𝑟 and 𝑣, while a child of 𝑣 is any neighbor of 𝑣 in 𝑇𝑟 other than its parent. The set of children of 𝑣 is denoted by 𝐶 (𝑣). A descendant of 𝑣 is a vertex 𝑢 ≠ 𝑣 such that the unique (𝑟, 𝑢)-path contains 𝑣, while an ancestor of 𝑣 is a vertex 𝑢 ≠ 𝑣 that belongs to the (𝑟, 𝑣)-path in 𝑇𝑟 . In particular, every child of 𝑣 is a descendant of 𝑣 while the parent of 𝑣 is an ancestor of 𝑣. A grandchild of 𝑣 is a descendant of 𝑣 at distance 2 from 𝑣. We let 𝐷 (𝑣) denote the set of descendants of 𝑣, and we define 𝐷 [𝑣] = 𝐷 (𝑣) ∪ {𝑣}. The maximal subtree at 𝑣, denoted 𝑇𝑣 , is the subtree of 𝑇𝑟 induced by 𝐷 [𝑣]. The depth of a vertex 𝑣 in 𝑇𝑟 equals d(𝑟, 𝑣), and the height of 𝑣, denoted ht(𝑣), is the maximum distance from 𝑣 to  a descendant of 𝑣. Thus, ht(𝑣) = max d(𝑣, 𝑤) : 𝑤 is a descendant of 𝑣 .

A.2.3 Graph Constructions Given a graph 𝐺 = (𝑉, 𝐸), the complement of 𝐺 is the graph 𝐺 = (𝑉, 𝐸), where 𝑢𝑣 ∈ 𝐸 if and only if 𝑢𝑣 ∉ 𝐸. Thus, the complement 𝐺 of a graph 𝐺 is formed by taking the vertex set of 𝐺 and joining two vertices by an edge whenever they are not adjacent in 𝐺. Notice that if 𝐻 = 𝐾𝑛 is a complete graph of order 𝑛 and 𝑉 (𝐻) = 𝑉 (𝐺) = 𝑉 where 𝐺 ≠ 𝐾𝑛 , then the edge set 𝐸 (𝐻) of 𝐻 can be partitioned into the two sets 𝐸 and 𝐸, where 𝐺 = (𝑉, 𝐸) and 𝐺 = (𝑉, 𝐸). By a graph product 𝐺 ⊗ 𝐻 on graphs 𝐺 and 𝐻, we mean the graph whose vertex set is the Cartesian product of the vertex sets of 𝐺 and 𝐻 (that is, 𝑉 (𝐺 ⊗ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻)) and whose edge set is determined by the adjacency relations of 𝐺 and 𝐻. Exactly how it is determined depends on what kind of graph product we are considering. The Cartesian product 𝐺 □ 𝐻 of two graphs 𝐺 and 𝐻 is the graph with vertex set 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢 1 , 𝑣 1 ) and (𝑢 2 , 𝑣 2 ) are adjacent if and only if either 𝑢 1 = 𝑢 2 and 𝑣 1 𝑣 2 ∈ 𝐸 (𝐻) or 𝑣 1 = 𝑣 2 and 𝑢 1 𝑢 2 ∈ 𝐸 (𝐺). The direct product (also known as the cross product, tensor product, categorical product, and conjunction) 𝐺 × 𝐻 of two graphs 𝐺 and 𝐻 is the graph with vertex set 𝑉 (𝐺) × 𝑉 (𝐻) where two vertices (𝑢 1 , 𝑣 1 ) and (𝑢 2 , 𝑣 2 ) are adjacent in 𝐺 × 𝐻 if and only if 𝑢 1 𝑢 2 ∈ 𝐸 (𝐺) and 𝑣 1 𝑣 2 ∈ 𝐸 (𝐻). Given a graph 𝐺 = (𝑉, 𝐸), the subdivision of an edge 𝑢𝑣 ∈ 𝐸 consists of deleting the edge 𝑢𝑣 from 𝐸, adding a new vertex 𝑤 to 𝑉, and adding the new edges 𝑢𝑤 and 𝑤𝑣 to 𝐸. In this case, we say that the edge 𝑢𝑣 has been subdivided. In general, for

Appendix A. Glossary

560

an edge 𝑢𝑣 ∈ 𝐸 to be subdivided 𝑘 ≥ 1 times, we mean that edge 𝑢𝑣 is removed and replaced by a (𝑢, 𝑣)-path of length 𝑘 + 1. The subdivision graph 𝑆(𝐺) is the graph obtained from 𝐺 by subdividing every edge of 𝐺 exactly once.
Given a graph 𝐺 = (𝑉, 𝐸), the line graph 𝐿 (𝐺) = 𝐸, 𝐸 (𝐿(𝐺)) , is the graph whose vertices correspond 1-to-1 with the edges in 𝐸, and two vertices are adjacent in 𝐿 (𝐺) if and only if the corresponding edges in 𝐺 have a vertex in common, that is, if and only if the corresponding two edges are adjacent. The corona 𝐺 ◦ 𝐾1 of a graph 𝐺, also denoted cor(𝐺) in the literature, is the graph obtained from 𝐺 by adding for each vertex 𝑣 ∈ 𝑉 a new vertex 𝑣 ′ and the edge 𝑣𝑣 ′ . The edge 𝑣𝑣 ′ is called a pendant edge. For 𝑘 ≥ 2, the 𝑘-corona 𝐺 ◦ 𝑃 𝑘 of 𝐺 is the graph of order (𝑘 + 1)|𝑉 (𝐺)| obtained from 𝐺 by attaching a path of length 𝑘 to each vertex of 𝐺 so that the resulting paths are vertex-disjoint; that is, for each vertex 𝑣 of 𝐺 we add a path 𝑃 𝑘+1 of length 𝑘 and identify the vertex 𝑣 with one end of the path. For any integer 𝑟 ≥ 1, the generalized corona cor(𝐺, 𝑟) of a graph 𝐺 is the graph obtained from 𝐺 by adding 𝑟 pendant edges to each vertex of 𝐺, that is, for each vertex 𝑣 of 𝐺, we add 𝑟 new vertices and add an edge from each new vertex to the vertex 𝑣. In particular, if 𝑟 = 1, then cor(𝐺, 𝑟) is the corona 𝐺 ◦ 𝐾1 . The clique graph 𝐾 (𝐺) of a graph 𝐺 has as its vertices the set of maximal cliques in 𝐺 and two vertices of 𝐾 (𝐺) are adjacent if and only if they intersect as cliques of 𝐺.    The circulant graph 𝐶𝑛 ⟨𝐿⟩ with a given list 𝐿 ⊆ 1, 2, . . . , 21 𝑛 is the graph on 𝑛 ≥ 3 vertices in which the 𝑖 th vertex is adjacent to the (𝑖 + 𝑗) th and (𝑖 − 𝑗) th vertices for each 𝑗 in the list is taken modulo 𝑛. More precisely, if 𝐿 = { 𝑗 1 ,  𝐿 and where  addition 𝑗2 , . . . , 𝑗𝑟 } ⊆ 1, 2, . . . , 12 𝑛 , then the circulant graph 𝐶𝑛 ⟨𝐿⟩ is the graph with  vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑛−1 } and edge set 𝑣 𝑖 𝑣 𝑖+ 𝑗 (mod 𝑛) : 𝑖 ∈ [𝑛 − 1] 0 and 𝑗 ∈ { 𝑗 1 , 𝑗 2 , . . . , 𝑗𝑟 } . For 𝑝 ≥ 3, 𝑝 > 𝑘 ≥ 1, and gcd( 𝑝, 𝑘) = 1, a generalized Petersen graph 𝑃( 𝑝, 𝑘) is the graph with vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑝−1 } ∪ {𝑤 0 , 𝑤 1 , . . . , 𝑤 𝑝−1 } and edges 𝑣 𝑖 𝑤 𝑖 , 𝑣 𝑖 𝑣 𝑖+1 and 𝑤 𝑖 𝑤 𝑖+𝑘 for every 𝑖 ∈ [ 𝑝 − 1] 0 and the subscript sum is taken modulo 𝑝.

A.3

Graph Parameters

In this section, we define most of the graph parameters which appear in this book.

A.3.1

Connectivity and Subgraph Numbers

The following parameters are related to connectivity in graphs: (a) girth 𝑔(𝐺), minimum length of a cycle in 𝐺. If a graph 𝐺 has no cycles, we say that 𝑔(𝐺) = 0. (b) clique number 𝜔(𝐺), maximum order of a clique of 𝐺. (c) components 𝑐(𝐺), number of maximal connected subgraphs of 𝐺.

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(d) A vertex cut of a connected graph 𝐺 = (𝑉, 𝐸) is a subset 𝑆 ⊆ 𝑉 with the property that 𝐺 − 𝑆 is disconnected (has more than one component). A vertex cut 𝑆 is a 𝑘-vertex cut if |𝑆| = 𝑘. (e) vertex connectivity 𝜅(𝐺), is the minimum cardinality of a vertex cut of 𝐺 if 𝐺 ≠ 𝐾𝑛 and 𝜅(𝐺) = 𝑛 − 1 if 𝐺 is a complete graph 𝐾𝑛 of order 𝑛. A graph 𝐺 is 𝑘-vertex-connected (or 𝑘-connected) if 𝜅(𝐺) ≥ 𝑘 for some integer 𝑘 ≥ 0. Thus, 𝜅(𝐺) is the smallest number of vertices whose deletion from 𝐺 produces a disconnected graph or trivial graph 𝐾1 . A nontrivial graph has vertex connectivity 0 if and only if it is disconnected. (f) An edge cut of a nontrivial connected graph 𝐺 = (𝑉, 𝐸) is a nonempty subset 𝐹 ⊂ 𝐸 with the property that 𝐺 − 𝐹 is disconnected (has more than one component). Thus, the deletion of the edges in an edge cut 𝐹 from the connected graph 𝐺 results in a disconnected graph. An edge cut 𝐹 is a 𝑘-edge cut if |𝐹 | = 𝑘. (g) edge connectivity 𝜆(𝐺), is the minimum cardinality of an edge cut of 𝐺 if 𝐺 is nontrivial, while 𝜆(𝐾1 ) = 0. A graph 𝐺 is 𝑘-edge-connected if 𝜆(𝐺) ≥ 𝑘 for some integer 𝑘 ≥ 0. Thus, 𝜆(𝐺) is the smallest number of edges whose deletion from 𝐺 produces a disconnected graph or trivial graph 𝐾1 . Hence, 𝜆(𝐺) = 0 if and only if 𝐺 is disconnected or trivial.

A.3.2

Distance Numbers

The following parameters are related to the distances 𝑑 (𝑢, 𝑣) = 𝑑𝐺 (𝑢, 𝑣) between vertices in a graph 𝐺. (a) distance 𝑑 (𝑢, 𝑣), between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 is the minimum length of a (𝑢, 𝑣)-path in 𝐺. (b) eccentricity ecc(𝑣) = ecc𝐺 (𝑣), of a vertex 𝑣 in a connected graph 𝐺 is the maximum of the distances from 𝑣 to the other vertices of 𝐺; that is, ecc(𝑣) = max 𝑑 (𝑢, 𝑣) : 𝑢 ∈ 𝑉 .  (c) diameter diam(𝐺) = max 𝑑 (𝑢, 𝑣) : 𝑢, 𝑣 ∈ 𝑉 , is the maximum distance  among all pairs of vertices of 𝐺. Equivalently, diam(𝐺) = max ecc(𝑣) : 𝑣 ∈ 𝑉 is the maximum eccentricity taken over all vertices of 𝐺. (d) radius rad(𝐺), isthe minimum eccentricity taken over all vertices of 𝐺; that is, rad(𝐺) = min ecc(𝑣) : 𝑣 ∈ 𝑉 . (e) The center of a graph 𝐺, denoted 𝐶 (𝐺), is the subgraph of 𝐺 induced by the vertices in 𝐺 whose eccentricity equals the radius of 𝐺. A vertex 𝑣 ∈ 𝐶 (𝐺) is called a central vertex of 𝐺.

