Research Trends in Graph Theory and Applications (Association for Women in Mathematics Series, 25) 3030779823, 9783030779825

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Association for Women in Mathematics Series

Daniela Ferrero Leslie Hogben Sandra R. Kingan Gretchen L. Matthews   Editors

Research Trends in Graph Theory and Applications

Association for Women in Mathematics Series Volume 25

Series Editor Kristin Lauter Microsoft Research Redmond, WA, USA

Association for Women in Mathematics Series

Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer’s Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics, as well as in the areas of mathematical education and history. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community in the United States and around the world. Titles from this series are indexed by Scopus.

More information about this series at http://www.springer.com/series/13764

Daniela Ferrero • Leslie Hogben • Sandra R. Kingan Gretchen L. Matthews Editors

Research Trends in Graph Theory and Applications

Editors Daniela Ferrero Department of Mathematics Texas State University San Marcos, TX, USA Sandra R. Kingan Department of Mathematics Brooklyn College and the Graduate Center City University of New York New York, NY, USA

Leslie Hogben Department of Mathematics Iowa State University Ames, IA, USA American Institute of Mathematics San Jose, CA, USA Gretchen L. Matthews Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA, USA

ISSN 2364-5733 ISSN 2364-5741 (electronic) Association for Women in Mathematics Series ISBN 978-3-030-77982-5 ISBN 978-3-030-77983-2 (eBook) https://doi.org/10.1007/978-3-030-77983-2 Mathematics Subject Classification: 05C, 05E, 94C, 94B, 68R © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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

Foreword

This book contains a selection of research at the frontiers of current knowledge in a broad variety of different areas of graph theory and its applications. Research topics include edge-density conditions in multipartite graphs, graph searching, metric dimension, reconfiguration problems, creating chaos by breaking symmetries, and coding theory problems for the protection and accessibility of distributed data storage. The research in this volume was performed by teams formed at the Workshop for Women in Graph Theory and Applications held at the Institute for Mathematics and its Applications (University of Minnesota–Minneapolis) on August 19–23, 2019. During this workshop, 42 participants performed collaborative research in 6 teams, each focused on a different area of graph theory and its applications. Teams included experts in each area, who helped participants identify open problems to work on. After the workshop, all teams continued their collaborations remotely and solved some of the problems they started researching during the workshop. This book contains six chapters, and each of them is written by one of the research collaboration teams formed at the Workshop for Women in Graph Theory and Applications. Chapter 1: Finding Long Cycles in Balanced Tripartite Graphs: A First Step Gabriela Araujo-Pardo, Zhanar Berikkyzy, Jill Faudree, Kirsten Hogenson, Rachel Kirsch, Linda Lesniak, and Jessica McDonald Intuitively, in a graph with no loops and no parallel edges, a high level of edge density implies the existence of long cycles. This observation leads to the problem of determining the minimum number of edges q such that an arbitrary graph with exactly p vertices and q edges necessarily has a cycle of length at least , a question especially interesting for large values of . In the case  = p this question asks for the minimum number of edges such that every graph with order p has a cycle containing every single vertex exactly once, or a Hamiltonian cycle. The problem considered in this chapter is the existence of long cycles in balanced tripartite graphs. This chapter starts with a survey of the relevant literature, namely degree and edge conditions for the existence of long cycles in graphs, including the case of v

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Foreword

Hamiltonian cycles as well as improved conditions obtained in the case of bipartite and k-partite results where they exist. After this survey, the authors prove that if G is a balanced tripartite graph on 3n vertices, G must contain a cycle of length at least 3n − 1 provided that e(G) ≥ 3n2 − 4n + 5 and n ≥ 14. Chapter 2: Product Throttling Sarah E. Anderson, Karen L. Collins, Daniela Ferrero, Leslie Hogben, Carolyn Mayer, Ann N. Trenk, and Shanise Walker Propagation processes on graphs, such as zero forcing and power domination, and pursuit-evasion games, such as cops and robbers, involve an element of graph searching. Parameters indicating the amount of time that it takes to finish the process or search have been independently proposed in each case, and their study has produced interesting results. In this chapter the authors study the cost trade-off between time and resources when the process uses more than the minimum possible number of resources. Throttling addresses the question of minimizing the sum or the product of the resources used in a graph searching process and the time needed to complete the process. The study of throttling began with the study of sum throttling, and the forms of graph searching that have been studied include various types of zero forcing, power domination, and cops and robbers. Recently, two different definitions of product throttling have been introduced for cops and robbers and power domination. This chapter presents a summary of prior results for these two cases and introduces universal versions of the two definitions. Each of the definitions is then applied and studied for each of the following graph searching processes: standard zero forcing, positive semidefinite zero forcing, power domination, and cops and robbers. Chapter 3: Analysis of Termatiko Sets in Measurement Matrices Katherine F. Benson, Jessalyn Bolkema, Kathryn Haymaker, Christine Kelley, Sandra R. Kingan, Gretchen L. Matthews, and Esmeralda L. N˘astase Termatiko sets are combinatorial structures that have been shown to hinder the success of the interval-passing algorithm in compressed sensing. In this chapter, the authors study how termatiko sets relate to other combinatorial structures in graphs representing measurement matrices that are also known to cause failure in similar iterative algorithms, including stopping, trapping, and absorbing sets. Results include bounds on the sizes of termatiko sets of measurement matrices based on finite geometries, as well as on the effect of the redundancy of the matrices on the number of these sets. Chapter 4: The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey Nadia Benakli, Novi H. Bong, Shonda Dueck (Gosselin), Beth Novick, and Ortrud R. Oellermann In a connected graph G with at least three vertices, a vertex w is said to resolve two vertices u and v if the distance from u to w does not equal the distance from v to w. If there is either a shortest u − w path that contains v or a shortest v − w

Foreword

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path that contains u, then w strongly resolves u and v. A set W of vertices of G is a resolving set (strong resolving set), if every pair of vertices of G is resolved (respectively, strongly resolved) by some vertex of W . A smallest resolving set (strong resolving set) of a graph is called a basis (respectively, a strong basis) and its cardinality, denoted β(G) (respectively, βs (G)), the metric dimension (respectively, the strong dimension) of G. The threshold dimension (respectively, threshold strong dimension) of a graph G, denoted τ (G) (respectively, τs (G)), is the smallest metric dimension (respectively, strong dimension) among all graphs having G as a spanning subgraph. In this chapter, the authors present and analyze results on these parameters concluding with several open problems about them. Chapter 5: Symmetry Parameters for Mycielskian Graphs Debra Boutin, Sally Cockburn, Lauren Keough, Sarah Loeb, K. E. Perry, and Puck Rombach A coloring of the vertices of a graph G is called a distinguishing coloring if no non-trivial automorphism of G preserves the color classes. A graph is called d-distinguishable if it has a distinguishing coloring using exactly d colors. The distinguishing number of G, is the smallest number of colors in a distinguishing coloring of G. A subset S of vertices of G is a determining set for G