A.3.3 Covering, Packing, Independence, and Matching Numbers A set 𝑆 ⊆ 𝑉 of vertices is called independent if no two vertices in 𝑆 are adjacent. Two edges in a graph 𝐺 are independent if they are not adjacent in 𝐺; that is, they do not have a vertex in common. A set of pairwise independent edges of 𝐺 is called a matching in 𝐺, while a matching of maximum cardinality is a maximum matching. Given a matching 𝑀,

562

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we denote by 𝑉 [𝑀] the set of vertices in 𝐺 incident with an edge in 𝑀. A perfect matching is a matching 𝑀 in which every vertex 𝑣 ∈ 𝑉 is incident with exactly one edge of 𝑀. Thus, if 𝐺 has a perfect matching 𝑀, then 𝐺 has even order 𝑛 = 2𝑘 for some 𝑘 ≥ 1 and |𝑀 | = 𝑘. A vertex and an edge are said to cover each other in a graph 𝐺 if they are incident in 𝐺. A vertex cover in 𝐺 is a set of vertices that covers all the edges of 𝐺, while an edge cover in 𝐺 is a set of edges that covers all the vertices of 𝐺. Thus, a vertex cover in 𝐺 is a set 𝑆 of vertices such that every edge is incident to a vertex in 𝑆. A subset 𝑆 of vertices in 𝐺 is a packing if the closed neighborhoods of vertices in 𝑆 are pairwise disjoint. Equivalently, 𝑆 is a packing in 𝐺 if for any two vertices 𝑢 and 𝑣 in 𝑆, 𝑑 (𝑢, 𝑣) > 2. Thus, if 𝑆 is a packing in 𝐺, then |N𝐺 [𝑣] ∩ 𝑆| ≤ 1 for every vertex 𝑣 ∈ 𝑉 (𝐺). A packing is also called a 2-packing in the literature. More generally, for 𝑘 ≥ 2 a 𝑘-packing in 𝐺 is a set of vertices 𝑆 in 𝐺 such that for any two vertices 𝑢 and 𝑣 in 𝑆, the distance 𝑑 (𝑢, 𝑣) > 𝑘. A subset 𝑆 of vertices in 𝐺 is an open packing if the open neighborhoods of vertices in 𝑆 are pairwise disjoint. Thus, if 𝑆 is an open packing in 𝐺, then |N𝐺 (𝑣) ∩ 𝑆| ≤ 1 for every vertex 𝑣 ∈ 𝑉 (𝐺). All of the parameters in this subsection have to do with sets that are independent or cover other sets. These include some of the most basic of all parameters in graph theory. (a) vertex independence numbers 𝑖(𝐺) and 𝛼(𝐺), minimum and maximum cardinality of a maximal independent set in 𝐺. The lower vertex independence number 𝑖(𝐺) is the independent domination number of 𝐺, while the upper vertex independence number 𝛼(𝐺) is the independence number of 𝐺. While the notation 𝑖(𝐺) is fairly standard for the independent domination number, we remark that the independence number is also denoted by 𝛽0 (𝐺) in the literature. (b) vertex cover numbers 𝛽(𝐺) and 𝛽+ (𝐺), minimum and maximum cardinality of a minimal vertex cover in 𝐺. We remark that the vertex cover number 𝛽(𝐺) is also denoted by 𝜏(𝐺) or by 𝛼0 (𝐺) in the literature. (c) edge covering numbers 𝛽′ (𝐺) and 𝛽′+ (𝐺), minimum and maximum cardinality of a minimal edge cover in 𝐺. (d) 𝑘-packing numbers 𝑝 𝑘 (𝐺) and 𝑃 𝑘 (𝐺), minimum and maximum cardinality of a maximal 𝑘-packing in 𝐺 for 𝑘 ≥ 2. When 𝑘 = 2, the (upper) 2-packing number 𝑃2 (𝐺) is called the packing number of 𝐺, and is denoted by 𝜌(𝐺). Thus, 𝜌(𝐺) is the maximum cardinality of a 2-packing in 𝐺. (e) open 𝑘-packing numbers 𝑝 o𝑘 (𝐺) and 𝑃o𝑘 (𝐺), minimum and maximum cardinality of a maximal open 𝑘-packing in 𝐺 for 𝑘 ≥ 2. When 𝑘 = 2, the (upper) open 2-packing number 𝑃2o (𝐺) is called the open packing number of 𝐺, and is denoted by 𝜌 o (𝐺). Thus, 𝜌 o (𝐺) is the maximum cardinality of an open 2-packing in 𝐺. (f) matching numbers 𝛼′− (𝐺) and 𝛼′ (𝐺), minimum and maximum cardinality of a maximal matching in 𝐺. The upper matching number 𝛼′ (𝐺) is simply called the matching number of 𝐺. Thus, the matching number 𝛼′ (𝐺) is the maximum cardinality of a matching in 𝐺. It should be noted that by a

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well-known theorem of Gallai, if 𝐺 is an isolate-free graph of order 𝑛, then 𝛼(𝐺) + 𝛽(𝐺) = 𝑛 = 𝛼′ (𝐺) + 𝛽′ (𝐺). We remark that the matching number is also denoted by 𝛽1 (𝐺) in the literature.

A.3.4

Core Domination Numbers

A dominating set in a graph 𝐺 is a set 𝑆 of vertices such that every vertex not in 𝑆 has a neighbor in 𝑆. Thus, if 𝑆 is a dominating set of 𝐺, then N𝐺 [𝑆] = 𝑉 (𝐺) and every vertex in 𝑉 (𝐺) \ 𝑆 is adjacent to at least one vertex in 𝑆. For subsets 𝑋 and 𝑌 of vertices of 𝐺, if 𝑌 ⊆ N𝐺 [𝑋], then the set 𝑋 dominates the set 𝑌 in 𝐺. In particular, if 𝑋 dominates 𝑉 (𝐺), then 𝑋 is a dominating set of 𝐺. If no proper subset of a dominating set 𝑆 is a dominating set of 𝐺, then 𝑆 is a minimal dominating set. The many variations of dominating sets in a graph 𝐺 = (𝑉, 𝐸) are based on (i) conditions which are placed on the subgraph 𝐺 [𝑆] induced by a dominating set 𝑆, (ii) conditions which are placed on the vertices in 𝑉 \ 𝑆, or (iii) conditions which are placed on the edges between vertices in 𝑆 and vertices in 𝑉 \ 𝑆. We mention only the core domination numbers that comprise the study in this book. An independent dominating set, abbreviated ID-set, in a graph 𝐺 is a set 𝑆 of vertices that is both dominating and independent. Thus, a subset 𝑆 ⊆ 𝑉 (𝐺) is an ID-set in 𝐺 if 𝑆 is a dominating set of 𝐺 and 𝑆 is an independent set. A total dominating set, abbreviated TD-set, in an isolate-free graph 𝐺 is a set 𝑆 of vertices such that every vertex in 𝑉 (𝐺) is adjacent to at least one vertex in 𝑆. Thus, a subset 𝑆 ⊆ 𝑉 (𝐺) is a TD-set in 𝐺 if N𝐺 (𝑆) = 𝑉 (𝐺). If no proper subset of 𝑆 is a TD-set of 𝐺, then 𝑆 is a minimal TD-set of 𝐺. Every isolate-free graph has a TD-set since 𝑆 = 𝑉 (𝐺) is such a set. If 𝑋 and 𝑌 are subsets of vertices in 𝐺, then the set 𝑋 totally dominates the set 𝑌 in 𝐺 if 𝑌 ⊆ N𝑌 (𝑋). In particular, if 𝑋 totally dominates 𝑉 (𝐺), then 𝑋 is a TD-set. (a) domination numbers 𝛾(𝐺) and Γ(𝐺), minimum and maximum cardinalities of a minimal dominating set in 𝐺. The parameters 𝛾(𝐺) and Γ(𝐺) are referred to as the domination number and upper domination number of 𝐺, respectively. A dominating set of 𝐺 of cardinality 𝛾(𝐺) is called a 𝛾-set of 𝐺, while a minimal dominating set of cardinality Γ(𝐺) is called a Γ-set of 𝐺. (b) independent domination number 𝑖(𝐺), minimum cardinality of an ID-set in 𝐺, that is, a set that is both dominating and independent. An ID-set of 𝐺 of cardinality 𝑖(𝐺) is called an 𝑖-set of 𝐺. (c) independence number 𝛼(𝐺), maximum cardinality of an independent set. An independent set of 𝐺 of cardinality 𝛼(𝐺) is called an 𝛼-set of 𝐺. (d) total domination numbers 𝛾t (𝐺) and Γt (𝐺), minimum and maximum cardinalities of a minimal total dominating set of 𝐺. The parameters 𝛾t (𝐺) and Γt (𝐺) are referred to as the total domination number and upper total domination number of 𝐺, respectively. A TD-set of 𝐺 of cardinality 𝛾t (𝐺) is called a 𝛾t -set of 𝐺, while a minimal TD-set of cardinality Γt (𝐺) is called a Γt -set of 𝐺.

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A.3.5 Domatic Partitions A domatic 𝑘-partition of a graph 𝐺 is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉 (𝐺) into 𝑘 dominating sets. In general, a domatic partition is a domatic 𝑘-partition for some unspecified integer 𝑘, and the domatic number dom(𝐺) is the maximum order 𝑘 of any domatic 𝑘-partition of 𝐺. A vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of a graph 𝐺 such that for every 𝑖, 𝑗 with 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, either 𝑉𝑖 dominates 𝑉 𝑗 or 𝑉 𝑗 dominates 𝑉𝑖 , or both, is called an upper domatic partition of 𝐺. The upper domatic number, denoted Dom(𝐺), equals the maximum order of an upper domatic partition of 𝐺. A graph 𝐺 is indominable or idomatic if it has a domatic partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } in which every subset 𝑉𝑖 , for 𝑖 ∈ [𝑘], is an ID-set, in which case 𝜋 is called an independent domatic partition. Thus, an independent domatic partition of 𝐺 is a collection of ID-sets and is also a proper coloring of 𝐺. The idomatic number, denoted idom(𝐺), equals the maximum order of an independent domatic partition of 𝐺. If a graph 𝐺 does not have an independent domatic partition, then we define idom(𝐺) = 0. A total domatic 𝑘-partition of an isolate-free graph 𝐺 = (𝑉, 𝐸) is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉 into 𝑘 TD-sets. In general, a total domatic partition is a total domatic 𝑘-partition for some unspecified integer 𝑘. A total domatic partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } is a vertex partition such that every vertex 𝑣 ∈ 𝑉 is adjacent to at least one vertex in every set 𝑉𝑖 in 𝜋. The total domatic number of a graph 𝐺, denoted tdom(𝐺), equals the maximum order of a total domatic partition of 𝐺. Thus, tdom(𝐺) is the maximum number of TD-sets into which the vertex set of 𝐺 can be partitioned. A graph 𝐺 is a TI-graph if its vertex set can be partitioned into a TD-set and an ID-set. A graph whose vertex set can be partitioned into two TD-sets is called a total dominating partitionable graph, abbreviated TDP-graph.

A.3.6

Perfect and Efficient Dominating Sets

A perfect dominating set is a dominating set 𝑆 in a graph 𝐺 such that every vertex not in 𝑆 has exactly one neighbor in 𝑆, that is, for every vertex 𝑢 ∈ 𝑉 \ 𝑆, |N(𝑢) ∩ 𝑆| = 1. Let 𝛾p (𝐺) and Γp (𝐺) denote the minimum and maximum cardinalities of a minimal perfect dominating set in 𝐺. Trivially, the entire vertex set 𝑉 is a perfect dominating set for any graph 𝐺, but for some graphs the set 𝑉 is a minimal perfect dominating set. An example of such a graph is the complete tripartite graph 𝐾2,2,2 of order 𝑛 = 6. For this graph, 𝛾p (𝐾2,2,2 ) = 𝑛 = 6. An efficient dominating set is a dominating set 𝑆 in a graph 𝐺 such that every vertex has exactly one neighbor in 𝑆, that is, for every vertex 𝑣 ∈ 𝑉, |N[𝑣] ∩ 𝑆| = 1. It is well known that not every graph has an efficient dominating set (e.g. the cycle 𝐶5 ), but if a graph 𝐺 has an efficient dominating set, then every efficient dominating set in 𝐺 has the same cardinality and it equals 𝛾(𝐺). We remark that efficient dominating sets are also known as perfect codes, the first mention of which is due to Biggs in 1973 [76].

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A perfect total dominating set is a TD-set 𝑆 in a graph 𝐺 such that every vertex not in 𝑆 has exactly one neighbor in 𝑆, that is, for every vertex 𝑢 ∈ 𝑉 \ 𝑆, |N(𝑢) ∩ 𝑆| = 1. A set 𝑆 is an efficient total dominating set, also called an open efficient dominating set, if for every vertex 𝑣 ∈ 𝑉, |N(𝑣) ∩ 𝑆| = 1, or equivalently, the open neighborhood N(𝑣) of 𝑣 contains exactly one vertex in 𝑆.

A.3.7

Enclaveless Sets

For 𝑆 ⊆ 𝑉, a vertex 𝑣 ∈ 𝑆 is called an enclave of 𝑆 if N[𝑣] ⊆ 𝑆. A set is said to be enclaveless if it does not contain any enclaves. Thus, if 𝑆 is an enclaveless set, then every vertex in 𝑆 has a neighbor in 𝑉 \ 𝑆. The enclaveless number Ψ(𝐺) is the maximum cardinality of an enclaveless set of 𝐺 and 𝜓(𝐺) is the minimum cardinality of a maximal enclaveless set.

A.3.8

Grid Graphs

A grid graph, or a grid, is a Cartesian product of the  form 𝑃𝑚 □ 𝑃𝑛 for integers 𝑚, 𝑛 ≥ 1. An 𝑚 × 𝑛 grid 𝐺 𝑚,𝑛 has a vertex set 𝑉 = (𝑖, 𝑗) : 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛] with vertex (𝑖, 𝑗) adjacent to vertex (𝑘, ℓ) if 𝑖 = 𝑘 and | 𝑗 − ℓ| = 1 or 𝑗 = ℓ and |𝑖 − 𝑘 | = 1. For a fixed value of 𝑖, the set of vertices of the form (𝑖, 𝑗), 𝑗 ∈ [𝑛], is called the 𝑖 th row of 𝐺 𝑚,𝑛 , and for a fixed value of 𝑗, the set of vertices of the form (𝑖, 𝑗), 𝑖 ∈ [𝑚], is called the 𝑗 th column of 𝐺 𝑚,𝑛 . Thus, the subgraph induced by the vertices in any given row 𝑖 is a copy of the path 𝑃𝑛 , while the subgraph induced by the vertices in any given column 𝑗 is a copy of the path 𝑃𝑚 . A cylindrical graph, or a cylinder, is a Cartesian product of the form 𝑃𝑚 □ 𝐶𝑛 or 𝐶𝑚 □ 𝑃𝑛 , for integers 𝑚 ≥ 1 and 𝑛 ≥ 3, or 𝑚 ≥ 3 and 𝑛 ≥ 1, respectively. A toroidal graph, or a torus, is a Cartesian product of the form 𝐶𝑚 □ 𝐶𝑛 for integers 𝑚, 𝑛 ≥ 3.

A.4

Hypergraph Terminology and Concepts

For hypergraph theory terminology and notation, we follow [490]. Hypergraphs are systems of sets which are conceived as natural extensions of graphs. A hypergraph 𝐻 = (𝑉, 𝐸) consists of a finite set 𝑉 = 𝑉 (𝐻) of elements, called vertices, together with a finite multiset 𝐸 = 𝐸 (𝐻) of subsets of 𝑉, called hyperedges or simply edges. The order of 𝐻 is 𝑛(𝐻) = |𝑉 | and the size of 𝐻 is 𝑚(𝐻) = |𝐸 |. For simplicity, we sometimes denote 𝑛(𝐻) and 𝑚(𝐻) by 𝑛 𝐻 and 𝑚 𝐻 , respectively. The size of a hyperedge or edge equals the number of vertices it contains. A 𝑘-edge in 𝐻 is an edge of size 𝑘. A hypergraph 𝐻 is said to be 𝑘-uniform if every edge of 𝐻 is a 𝑘-edge. Every (simple) graph is a 2-uniform hypergraph. Thus, graphs are special hypergraphs. For 𝑖 ≥ 2, we denote the number of edges in 𝐻 of size 𝑖 by 𝑒 𝑖 (𝐻). The degree of a vertex 𝑣 in 𝐻, denoted by deg 𝐻 (𝑣), is the number of edges of 𝐻 which contain 𝑣. A degree-𝑘 vertex is a vertex of degree 𝑘. The minimum and maximum degrees among the vertices of 𝐻 are denoted by 𝛿(𝐻) and Δ(𝐻), respectively.

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Two vertices 𝑥 and 𝑦 of 𝐻 are adjacent if there is an edge 𝑒 ∈ 𝐸 (𝐻) such that {𝑥, 𝑦} ⊆ 𝑒. The neighborhood of a vertex 𝑣 in 𝐻, denoted N 𝐻 (𝑣) or simply N(𝑣) if 𝐻 is clear from the context, is the set of all vertices different from 𝑣 that are adjacent to 𝑣. Two vertices 𝑥 and 𝑦 of 𝐻 are connected if there is a sequence 𝑥 = 𝑣 0 , 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 = 𝑦 of vertices of 𝐻 in which 𝑣 𝑖−1 is adjacent to 𝑣 𝑖 for 𝑖 ∈ [𝑘]. A connected hypergraph is a hypergraph in which every pair of vertices are connected. A maximal connected subhypergraph of 𝐻 is a component of 𝐻. Thus, no edge in 𝐻 contains vertices from different components. A hypergraph 𝐻 is called an intersecting hypergraph if every two distinct edges of 𝐻 have a non-empty intersection. A subset 𝑇 of vertices in a hypergraph 𝐻 is a transversal (also called vertex cover or hitting set in many papers) if 𝑇 has a nonempty intersection with every edge of 𝐻. The transversal number 𝜏(𝐻) of 𝐻 is the minimum size of a transversal in 𝐻. A transversal of size 𝜏(𝐻) is called a 𝜏-transversal of 𝐻. A hypergraph 𝐻 is called a linear hypergraph if every two distinct edges of 𝐻 intersect in at most one vertex. We say that two edges in 𝐻 overlap if they intersect in at least two vertices. A linear hypergraph therefore has no overlapping edges. Given a hypergraph 𝐻 and subset 𝑋 ⊆ 𝑉 (𝐻) of vertices, we let 𝐻 − 𝑋 denote the hypergraph obtained from 𝐻 by removing the vertices 𝑋 from 𝐻, removing all edges that intersect 𝑋, and removing all resulting isolated vertices, if any. Further, if 𝑋 = {𝑥}, we simply write 𝐻 − 𝑥.

Appendix B

Books Containing Information on Domination in Graphs In this appendix, we list, in chronological order, books containing information on domination in graphs. We include books with at least one chapter on domination, but we do not include the many books of conference proceedings containing papers on domination. 1. W.W. Rouse Ball, Mathematical Recreations and Problems of Past and Present Times (MacMillan, London, 1892), xii + 232 pp 2. D. Kőnig, Theorie der endlichen und unendlichen graphen, Kombinatorische Topologie der Streckenkomplexe (Akad. Verlagsges., Leipzig, 1936), x + 258 pp, Chapter VII Basisprobleme für gerichtete Graphen 3. C. Berge, The Theory of Graphs and Its Applications (Methuen & Co. Ltd., London, 1962), x + 247 pp, Chapter 4 The Fundamental Numbers of the Theory of Graphs; Chapter 5 Kernels of a Graph 4. Ø. Ore, Theory of Graphs. Amer. Math. Soc. Colloq. Publ. 38 (Amer. Math. Soc., Providence, RI, 1962), 206–212, Chapter 13 Dominating Sets, Covering Sets, and Independent Sets 5. A.M. Yaglom, I.M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I: Combinatorial Analysis and Probability Theory (Holden-Day, San Francisco, 1964), viii + 231 pp 6. C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam-London, 1973), xiv + 528 pp, Chapter 13 Stability Number; Chapter 14 Kernels and Grundy Functions 7. S.T. Hedetniemi, R.C. Laskar (Eds), Topics on Domination. Annals of Discrete Mathematics 48 (North Holland, 1991), 280 pp 8. G. Chartrand, L. Lesniak, Graphs and Digraphs (Chapman & Hall, London, 1996), x + 422 pp, Chapter 10 Domination in Graphs 9. T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds), Domination in Graphs: Advanced Topics. Monographs in Pure and Applied Mathematics 209 (Marcel Dekker, New York, 1998), xii + 497 pp © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

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568

Appendix B. Books Containing Information on Domination in Graphs

10. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs. Monographs in Pure and Applied Mathematics 208 (Marcel Dekker Inc., New York, 1998), ix + 446 pp 11. J.J. Watkins, Across the Board: The Mathematics of Chessboard Problems (Princeton University Press, Princeton, NJ, 2004), xii + 257 pp, Chapter 7 Domination; Chapter 8 Queens Domination; Chapter 9 Domination on Other Surfaces; Chapter 10 Independence 12. G. Agnarson, R. Greenlaw, Graph Theory: Modeling, Applications, and Algorithms (Prentice Hall, Upper Saddle River, NJ, 2007), xviii + 446 pp, Chapter 10 Independence, Dominance, and Matchings 13. W. Imrich, S. Klavžar, D.F. Rall, Topics in Graph Theory: Graphs and Their Cartesian Product (A.K. Peters, Ltd., Wellesley, MA, 2008), xiv + 205 pp, Chapter 10 Domination 14. G. Chartrand, P. Zhang, Chromatic Graph Theory (CRC Press, Boca Raton, FL, 2009), 504 pp, Chapter 14 Colorings, Distance, and Domination 15. V.R. Kulli, Theory of Domination in Graphs (Vishwa International Publ., Gulbarga, 2010), xii + 284 pp 16. D.-Z. Du, P.-J. Wan, Connected Dominating Set: Theory and Applications. Springer Optimization and Its Applications (Springer, New York, NY, 2012), x + 203 pp 17. M.A. Henning, A. Yeo, Total Domination in Graphs. Springer Monographs in Mathematics (Springer, New York, NY, 2013), xiv + 178 pp 18. J.L. Gross, J. Yellen, P. Zhang (Eds), Handbook of Graph Theory (Chapman & Hall/CRC Press, Boca Raton, FL, 2014), xix + 1610 pp, Section 9.2 Domination in Graphs, by T.W. Haynes, M.A. Henning 19. R. Gera, S.T. Hedetniemi, C. Larson (Eds), Graph Theory, Favorite Conjectures and Open Problems. Problem Books in Mathematics 1 (Springer, Cham, 2016), ix + 291 pp, Chapter 5 All My Favorite Conjectures are Critical, by T.W. Haynes; Chapter 8 My Top 10 Graph Theory Conjectures and Open Problems, by S.T. Hedetniemi; Chapter 14 It Is All Labeling, by P.J. Slater; Chapter 15 My Favorite Domination Conjectures in Graph Theory Are Bounded, by M.A. Henning 20. R. Gera, T.W. Haynes, S.T. Hedetniemi (Eds), Graph Theory, Favorite Conjectures and Open Problems. Problem Books in Mathematics 2 (Springer, Cham, 2018), vi + 281 pp, Chapter 12 My Favorite Domination Game Conjectures, by M.A. Henning; Chapter 14 An Annotated Glossary of Graph Theory Parameters, with Conjectures, by R. Gera, T.W. Haynes, S.T. Hedetniemi, M.A. Henning 21. K.H. Rosen, D.R. Shier, W. Goddard (Eds), Handbook of Discrete and Combinatorial Mathematics. Discrete Mathematics and its Applications (Boca Raton) (CRC Press, Boca Raton, FL, 2018), 1590 pp, Section 8.6 Graph Colorings, Labelings, & Related Parameters, by A.T. White, T.W. Haynes, M.A. Henning, G. Hurlbert, J.A. Gallian 22. G. Chartrand, T.W. Haynes, M.A. Henning, P. Zhang, From Domination to Coloring, Stephen Hedetniemi’s Graph Theory and Beyond. SpringerBriefs in Mathematics (Springer, Cham, 2019), x + 94 pp, Chapter 1 Pioneer of Domination

Appendix B. Books Containing Information on Domination in Graphs

23. 24. 25. 26.

569

in Graphs; Chapter 2 Key Domination Parameters; Chapter 3 Dominating Functions; Chapter 4 Domination Related Parameters and Applications T.W. Haynes, S.T. Hedetniemi, M.A. Henning (Eds), Topics in Domination in Graphs. Developments in Mathematics, viii + 545 pp (Springer, Cham, 2020) I.G. Yero (Ed), Distances and Domination in Graphs (MDPI AG, Basel, 2020), ix + 133 pp B. Brešar, M.A. Henning, S. Klavžar, D.F. Rall, Domination Games Played on Graphs. SpringerBriefs in Mathematics (Springer, Cham, 2021), x + 122 pp T.W. Haynes, S.T. Hedetniemi, M.A. Henning (Eds), Structures of Domination in Graphs. Developments in Mathematics, iv + 536 pp (Springer, Cham, 2021)

Appendix C

Surveys Containing Information on Domination in Graphs In this appendix, we list selected surveys containing information on domination in graphs. 1. A. Agrawal, R.E. Barlow, A survey of network reliability and domination theory. Oper. Res. 32 (1984), no. 3, 478–492 2. N. Ananchuen, W. Ananchuen, M.D. Plummer, Domination in graphs. Structural Analysis of Complex Networks, ed. M. Dehmer, 73–104 (Birkhäuser/Springer, New York, 2011) 3. C. Bazgan, L. Brankovic, K. Casel, H. Fernau, On the complexity landscape of the domination chain. Algorithms and Discrete Applied Mathematics, 61–72. Lecture Notes in Comput. Sci. 9602 (Springer, Cham, 2016) 4. A. Brandstädt, Efficient domination and efficient edge domination: a brief survey. Algorithms and Discrete Applied Mathematics, 1–14. Lecture Notes in Comput. Sci. 10743 (Springer, Cham, 2018) 5. B. Brešar, P. Dorbec, W. Goddard, B.L. Hartnell, M.A. Henning, S. Klavžar, Vizing’s conjecture: a survey and recent results. J. Graph Theory 69 (2012), 46–76 6. M. Chellali, O. Favaron, A. Hansberg, L. Volkmann, 𝑘-domination and 𝑘independence in graphs: a survey. Graphs Combin. 28 (2012), no. 1, 1–55 7. M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, L. Volkmann, A survey on Roman domination parameters in directed graphs. J. Combin. Math. Combin. Comput. 115 (2020), 141–171 8. M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, L. Volkmann, Varieties of Roman domination II. AKCE Int. J. Graphs Comb. 17 (2020), no. 3, 966–984 9. E.J. Cockayne, Domination of undirected graphs—a survey. Theory and Applications of Graphs (Proc. Internat. Conf. Western Mich. Univ., Kalamazoo, MI, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

571

572

10. 11. 12. 13. 14. 15. 16.

17. 18.

19. 20. 21. 22. 23. 24. 25.

26. 27.

Appendix C. Surveys Containing Information on Domination in Graphs 1976), eds. Y. Alavi, D.R. Lick, 141–147. Lecture Notes in Math. 642 (Springer, Berlin, 1978) E.J. Cockayne, Chessboard domination problems. Discrete Math. 86 (1990), no. 1-3, 13–20 W.J. Desormeaux, M.A. Henning, Paired domination in graphs: a survey and recent results. Util. Math. 94 (2014), 101–166 W. Goddard, M.A. Henning, Independent domination in graphs: a survey and recent results. Discrete Math. 313 (2013), no. 7, 839–854 F. Harary, T.W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs. Discrete Math. 155 (1996), no. 1-3, 99–105 J. Hattingh, Restrained and total restrained domination in graphs. Not. S. Afr. Math. Soc. 41 (2010), no. 1, 2–15 T.W. Haynes, Domination in graphs: a brief overview. J. Combin. Math. Combin. Comput. 24 (1997), 225–237 T.W. Haynes, M.A. Henning, L.C. van der Merwe, A. Yeo, Progress on the Murty-Simon Conjecture on diameter-2 critical graphs: a survey. J. Comb. Optim. 30 (2015), no. 3, 579–595 T.W. Haynes, M.A. Henning, P. Zhang, A survey of stratified domination in graphs. Discrete Math. 309 (2009), no. 19, 5806–5819 S.M. Hedetniemi, S.T. Hedetniemi, R. Laskar, A. Majumdar, Domination, independence and irredundance in total graphs: a brief survey. Graph Theory, Combinatorics and Applications, Vol. 1, 2 (Kalamazoo, MI, 1992), eds. Y. Alavi, A. Schwenk, 671–683 (Wiley-Intersci. Publ., New York, 1995) M.A. Henning, Domination in graphs: a survey, 139–172. Surveys in Graph Theory 116 (San Francisco, CA, 1996) M.A. Henning, Recent bounds on domination parameters. J. Combin. Math. Combin. Comput. 34 (2000), 177–196 M.A. Henning, A survey of selected recent results on total domination in graphs. Discrete Math. 309 (2009), no. 1, 32–63 M.A. Henning, Bounds on domination parameters in graphs: a brief survey. Discuss. Math. Graph Theory 42 (2022), no. 3, 665–708 M.A. Henning, O.R. Oellermann, H.C. Swart, The diversity of domination. Discrete Math. 161 (1996), no. 1-3, 161–173 W.F. Klostermeyer, A taxonomy of perfect domination. J. Discrete Math. Sci. Cryptogr. 18 (2015), no. 1-2, 105–116 R. Laskar, J. Jacob, J. Lyle, Variations of graph coloring, domination, and combinations of both: a brief survey. Advances in Discrete Mathematics and Applications: Mysore, 133–152. Ramanujan Math. Soc. Lect. Notes Ser. 13 (Ramanujan Math. Soc., Mysore, 2010) M.D. Plummer, Well-covered graphs: a survey. Questions Math. 16 (1993), no. 3, 253–287 A. Rana, A survey on the domination of fuzzy graphs. Discrete Math. Algorithms Appl. 13 (2021), no. 1, Paper No. 2130001, 15 pp

Appendix C. Surveys Containing Information on Domination in Graphs

573

28. R.R. Rubalcaba, A. Schneider, P.J. Slater, T.W. Haynes, A survey on graphs which have equal domination and closed neighborhood packing numbers. AKCE Int. J. Graphs Comb. 3 (2006), no. 2, 93–114 29. E. Sampathkumar, On some new domination parameters of a graph—a survey. Proceedings of the Symposium on Graph Theory and Combinatorics (Cochin, 1991), 7–13. Publication (Centre for Mathematical Sciences (Trivandrum, India)) 21 (Centre Math. Sci., Trivandrum, 1991) 30. D. Sumner, Critical concepts in domination. J. Combin. Theory Ser. B 86 (1990), no. 1-3, 33–46 31. T. Tamizh Chelvam, T. Asir, K. Selvakumar, On domination in graphs from commutative rings: a survey. Algebra and Its Applications, eds. S.T. Rizvi, A. Ali, V.D. Filippis, 363–379. Springer Proc. Math. Stat. 174 (Springer, Singapore, 2016) 32. T. Tamizh Chelvam, M. Sivagami, Domination in Cayley graphs: a survey. AKCE International Journal of Graphs and Combinatorics 16 (2019), no. 1, 27–40 33. J.-M. Xu, On bondage numbers of graphs: a survey with some comments. Int. J. Comb. (2013), Art. ID 595210, 34 pp

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Index Symbol index Symbols

C

[𝑋, 𝑌 ] - set of edges incident to a vertex in 𝑋 and a vertex in 𝑌 , 195 [𝑘] - {1, 2, . . . , 𝑘 }, 3 [𝑘] 0 - {0, 1, . . . , 𝑘 }, 3

𝐶 (𝐺) - coalition number of 𝐺, 360 𝑐(𝐺) - minimum order of a maximal clique in 𝐺, 363 Cay(Γ, 𝑆) - Cayley graph on group Γ relative to connection set 𝑆 ⊂ Γ, 268 CEA - 𝛾(𝐺 + 𝑒) ≠ 𝛾(𝐺) for all 𝑒 ∈ 𝐸 ( 𝐺), 384 CER - 𝛾(𝐺 − 𝑒) ≠ 𝛾(𝐺) for all 𝑒 ∈ 𝐸 (𝐺), 384 𝜒(𝐺) - chromatic number of 𝐺, 25 𝐶𝑛 , 𝑛-cycle - cycle of order 𝑛, 4 CVR - 𝛾(𝐺 − 𝑣) ≠ 𝛾(𝐺) for all 𝑣 ∈ 𝑉 (𝐺), 384

A

𝛼(𝐺) - vertex independence number of 𝐺, 3 𝛼′ (𝐺) - matching number of 𝐺, 3 𝛼ap (𝐺) - maximum cardinality of an independent eap-set in 𝐺, 43 𝛼★ max (𝐺) - maximum order of a spanning star partition of 𝐺, 38 𝛼★ min (𝐺) - minimum order of a spanning star partition of 𝐺, 38 AG(2, 𝑞) - affine plane of dimension 2 and order 𝑞, 304 B

𝛽(𝐺), 𝜏(𝐺) - vertex covering number of 𝐺, 3 𝑏(𝐺) - maximum number of vertex disjoint ID-sets in 𝐺, 363 𝛽′ (𝐺) - edge covering number of 𝐺, 3 𝜕 (𝑆) - boundary of 𝑆, 427

D

𝑑 (𝑢, 𝑣), 𝑑𝐺 (𝑢, 𝑣) - distance between vertices 𝑢 and 𝑣 in graph 𝐺, 4 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ) - daisy, 132 deg(𝑣), deg𝐺 (𝑣) - degree of vertex 𝑣 in graph 𝐺, 2 deg𝑋 (𝑣) - number of neighbors of 𝑣 in set 𝑋, 192 Δ(𝐺) - maximum degree of a vertex in 𝐺, 2 𝛿(𝐺) - minimum degree of a vertex in 𝐺, 2 Δ 𝑘 (𝐺) - generalized maximum degree of 𝐺, 78

© Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

623

Index

624 diam(𝐺) - diameter of 𝐺, 4 Dom(𝐺) - upper domatic number of 𝐺, 360 dom(𝐺) - domatic number of 𝐺, 24, 353 𝐷 (𝑛1 , 𝑛2 , 𝑘) - dumbbell, 133 E

𝐸, 𝐸 (𝐺)  - edge set of graph 𝐺, 2 𝐸 − - 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 + 𝑢𝑣) < 𝛾(𝐺) , 382  𝐸 0 - 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 − 𝑢𝑣) = 𝛾(𝐺) , 382  𝐸 0 - 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 + 𝑢𝑣) = 𝛾(𝐺) , 382 ecc(𝑣), ecc𝐺 (𝑣) - eccentricity of vertex 𝑣 in graph 𝐺, 4 𝜀 𝑓 (𝐺) - maximum number of pendant edges in a spanning forest in 𝐺, 36 Ψ(𝐺) - maximum cardinality of an enclaveless set in 𝐺, 37 𝜓(𝐺) - minimum cardinality of a maximal enclaveless set in 𝐺, 37  𝐸 + - 𝑢𝑣 ∈ 𝐸 (𝐺) : 𝛾(𝐺 − 𝑢𝑣) > 𝛾(𝐺) , 382 G

𝐺 + 𝑒 - graph obtained from 𝐺 by adding edge 𝑒, 382 𝐺 − 𝑒 - graph obtained from 𝐺 by deleting edge 𝑒, 382 𝐺 − 𝑣 - graph obtained from 𝐺 by deleting vertex 𝑣, 382 𝐺 [𝑆] - subgraph induced by set 𝑆, 2 Γ(𝐺) - upper domination number of 𝐺, 3 𝛾(𝐺) - domination number of 𝐺, 2 Γap (𝐺) - maximum cardinality of a minimal dominating eap-set in 𝐺, 43 𝛾 ap (𝐺) - minimum cardinality of a dominating eap-set in 𝐺, 43

𝛾 ap (𝐺) - minimum cardinality of an externally almost perfect minimal dominating set in 𝐺, 45 ap 𝛾d (𝐺) - minimum cardinality of an externally almost perfect dominating set in 𝐺, 45 𝛾f (𝐺) - fractional domination number of 𝐺, 36 𝛾 𝐻 (𝐺) - 𝐻-forming number of 𝐺, 34 𝛾 (𝑖, 𝑗 ) (𝐺) - (𝑖, 𝑗)-domination number of 𝐺, 41 𝛾in (𝐷) - in-domination number of digraph 𝐷, 65 Γ≤ 𝑘 (𝐺) - upper distance-𝑘 domination number of 𝐺, 35 Γ𝑘 (𝐺) - upper 𝑘-domination number of 𝐺, 34 𝛾 ≤ 𝑘 (𝐺) - distance-𝑘 domination number of 𝐺, 35 𝛾 𝑘 (𝐺) - 𝑘-domination number of 𝐺, 34 𝛾-set - dominating set with cardinality 𝛾(𝐺), 2 Γt (𝐺) - upper total domination number of 𝐺, 3 𝛾t (𝐺) - total domination number of 𝐺, 3 𝛾t -set-total dominating set with cardinality 𝛾(𝐺), 3 𝛾tf (𝐺) - fractional total domination number of 𝐺, 36 𝐺 - complement of graph 𝐺, 2 G(𝑛, 𝑝) - random graph of order 𝑛 and edge probability 𝑝, 217 I

𝐼 (𝐺) - independence graph of 𝐺, 25 𝑖(𝐺) - independent domination number of 𝐺, 3 𝑖 ap (𝐺) - minimum cardinality of a maximal independent eap-set in 𝐺, 43

Symbol index 𝑖-set - independent dominating set of cardinality 𝑖(𝐺), 3 ID-set - independent dominating set, 3 idom(𝐺) - idomatic number of 𝐺, 24, 362 IR(𝐺) - upper irredundance number of 𝐺, 25 ir(𝐺) - lower irredundance number of 𝐺, 25 IRap (𝐺) - maximum cardinality of an irredundant eap-set in 𝐺, 43 irap (𝐺) - minimum cardinality of a maximal irredundant eap-set in 𝐺, 43 𝐼 𝑣 - independent set of maximum cardinality containing 𝑣, 491

625 O

ONH(𝐺) - open neighborhood hypergraph of graph 𝐺, 178 P

𝜌(𝐺) - packing number of 𝐺, 3 pc(𝐺) - path covering number of 𝐺, 94 𝜋nd (𝐺) - minimum order of a non-dominating partition of 𝐺, 39 𝑃𝑛 - path of order 𝑛, 4 𝜌 o (𝐺) - open packing number of 𝐺, 73 Q

𝑄 𝑛 - 𝑛-dimensional cube graph, 365 Q𝑛 - 𝑛 × 𝑛 queens graph, 6 R

K

𝐾 (𝐺) - clique graph of 𝐺, 423 𝐾1,𝑠 , star, 3 𝐾𝑛 - complete graph of order 𝑛, 2 𝐾 𝑛 - empty graph of order 𝑛, 2 𝐾𝑟 ,𝑠 -complete bipartite graph, 3 L

𝐿(𝐺) - line graph of 𝐺, 423 M

𝑀 (𝐺) graph of 𝐺, 443 - middle 𝑚, 𝐸 (𝐺) - number of edges in 𝐺, 2 N

𝑛, 𝑉 (𝐺) - number of vertices in 𝐺, 2 N(𝑆), N𝐺 (𝑆) - open neighborhood of set 𝑆, 2 N(𝑣), N𝐺 (𝑣) - open neighborhood of vertex 𝑣, 2 N[𝑆], N𝐺 [𝑆] - closed neighborhood of set 𝑆, 2 N[𝑣], N𝐺 [𝑣] - closed neighborhood of vertex 𝑣, 2

rad(𝐺) - radius of 𝐺, 4 S

𝑆(𝐺) - subdivision graph of 𝐺, 443 𝑆(𝑟, 𝑠)-double star, 3 sl(𝐺) - Slater number of 𝐺, 75 T

𝜏(𝐺) - transversal number of 𝐺, 431 𝜏(𝐻) - transversal number of hypergraph 𝐻, 178 TD-set - total dominating set, 3 tdom(𝐺) - total domatic number of 𝐺, 364 Θ(𝐺) - upper perfect neighborhood number of 𝐺, 43 𝜃 (𝐺) - perfect neighborhood number of 𝐺, 43 Θap (𝐺) - maximum cardinality of an ap-set in 𝐺, 42 Θap (𝐺) - maximum cardinality of a minimal eap-set in 𝐺, 43 ap Θp (𝐺) - maximum cardinality of an independent perfect neighborhood set in 𝐺, 43

Index

626 𝜃 ap (𝐺) - the minimum cardinality of a maximal ap-set in 𝐺, 42 𝜃 ap (𝐺) - minimum cardinality of an eap-set in 𝐺, 43 ap 𝜃 p (𝐺) - minimum cardinality of an independent perfect neighborhood set in 𝐺, 43 Tr(𝐺) - transitivity number of 𝐺, 359 U

UEA - 𝛾(𝐺 + 𝑒) = 𝛾(𝐺) for all 𝑒 ∈ 𝐸 ( 𝐺), 384 UER - 𝛾(𝐺 − 𝑒) = 𝛾(𝐺) for all 𝑒 ∈ 𝐸 (𝐺), 384

UVR - 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) for all 𝑣 ∈ 𝑉 (𝐺), 384 V  𝑉 + - 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) > 𝛾(𝐺) , 382  𝑉 − - 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) < 𝛾(𝐺) , 382  𝑉 0 - 𝑣 ∈ 𝑉 (𝐺) : 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) , 382 W

𝑊𝑛 - wheel graph on 𝑛 + 1 vertices, 365

Subject index Symbols

(𝜆, 𝜇)-graphs, 436 (𝜆, 𝜇)-perfect graphs, 442 (𝛾, 𝑖)-trees, 124 (𝛾, 𝛾t )-trees, 124 (𝛾, Γ)-graphs, 436 (𝜌, 𝑖)-trees, 124 (𝜌, 𝛾t )-trees, 126 (𝛾t , Γt )-graphs, 436 (𝑖, 𝑗)-set, 41 (𝑢, 𝑣)-path, 3 𝛾-excellent graph, 114 𝛾-insensitive graph, 386 𝛾-set of 𝐺, 2, 29 𝛾t -set of 𝐺, 3, 33 Γ-set of 𝐺, 29, 412 Γt -set of 𝐺, 33, 412 Vizing’s Conjecture, 525 1-maximal P-set, 28 1-minimal P-set, 28 2-coloring, 375 2 5 -minimal graph, 130 4 7 -minimal graph, 170 A

𝛼-set of 𝐺, 412 absorption number, 21 accepting path, 142 acceptor, 142 in-, 143 adjacent edges, 2 adjacent vertices, 2 affine plane, 304 algorithm deterministic, 61 exact exponential, 71 greedy algorithm for dominating set, 210 greedy approximation, 187 heuristic total domination algorithm, 187 minimum dominating set, 61

minimum independent domination set, 63 minimum total dominating set, 64 nondeterministic, 51 randomized greedy, 226 randomized independent dominating set, 226 almost perfect, 42 externally, 43 internally, 42 Art Gallery Problem, 334 B

bad-4-cut-vertex, 292 basis in a digraph, 19 BG-graph, 527 bipartite complement, 182 bipartite graph, 3, 32 bishops domination, 11 block, 330 block graph, 436 boundary, 427 branch vertex, 112 Brooks’ Coloring Theorem, 175 C

candidate solution, 51 capacitated domination, 15 Caro-Wei Theorem, 430 Cartesian product, 23 caterpillar, 99 code, 99 spine, 99 Cauchy-Schwarz Inequality, 431 Cayley graph, 268 central vertex, center, 4 characteristic function, 35 chord circulants, 270 chromatic number, 25 Chvátal’s Watchman Theorem, 334–336 class NP, 50

© Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

627

Index

628 class NPc, 50 class P, 50 claw-free, 73, 291 clique, 2 clique cover number, 527 co-NP-complete, 437 coalition, 360 number, 360 partition, 360 coefficient of external stability, 21 coefficient of internal stability, 21 complement of 𝐺, 2 complete bipartite graph, 3 complete graph, 2 complexity results, 58 connected dominating set, 13 connected graph, 4 contraction graph, 374 corona, 73 coupon coloring problem, 364 critical graphs 𝑘-edge-, 397 𝑘-vertex-, 393 cubic graph, 145 cycle, 4 D

decision problem, 52 3-SATISFIABILITY, 53 BIPARTITE DOMINATING SET, 56 BIPARTITE TOTAL DOMINATING SET, 57 DOMINATING SET, 52 EFFICIENT DOMINATING SET, 283 EFFICIENT DOMSET, 260 EFFICIENT TOTAL DOMINATING SET, 283 EXACT COVER BY 3-SETS,

285 GRIDTOTALDOMSET, 515 INDEPENDENT DOMINATING SET, 54

PERFECT DOMINATING SET,

283 PERFECT TOTAL DOMINATING SET, 283 THREE-DIMENSIONAL MATCHING, 284 TOTAL DOMINATING SET, 55 3-SATISFIABILITY, 53 DOMINATING SET, 52 PLANAR DOMINATING SET,

326 decomposable, 527 degree of a vertex, 2 degree sequence, 75 diameter, 4 diamond-necklace, 320 directed graph, 19 distance, 4 distance-𝑘 dominating set, 35 domatic number, 24, 47 domatic partition, 353 domatically full graph, 356 dominating bipartition, 368 dominating pair of vertices, 407 dominating set, 2, 29 dominating vertex, 2 Domination Chain, 24, 32 domination critical edge, 382 vertex, 382 domination edge-critical graph, 383 domination number, 2 domination stable edge, 382 vertex, 382 domination vertex-critical graph, 383 double star, 3 double-projection argument, 527 dumbbell, 133 E

eccentric vertex, 126 eccentricity, 4 edge set, 2 efficient dominating set, 34, 73, 259

Subject index efficient partition, 259 efficient total dominating set, 73, 260 empty graph, 2 enclave, 37 enclaveless number, 37 enclaveless set, 37 endblock, 330 endvertex, 141 Erdős-Rényi random graph model, 217 F

𝐹-component, 368 factor, 480 factor-critical graph, 396 factorable, 480 Fano plane, 178 forest, 23, 36 Four Color Theorem, 348 fractional domination, 35 fractional total domination, 36 framework to Vizing’s Conjecture, 527 G

𝐺-fiber, 525 Gallai’s Theorem, 22 (𝛾, 𝑖)-graphs, 441 generalized corona, 87 generalized maximum degree, 78 generalized Petersen graph, 149 girth, 296 graph, 2 (𝐶4 , 𝐶6 )-free, 303 2-corona, 84 𝐶4 -free, 306 balloon, 98 BG-graph, 527 bipartite, 3, 32 block, 436 block-cactus, 356 bowtie, 371 𝐶6 -free, 323 (𝐶4 , 𝐶5 )-free, 291 cactus, 367 Cayley, 268

629 circulant, 268, 270 circular-arc graph, 456 claw-free, 73, 291 clique, 423 complete bipartite, 3 connected, 4 corona, 73 cross, 512 cube-connected, 264 cube-connected cycle, 266 cubic, 145 cylinder, 505 cylindrical, 275 daisy, 132, 547 de Bruijn, 264 decomposable, 527 diameter-2, 219 diameter-2-critical, 404 diamond-free, 291 domination edge-critical, 397 domination perfect, 128 double star, 3 dumbbell, 133 efficient, 259 expansion, 203 (𝐹1 , 𝐹2 , . . . , 𝐹𝑘 )-free, 291 factor-critical, 404 galaxy, 3, 384 generalized corona, 87 grid, 264, 505 Heawood, 178 hypercube, 269 inefficient, 260 isolate-free, 2 𝐾1,𝑘+1 -free, 419 𝑘-chordal, 372 lattice, 367 line, 423 middle, 443 Möbius ladder, 268 𝑛-cube, 262 𝑛-dimensional cube, 365 nontrivial, 2 outerplanar, 326 petals, 547

Index

630 planar, 325 prism, 268 random, 217 series-parallel, 58 simplicial, 436 splitting, 40 star, 3, 384 strongly perfect, 456 subcubic, 149 subdivision, 4, 443 toroidal, 264, 505 total efficient, 261 triangle-free, 291 trivial, 2 2-chord circulant, 271 underlying, 299 vertex-transitive, 267 wheel, 365 graph product direct product, 282 disjunctive product, 281 lexicographic, 271 strong product, 281 graphs satisfying Vizing’s Conjecture, 527 Grundy function, 20 guarded commands, 60 H

𝐻-forming set, 34 Hamilton-connected, 399 Hamming distance-1, 262 handle, 135 Heawood graph, 178 bipartite complement, 182 hereditary property, 28 ht-dominated, 537 hypercube, 269 hypergraph, 177 complement, 181 hyperedge, 178 incidence bipartite graph, 182 𝑘-uniform, 178 linear, 304 maximum degree, 178

minimum degree, 178 open neighborhood hypergraph, 178 order, 178 size, 178 transversal, 178 I

𝑖-set of 𝐺, 3, 29 idomatic graph, 361 idomatic number, 24 independence graph, 25 independence number, 3 independent domatic partition, 362 independent dominating set, 3, 21 independent dominating set of edges, 20 independent domination number, 3 independent perfect dominating set, 260 independent set of vertices, 3, 27 indicator random variable, 210 induced subgraph, 2 influence domination number, 17 influence set, 17 inner grid, 511 internal vertex, 141 IR-set of 𝐺, 425 irredundance number, 25, 31 irredundant set, 25, 31 isolate-free graph, 2 isolated vertex, 2 J

Jaeger-Payan Theorem, 468 Jordan Closed Curve Theorem, 338 K

𝑘-regular, 2 𝑘-dominating set, 33 𝑘 t -edge-supercritical graphs, 402 kernel, 20 Klein bottle, 329 Kneser graph, 367 knights domination, 11 𝑘 t -edge-critical graphs, 401

Subject index L

layers, 522 leaf, 2 length cycle, 4 path, 4 lexicographic product, 280 linearity of expectation, 211 linkage, 135 lower matching number, 102 M

matching, 3 matching number, 38 maximal independent set, 3 maximal outerplanar graph (mop), 326 maximal P-set, 28 maximum degree of 𝐺, 2 Meir-Moon Theorem, 115 minimal dominating set, 3 minimal domination imperfect graphs, 446 minimal P-set, 28 minimal total dominating set, 3 minimum degree of 𝐺, 2 Murty-Simon Conjecture, 404 N

𝑛-cube, 262 near perfect matching, 395 near-triangulation, 325 neighbor, 2 neighborhood of a set closed, 2 open, 2 neighborhood of a vertex closed, 2 open, 2 non-dominating partitions, 39 non-dominating set, 39 non-total dominating set, 39 nontrivial graph, 2 Nordhaus-Gaddum bounds, 467 Nordhaus-Gaddum Theorem, 467 NP-complete, 51, 52

631 NP-hard, 52 O

odd component, 96 optimal vpd-cover, 141 optional domination, 59 optional total dominating set, 63 order, 2 order of a partition, 353 Ore’s Lemma, 80 Ore’s Theorem, 23 orientable surface, 340 out-endvertex, 141 outerplanar graph, 326 P

𝑝-clique, 490 𝑝-clique-minimal, 490 P-set, 27 packing, 3, 73 number, 73 open, 73 perfect, 73 packing number open, 120 paired dominating set, 36 partition loose, 501 partners, 459, 522 path, 4 path covering, 94 pendant edge, 36 perfect, 42, 73 𝑆-, 42 completely, 43 externally, 43 internally, 42 perfect [1, 2]-factor, 438 perfect code, 44 perfect dominating set, 34, 261 connected, 286 independent, 286 weakly, 261 perfect matching, 395 perfect neighborhood set, 43

Index

632 independent, 43 perfect set, 261 perfect total dominating set, 261 peripheral vertices, 102 Petersen graph, 369 Pigeonhole Principle, 86 planar graph, 325 maximal planar, 326 planar triangulation, 326 plane graph, 325 adjacent faces, 325 boundary of a face, 325 external edge, 325 external vertex, 325 face, 325 inner face, 325 internal edge, 325 internal vertex, 325 outer face, 325 triangular face, 325 polynomial-time reducible, 51 polynomial-time verification, 51 power domination, 16 prism, 522 bipartite, 522 private neighbor, 30 𝑆-, 30 𝑆-external, 80 𝑆-internal, 82 open 𝑆-, 30 open 𝑆-external, 82 open 𝑆-internal, 82 private neighborhood 𝑆-, 30 𝑆-external, 30 𝑆-internal, 30 probabilistic method, 210 projection, 522, 525 natural, 522 projective plane, 329 property P, 27 pruning, 113 Q

queens domination number, 8

queens graph, 6 Queens Problems 𝑛 Queens Problem, 549 𝑛-Queens Problem, 6 Domination Problem, 549 Eight Queens Problem, 6 Five Queens Problem, 7 Independent, 550 Queens Domination, 7 Queens Independent Domination, 9 Queens Total Domination, 10 Total Domination, 550 R

(𝑟, 𝑑)-configuration, 14 𝑟-dynamic coloring, 349 (𝜌, 𝑖)-graphs, 442 radius, 4 random graph, 217 relative complement, 484 residual graph, 152 restricted domination, 300 restricted total domination, 300 rooted tree, 99 child, 99 descendant, 100 maximal subtree, 100 parent, 99 S

school bus routing, 15 semiperfect dominating set, 261 set 𝑋 dominates set 𝑌 , 29 set 𝑋 totally dominates set 𝑌 , 33 simple graph, 2 simplicial graph, 436 vertex, 423 size, 2 Slater graph, 31 Slater number, 75 social networks, 16 spanning subgraph, 2 special-cycle, 292

Subject index star, 3, 4 𝐾1,4 -star pattern, 511 strong equality, 128 strong support vertex, 99 strongly perfect, 456 subcubic graph, 149 subdivided edge, 4 subdivided star, 78 subgraph, 2 induced, 2 spanning, 2 superhereditary property, 28 support vertex, 99 T

torus, 264, 505 total domatic number, 364 total domatic partition, 363 total dominating set, 3, 32 total domination edge-critical graph, 401 total domination number, 3, 33 total efficient graph, 261 total efficient partition, 261 total efficient set, 260 total perfect set, 261 totally balanced matrices, 356 totally dominated, 537 vertically, 537 toughness, 400 trail, 3 transitivity of a graph, 359 tree, 4 triangle-diamond partition, 373 triangulated disc, 325 irreducible weak, 330 outward numbering, 326 reducible weak, 330 weak, 326, 330 trivial code, 262

633 trivial graph, 2 truncated tetrahedron, 366 Turán’s Theorem, 431 twins open, 203 U

underlying graph, 170, 299 union (of two graphs), 3 unit, 170, 373 diamond, 373 key, 170 triangle, 373 upper domatic number, 360 upper domination number, 3 upper domination perfect graph, 457 upper irredundance number, 31 upper irredundance perfect graph, 457 upper total domination number, 3, 33 V

verification algorithm, 51 vertex large, 135 small, 135 vertex cover, 3 vertex covering number, 21 vertex independence number, 3, 29 Vizing’s Conjecture, 23, 525 vt-dominated, 537 W

walk, 3 Watched Art Gallery Problem, 343 watched guard set, 343 weak near-triangulation, 326 weak partition, 382 well-covered graphs, 436 well-dominated graphs, 436, 438 well-total-dominated graphs, 436 wounded spider, 101

Author index A

Abay-Asmerom, G., 282 AbouEisha, H., 72 Abrishami, G., 287, 289, 312, 350–352 Acharya, B.D., 38, 74, 76, 385 Adhikari, A., 508, 509 Aharoni, R., 527 Ahrens, W., 7, 8 Aigner, M., 335 Akbari, S., 365 Alanko, S., 507 Alber, J., 72 Alizadeh, H., 440 Allan, R.B., 92, 314, 443 Alon, N., 191, 211, 372, 430 Alvarado, J.D., 344, 552 Ananchuen, N., 395–397, 399 Ananchuen, W., 397 Anders, Y., 237 Anderson, S.E., 439 Ao, S., 400 Aouchiche, M., 467 Aram, H., 365, 383 Archdeacon, D., 169, 175, 176, 179, 188, 320 Arnautov, V.I., 209, 210, 482 Arnborg, S., 59, 69 Arumugam, S., 470, 472, 504 Atallah, G.K., 69 Attiya, H., 13 Azarija, J., 522, 523 B

Babikir, A., 316, 317, 320, 446, 454 Badakhshian, L., 523, 524 Bahadir, S., 440 Bakker, E.M., 287 Balbuena, C., 399, 407–409 Balogh, J., 524 Bange, D.W., 34, 260, 283, 287, 288 Baogen, X., 82, 472, 473, 502, 504

Barbosa, R., 288, 438 Barcalkin, A.M., 527, 533, 534 Barkauskas, A.E., 34, 260, 283, 287, 288 Bauer, D., 385, 393 Bazgan, C., 17, 72 Belhoul, Y., 72 Benecke, S., 383 Berge, C., i, 1, 21, 22, 24, 25, 29, 74, 96, 101, 227 Berglund, P.G., 18 Bertolo, R., 521 Bertossi, A.A., 59, 67–69 Beyer, T., 59, 62, 66 Bezzel, M., 6, 7 Biggs, N., 262 Bird, W.H., 9 Biró, C., 212 Blank, M.M., 130, 131, 134, 139, 151, 167 Blass, U., 521 Blidia, M., 101, 112 Blitch, P., 384, 397, 398, 400 Blum, C., 72 Blumenthal, A., 204, 206 Bodlaender, H.L., 72 Bögl, M., 15 Bollobás, B., 80, 86, 87, 94, 129, 192, 373, 436 Bonato A., 225 Bondy, J.A., 405 Booth, K.S., 59, 67 Borowiecki, M., 468, 469 Bourgeois, N., 72 Bousquet, N.F., 72 Boyaci, A., 72 Boyar, J., 72 Brandstädt, A., 50, 59, 68, 263, 287–289, 457 Brankovic, L., 72 Brause, C., 313 Bregman, Z., 372

© Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5

634

Author index Brešar, B., 18, 23, 526, 527, 529, 531–536, 539, 542, 543, 546 Brigham, R.C., 85, 87, 88, 116, 169, 296, 315, 318, 383–386, 393, 394, 476 Broere, I., 371 Broersma, H., 59 Bujtás, C., 129, 152, 153, 156, 159, 160, 162, 163, 167, 168, 478, 479 Burton, T., 114, 383 C

Caccetta, L., 398, 404 Calkin, N., 359, 371 Campbell, S.R., 437 Campos, C.N., 336, 337 Cao, L., 405 Caro, Y., 211, 428, 430, 431, 437 Carreño, J.J., 508 Carrington, J.R., 85, 87, 116, 169, 384–386 Casel, K., 72 Castle, M., 282 Černý, A., 72 Chambers, E.W., 469 Chan, W.H., 314 Chang, G.J., 56, 59, 67, 68, 288, 356, 357 Chang, M.S., 59, 69, 70, 288 Chang, T.Y., 507, 511 Chellali, M., 101, 112, 116, 117 Chen, B., 364 Chen, C., 72 Chen, X.G., 117, 159, 282, 314 Chen, Y.J., 399, 400 Chen, Z.Q., 399 Cheng, T.C.E., 399, 400 Chengye, Z., 383 Chérifi, R., 507 Cheston, G.A., 69, 456 Chinn, P.Z., 88, 384, 393, 394, 476 Chiu, W.Y., 72 Chlebík, M., 72

635 Chlebíková, J., 72 Cho, E.K., 203 Choi, I., 203 Chopin, M., 17 Chudnovsky, M., 314 Chvátal, V., 179, 180, 188, 334–336, 383, 428, 431–434, 437 Clark, L.H., 225, 264 Clark, W.E., 211, 507, 527, 533, 534, 536 Cockayne, E.J., 1, 5, 22, 24, 25, 27, 28, 30–32, 34, 43, 45, 47, 59, 60, 66, 67, 77, 78, 80, 82, 84, 86, 87, 92, 94, 112, 115, 116, 118, 124, 129, 169, 188, 192, 263, 318, 353–356, 362–364, 373, 400, 435, 436, 443, 455, 458, 469, 472, 473, 485, 495, 498, 499, 501–504, 507, 508 Cook, W., 383 Coorg, S.R., 288 Corneil, D.G., 58, 59, 67–69 Cortés, M., 510 Courcelle, B., 288 Cowen, R., 274 Crevals, S., 72, 507, 510, 511, 519, 520 Currie, J.D., 437 Cyman, J., 451–453, 460–462 Czabarka, É., 212 D

Dailly, A., 406, 409 Damaschke, P., 59 Damian-Iordache, M., 72 Dankelmann, P., 212, 238, 256, 257, 359, 371 Dantas, S., 344, 552 Datta, A.K., 72 Davidson, P.P., 72 Dawes, R.M., 25, 32, 77, 78, 82, 84, 115, 116, 169, 188, 318, 364, 458, 485, 502, 503

Index

636 Daykin, D.E., 66 de Jaenisch, C.F., 8, 9 de Werra, D., 72 Dejter, I.J., 34, 275, 276 DeLaViña, E., 88–90, 94, 102 Delgado, P., 370, 371 Della Croce, F., 72 DeMaio, J., 282 Deng, F., 462 Desormeaux, W.J., 39, 40, 110, 111, 224, 370–373, 383 Dettlaff, M., 383, 451–453, 460–462 Dewdney, A.K., 59, 67 Ding, Y., 72 Doerner, K.F., 15 Domke, G.S., 101, 238, 256, 257 Dorbec, P., 23, 149, 310, 311, 313, 526, 539, 546–548 Dorfling, M., 124–128, 183, 184, 336, 337, 339–342, 344–346, 371, 441, 445, 458, 460 Downey, R.G., 72 Dreyer, P.A., 5 Dubickas, A., 224, 225 Duckworth, W., 72, 225, 226 Dunbar, J.E., 12, 13, 101, 383, 475–477 Durand, A., 8 Dutton, R.D., 88, 296, 315, 318, 384–386, 393, 394, 476 Dyer, M.E., 284 E

Egerváry, J., 20 Ekim, T., 440 El-Zahar, M.H., 508, 509, 526 Ellingham, M.N., 437, 438 Ellis-Monaghan, J., 169, 175, 176, 179, 188, 320 Elmallah, E.S., 69 Entringer, R.C., 490, 491 Erdős, P., 217, 404 Erkut, E., 17 Erlebach, T., 72 Erwin, D.J., 12, 13

Eschen, E.M., 287, 289 Escoffier, B., 72 Eustis, A., 183, 184, 188 F

Fajtlowicz, S., 428–430 Fan, G., 404 Farber, M., 59, 67, 68, 356 Faudree, R.J., 314, 419 Favaron, O., 32, 43, 86, 101, 122–124, 172, 173, 175, 177, 192, 193, 195, 200, 206, 207, 263, 319, 320, 322, 383, 398–400, 413, 421, 423–426, 443, 455, 495 Feige, U., 371 Fellows, M.R., 69, 72, 261, 283–285, 287, 338 Feng, L., 99 Feng, R.Q., 282 Fernau, H., 72 Fičur, P., 289 Fiedler, M., 366 Finbow, A.S., 437–440 Fink, J.F., 34, 383, 469 Finozhenok, D., 550 Firoozi, P., 397 Fischer, D., 169, 175, 176, 179, 188 Fischermann, M., 113, 119 Fisher, D.C., 149, 211, 236, 311, 320, 507, 510 Fisk, S., 334–336 Flach, P., 75 Flandrin, E., 314 Fomin, F.V., 71, 72 Forrester, R., 508 Foucaud, F., 406, 409 Fraughnaugh, K., 149, 236, 311 Frendrup, A., 440 Fricke, G.H., 42, 43, 69, 70, 114, 427, 456, 495, 550 Friese, E., 289 Frieze, A.M., 284 Froncek, D., 169, 175, 176, 179, 188, 320

Author index Fujita, S., 14 Fulman, J., 128, 230, 394, 395, 445 Füredi, Z., 404, 405 Furuya, M., 383, 398, 399 G

Gaddum, J.W., 467, 479–481, 484–486, 488–490, 493, 495, 498–501, 503 Gairing, M., 72 Gallai, T., 1, 19, 21, 22, 92, 431, 563 Garey, M.R., 52, 59, 66, 326 Garnick, D., 512 Gary, K., 282 Gasharov, V., 420 Gavlas, H., 272–274, 282 Gavril, F., 66, 67 Gera, R., 50 German, L.F., 527, 533, 534 Ghaleb, F.F.M., 72, 508 Giakoumakis, V., 287, 289 Gibbons, P.B., 8 Gimbel, J., 86, 495 Gionet Jr., T.J., 439 Glebov, N.I., 195 Glebov, R., 218 Glivjak, F., 404 Godbole, A., 218 Goddard, W., 15, 23, 36, 72, 124–128, 200, 202–204, 207, 238, 256, 257, 300, 312, 314, 338–340, 344–347, 349, 365, 366, 371, 441, 445, 451, 481–484, 490, 491, 493–495, 497, 519, 522, 526, 539, 551 Goldwasser, J.L., 274 Golumbic, M.C., 456 Goncalves, D., 68, 72, 507 Goodman, S., 24, 59, 60, 66 Gordon, V.S., 72 Gori, A., 69 Gould, R.J., 419 Gözüpek, D., 439, 440 Grandoni, F., 72

637 Gravier, S., 507, 512, 513, 515, 516 Grobler, P.J.P., 238, 256, 257, 397 Grundy, P.M., 1, 20 Guellati, N., 72 Guichard, D.R., 507 Gunther, G., 113, 114 Gutin, G., 457 H

Haddad, M., 72 Hagerup, T., 72 Häggkvist, R., 404 Hajian, M., 104–107, 109, 110, 116, 117 Hall, A.J., 383 Halldórsson, M.M., 72, 371 Hamada, T., 443 Hammack, R.H., 282 Hansberg, A., 101, 399, 406–409 Hansen, P., 467 Hanson, D., 394, 395, 399, 405, 406, 552 Harant, J., 212 Harary, F., 123, 263, 272, 338, 381, 384–386, 393, 441, 502, 521 Hare, E.O., 68, 507, 508 Hare, W.R., 507, 508 Hartnell, B.L., 23, 34, 113, 114, 383, 385, 389, 390, 437–440, 526, 527, 529, 531–536, 539 Hasni, R., 397 Hattingh, J.H., 28, 238, 256, 257, 336, 337, 341, 342, 371 Haviland, J., 79, 192, 193, 207, 314 Haynes, T.W., i, 1, 12, 13, 16, 25, 34, 39, 40, 42, 43, 50, 70, 71, 78, 82, 99, 110, 111, 114, 116, 117, 119, 128, 151, 224, 272, 307, 359–361, 366, 370–373, 383–386, 388–391, 397, 400, 402, 403, 406–409, 427, 435,

Index

638 472, 473, 475–477, 500, 502, 504, 552, 553, 555 Hechler, S.H., 274 Hedetniemi, J.T., 42, 44, 45, 123, 359–361 Hedetniemi, K.D., 42 Hedetniemi, S.M., 5, 12, 13, 16, 28, 35, 41–45, 59, 63, 68, 72, 114, 383, 427, 550 Hedetniemi, S.T., i, 1, 5, 9, 12, 13, 15, 16, 22, 24, 25, 27, 28, 30–32, 34, 35, 37, 38, 41–45, 47, 50, 59, 60, 62, 63, 66–72, 77, 78, 82, 84, 87, 92, 114–116, 151, 169, 188, 318, 353–356, 359–364, 366, 383, 397, 400, 427, 435, 443, 458, 469, 472, 473, 475–477, 485, 501–504, 507, 508, 550, 552, 553, 555 Heggernes, P., 370 Hell, P., 338 Hellwig, A., 476, 477 Henning, M.A., 16, 23, 25, 34–36, 39, 40, 42, 43, 50, 71, 79, 82, 94–96, 104–107, 109–112, 114, 116–119, 124–128, 140, 148, 149, 159, 163, 166, 167, 169–173, 175, 177, 180, 182–188, 200, 202–204, 207, 217–220, 224, 245, 252, 254, 255, 292–295, 298–313, 316, 317, 319, 320, 322, 338–340, 344–347, 349–352, 365, 366, 368, 369, 371–373, 375, 378, 383, 386, 388–391, 401, 402, 404, 406–409, 414, 415, 418, 419, 425–428, 433, 441, 445, 446, 448, 449, 451–454, 458, 460–462, 464, 479, 481–484, 488–491,

493–495, 497, 500, 522, 523, 526, 527, 529, 531–537, 539, 546–548, 550–553, 555 Hermelin, D., 72 Ho, P.H., 356 Ho, P.T., 537, 538 Hon, W.-K., 72 Honjo, T., 329 Hoover, M.N., 261, 283–285, 287 Host, L.H., 287 Hsieh, S.Y., 288 Huang, H., 282 Huff, J.L., 15 Hujdurović, A., 439 Hurink, J., 72 Hussain, S., 72 Hutson, K.R., 508 I

Ikeda, M., 72 Isopoussu, A., 507 Ivančo, J., 367 Iwama, K., 72 J

Jacobs, D.P., 69, 70, 72, 383 Jacobson, M.S., 34, 383, 419, 455, 469, 507, 550 Jaeger, F., 467, 468 Jafari Rad, N., 104–107, 109, 110, 116, 117, 212, 214–217, 383 Jiang, Y., 398 Johnson, D.S., 52–55, 59, 66–68, 225, 326 Johnson, J.H., 59 Johnson, S.M., 523 Jonck, E., 336, 337, 341, 342 Joseph, J.P., 470, 472 Joubert, E.J., 254, 488, 489 Juedes, D., 72 K

Kaemawichanurat, P., 398 Kakuda, Y., 59, 67

Author index Kakugawa, H., 72 Kameda, T., 14 Kamei, S., 72 Kang, L., 94–96, 245, 252 Kanj, I.A., 72 Karami, H., 486, 488 Kardoš, F., 18 Karthick, T., 289 Katona, G., 523, 524 Katrenič, J., 18 Kawarabayashi, K., 297, 329 Keil, J.M., 59, 68, 69 Kelmans, A., 150, 297, 299 Kennedy, J.W., 274 Kennedy, K.E., 72 Kéri, G., 521 Kheddouci, H., 72 Khelifi, S., 112 Khodkar, A., 486, 488 Kikuno, T., 59, 67 Kim, B., 15 Kim, J.H., 364 Kim, S.J., 349 Kinch, L.F., 383, 469, 507 King, E.L.C., 333, 439 Kinnersley, B., 469 Klavžar, S., 23, 159, 160, 162, 163, 167, 168, 266, 508, 509, 515, 522, 523, 526, 539 Klee, V., 334 Klein, K.-M., 72 Klobučar, A., 515, 519 Kloks, T., 59, 72 Klostermeyer, W.F., 261, 274 Kneser, M., 367 Knisely, J., 41, 383 Knisley, D., 99 Knor, M., 268, 446, 463 Koessler, D., 383 Kok, J., 383 Kőnig, D., i, 1, 18–22 Kortsarz, G., 371 Kostochka, A.V., 149, 150, 195, 297–299, 323, 551 Král, D., 297

639 Kratochvíl, J., 69, 287, 288 Kratsch, D., 59, 68, 69, 71, 72 Krishnamoorthy, V., 405 Kristiansen, P., 72 Krivánek, M., 287 Krzywkowski, M., 117 Kudahl, C., 72 Kuenzel, K., 439, 527, 529, 531–536 Kürschak, J., 19 Kuziak, D., 275, 276, 280–282 Kwon, C., 18 Kwon, Y.S., 282 Kyš, P., 404 L

Lagergren, L., 59 Lagraula, X., 507 Lam, P.C.B., 169, 175–177, 179, 188, 201, 320, 464 Lampis, M., 72 Langston, M.A., 383 Laskar, R.C., 22, 34, 59, 63, 67, 68, 92, 114, 314, 443, 456 Le, V.B., 50, 289, 457 Lee, J., 268, 269, 521 Lee, R.C.T., 286–288 Lee, S.J., 349 Lehel, J., 19 Leitert, A., 263, 288, 289 Lemańska, M., 103–105, 342–344, 383, 451–453, 460–462 Lesniak, L.M., 419 Levit, V.E., 439 Li, H., 419 Li, Z., 337 Liang, Y.D., 69, 288 Lichiardopol, N., 320 Liebenau, A., 218 Liedloff, M., 72 Lin, X., 398 Lin, Y.-L., 72, 288 Lindquester, T.E., 419 Linlin, S., 383 Linz, W., 524 Liu, C.L., 12, 462

Index

640 Liu, H.-H., 72, 333 Liu, H.Q., 507 Liu, J.-B., 99 Liu, Q., 89, 90, 94 Liu, Y.C., 288 Livingston, M., 123, 263, 265, 266, 287, 441, 507, 521 Loizeaux, M., 402 Lokshtanov, D., 289 Lovász, L., 19 Löwenstein, C., 297, 298, 312, 369, 373, 428 Lozano, J.A., 72 Lozin, V., 72 Lu, C.L., 286–288 Lu, T.L., 356 Lyle, J., 200, 207, 208, 314, 445 M

MacGillivray, G., 271, 337–340, 347–349, 394, 395, 400 Madden, J., 404 Maffray, F., 101 Maghout, K., 65 Makowsky, M.A., 288 Manacher, G.K., 69 Mantel, W., 404 Manuel, P.D., 287, 288 Mao, J.Z., 507 Marcon, A.J., 112 Marcu, D., 75 Markus, L.R., 78, 101, 113, 114 Martin, H.W., 550 Martínez, J.A., 508 Marx, D., 72 Masuzawa, T., 72 Matheson, L.R., 326–329, 332–334, 342, 551, 552 Matsumoto, N., 398, 399 McCoy, J., 345 McCuaig, W., 82, 129–131, 135, 139, 151, 167, 171, 293, 294, 323, 474, 475 McDiarmid, C., 179, 180, 188, 428, 431–434

McFall, J.D., 92, 122 McRae, A.A., 15, 28, 70–72, 285–287, 359–361, 550 Meir, A., 1, 24, 92, 114, 115, 441 Mertzios, G.B., 72 Michael, T.S., 343 Milanič, M., 287–289, 439 Miller, D.J., 24, 25, 27, 30–32, 45, 87, 435, 443 Miller, M., 287, 288 Minkowski, H., 19 Mitchell, S.L., 59, 62, 66, 67, 334, 443 Miyazaki, S., 72 Mnich, M., 72 Mohan, R., 71, 360, 361 Mohr, E., 552 Mojdeh, D.A., 397 Molloy, M., 225 Monnot, J., 72 Montassier, M., 149, 310, 311, 313 Moon, J.W., 1, 24, 92, 114, 115, 441 Moore, J.L., 442, 444 Morgenstern, O., 1, 20 Mosca, R., 289 Motiei, M., 365 Mozaffari, S., 365 Müller, H., 59 Murty, U.S.R., 404–408, 552 Mutherasu, S., 269 Mynhardt, C.M., 43, 101, 112, 113, 118, 124–128, 172, 173, 175, 177, 263, 383, 400, 402, 403, 441, 445, 495, 498, 499 N

Nakamoto, A., 329 Nandakumar, R., 405 Nandi, M., 508, 509 Natarajan, K.S., 66 Nauck, F., 7 Neeralagi, P.S., 384 Neggazi, B., 72 Nemhauser, G.L., 56, 59, 67, 68

Author index Nešetřil, J.J., 69, 366 Nevries, R., 288 Ng, C.P., 66 Ng, C.T., 399, 400 Nichterlein, A., 17 Nieberg, T., 72 Niedermeier, R., 72 Nieminen, J., 1, 23, 24, 36, 66, 92, 385, 393 Niepel, Ľ, 72 Nieuwejaar, N., 512 Nishizeki, T., 67 Nordhaus, E.A., 467, 479–481, 484–486, 488–490, 493, 495, 498–501, 503 Nowakowski, R.J., 92, 122, 437–439, 541, 542 O

O’Rourke, J., 342, 343 O, S., 96, 98, 449, 450 Obradović, N., 270, 271 Ore, Ø., 1, 22–24, 46, 47, 80, 81, 100, 101, 130, 151, 167, 296, 354, 368, 404, 412, 467, 498 Orlovich, Y.L., 72 Östergård, P.R.J., 9, 72, 507, 510, 511, 519–521, 550 P

Panda, B.S., 72 Pandey, A., 72 Pandu Rangan, C., 288 Pareek, C.M., 526 Pargas, R.P., 507 Paris, M., 397, 398 Park, B., 203 Park, J., 15 Park, W.J., 349 Parks, D.A., 71 Parragh, S.N., 15 Parui, S., 508, 509 Paschos, V.Th., 72 Paul, C., 72 Pauls, E., 7

641 Pavlič, P., 508 Payan, C., 32, 75, 80, 81, 100, 101, 130, 209, 210, 455, 467–470, 472, 482, 507 Pelsmajer, M.J., 333 Pemmaraju, S.V., 72 Pepper, R., 88–90, 94, 102 Perković, L., 72 Perl, Y., 59, 67 Peterin, I., 275, 276, 280–282, 375, 378 Peters, J., 270, 271 Peters, K., 68, 455 Pettersson, V., 507 Pfaff, J., 59, 63, 67, 68 Phillips, N., 359, 360 Pilipczuk, M., 289, 539, 540 Pinciu, V., 343 Pinlou, A., 68, 507 Plesník, J., 404 Plummer, M.D., 112, 297, 395–397, 399, 437, 438 Poon, S.-H., 72 Potočnik, P., 268 Prince, N., 469 Prisner, E., 437 Proskurowski, A., 59, 62, 66, 69 Pruchnewski, A., 212 Puech, J., 43, 124, 172, 173, 175, 177, 263 Puertas, M.L., 508 R

Rényi, A., 217 Raczek, J., 383, 451–453, 460–462 Rahbarnia, F., 287, 289, 350–352 Rall, D.F., 23, 41, 92, 113–115, 120, 383, 385, 389, 390, 425, 438, 439, 458, 460, 526, 527, 529, 531–537, 539, 541, 542, 546–548 Ramalingam, G., 69 Ramey, J.E., 437 Randerath, B., 82, 438, 472, 473 Rangan, C.P., 69

642 Rao, M., 68, 507 Rautenbach, D., 149, 231, 236–238, 263, 288, 296–298, 311, 312, 344, 356, 369, 445, 552 Ravi, S.S., 72 Ravindra, G., 437 Reed, B.A., 82, 129, 140, 141, 148–151, 159, 167, 225, 294, 296, 297, 299, 311, 316, 323, 476 Reji Kumar, K., 271 Repolusk, P., 508 ReVelle, C.S., 5 Rhee, C., 69 Rickett, S., 383 Ries, B., 72 Roberts, J., 383, 469 Roditty, Y., 211 Rosenfeld, M., 200, 465 Rosenkrantz, D.J., 72 Rosing, K.E., 5 Rotics, U., 288 Rouse Ball, W.W., 1, 7, 10 Roux, A., 397 Royle, G.F., 437 Ružić, G., 270, 271 Ryjáček, Z., 314, 419, 420 S

Saito, A., 297 Saito, N., 67 Sampathkumar, E., 74, 76, 384 Sanchis, L.A., 230, 231, 244, 245, 300 Sankaranarayana, R.S., 437 Sau, I., 72 Schaudt, O., 287, 288 Schiermeyer, I., 72, 140, 292–295, 419, 420 Schultz, K., 272–274, 282 Scott, H., 383 Seager, S.M., 72, 149, 169, 175, 176, 179, 188, 236, 311, 320 Sebő, A., 437

Index Seese, D., 59 Seier, E., 99 Seifter, N., 266, 508, 509 Semanišin, G., 18 Seyffarth, K., 337–340, 347–349 Seymour, P., 314 Sha, Y., 439 Shaheen, R.S., 72, 508, 509 Shan, E., 94–96, 148, 245, 252 Shangchao, Z., 82, 472, 473, 502, 504 Shao, Z., 337 Sheikholeslami, S.M., 365, 383, 486, 488 Shekhtman, B., 211 Shepherd, B., 82, 129–131, 135, 139, 151, 167, 171, 293, 294, 323, 474, 475 Shiu, W.C., 201, 314, 464 Sikora, F., 17 Simkin, M., 549 Simon, I.A., 404–408 Singh, H.G., 507 Škoda, P., 297 Škrekovski, R., 446, 463, 539, 540 Skukla, S.K., 72 Slater, P.J., i, 1, 25, 26, 31, 32, 34, 35, 37, 70, 75, 79, 92, 109, 110, 128, 151, 260, 272, 283, 287, 288, 366, 368, 397, 400, 426, 435, 437, 555 Smart, C.B., 287 Sohn, M.Y., 117, 148, 151, 153, 167, 282, 478 Southey, J., 149, 200, 207, 310–313, 320, 368, 369, 373, 414, 415, 418, 419, 428, 445, 448, 449, 488, 489 Špacapan, S., 329–332 Spalding, A., 507 Spencer, J.H., 430 Spinrad, J.P., 50, 457 Srimani, P.K., 72 Srinivasan, A., 371 Steinberg, A., 274 Stewart, I., 5

Author index Stewart, L.K., 58, 59, 69, 437 Stocker, C., 149, 150 Stodolsky, B.Y., 150, 297–299 Stout, Q.F., 263, 265, 266, 287, 507 Sudo, Y., 72 Suen, S., 211, 527, 533–536 Suffel, C.L., 385, 393 Sumner, D.P., 114, 128, 383, 384, 397, 398, 400, 442, 444, 445 Sun, L., 159, 169, 191, 195, 196, 201, 464, 527 Swart, H.C., 238, 256, 257, 481–484 Szabó, T., 218, 527 Székely, L., 212 T

Tait, M., 364 Takamizawa, K., 67 Takatou, M., 383 Taketomi, S., 72 Tamizh Chelvam, T., 269 Tanaka, H., 72 Tang, C.Y., 286–288 Tankus, D., 437, 439 Tarjan, R.E., 326–329, 332–334, 342, 551, 552 Tarr, J., 527, 533–535 Tarsi, M., 437 Taylor, D.T., 271, 282 Telle, J.A., 72, 370 Tepeh, A., 446, 463 Teschner, U., 383 Thilikos, D.M., 72 Thomason, A., 32, 455 Thomassé, S., 68, 72, 169, 179–183, 185, 186, 188, 191, 507, 551 Thomassen, C., 371 Thuraiswamy, A., 504 Tian, F., 399, 400 Tjandra, S.A., 17 Tokunaga, S., 336, 337 Topp, J., 383, 437–439, 441, 444 Tsai, S.-Y., 72

643 Turán, P., 404, 428, 431, 524 Turau, V., 72 Turner, J., 270 Tutte, W.T., 96 Tuza, Z., 179, 180, 186, 188, 523, 524 U

Ungerer, E., 371 Urrutia, J., 69 V

van der Merwe, L.C., 383, 400, 402, 403, 406, 407 van Leeuwen, E.J., 72, 287, 289 van Rooij, A., 443 van Vuuren, J.H., 383 van Wee, G.J.M., 521 van Wieren, D., 266 Verstraëte, J., 298, 323, 364, 551 Verter, V., 17 Vestergaard, P.D., 86, 186, 437, 438, 440, 495 Villanger, Y., 72 Virlouvet, C., 419 Vitray, R.P., 85, 87, 116, 169 Vizing, V.G., 1, 23, 228–231, 238, 245, 525, 526, 552 Voigt, M., 212 Volec, J., 297 Volkmann, L., 75, 82, 101, 296, 356, 365, 383, 437–439, 441, 444, 472, 473, 476–478 von Neumann, J., 1, 20 W

Wakabayashi, Y., 336, 337 Walikar, H.B., 38, 74, 76, 385 Waller, B., 88–90, 94, 102 Wallis, C.K., 550 Wang, C., 99, 225, 507 Wang, J., 72, 191, 195, 196 Wang, P., 405, 406, 552 Wang, S., 99 Wang, T., 395, 396, 406, 407 Wang, X., 287 Wang, Y.-L., 72

Index

644 Wash, K., 438 Weakley, W.D., 9, 521, 523, 550 Webb, J.A., 8 Weber, K., 217, 218, 225 Wei, B., 169, 175–177, 179, 188, 320, 399 Wei, V.K., 428, 430, 431 Weichsel, P.M., 34, 521 Welch, J., 13 West, D.B., 89, 90, 94, 96, 98, 449, 450, 469, 486, 488 White, L.J., 66 Whitney, H., 337 Wieland, B., 218 Wilf, H., 443 Wimer, T.V., 35, 68, 69, 507, 508 Woeginger, G.J., 72 Wojcicka, E., 397–400 Wormald, N.C., 72, 225, 226

Yazdanbod, S., 365 Yen, C.C., 286–288 Yeo, A., 25, 79, 94–96, 117, 140, 169, 177, 179–188, 191, 218–220, 224, 237, 238, 254, 255, 292–295, 299–309, 371, 372, 401, 402, 404, 406, 407, 419, 428, 433, 489, 551, 555 Yero, I.G., 275, 276, 280–282, 383 Yongsheng, Y., 462 Yoshida, N., 59, 67 Yoshimura, I., 443 Yu, Q., 395, 396 Yuan, X.D., 148, 478 Yuansheng, Y., 383 Yuster, R., 169, 175, 176, 179, 188, 320

X

Zamaraev, V., 72 Zelinka, B., 358, 359, 362, 364–367 Zerbib, S., 533, 535 Žerovnik, J., 508, 515 Zhang, L., 399, 400 Zhang, Y., 399 Zhao, C., 398 Zhou, S.M., 282 Zhu, E., 337, 462 Ziegler, G.M., 335 Zighem, I., 507 Zito, M., 72 Zoe, Y., 99 ZuaZua, R., 342–344 Zverovich, I.E., 72, 128, 444, 445, 458 Zverovich, V.E., 128, 444, 445, 457, 458 Żyliński, P., 342–344, 383

Xin, H., 398 Xing, H.M., 159 Xu, J., 337 Xu, K., 99 Xu, Z., 72 Xudong, Y., 151, 153, 167 Xue, Y.F., 399 Xuong, N.H., 80, 81, 100, 101, 130, 469, 470, 472 Y

Yaglom, A.M., 11 Yaglom, I.M., 11 Yahiaoui, S., 72 Yamashita, M., 14 Yan, G.Y., 405 Yang, Y., 398 Yannakakis, M., 67

Z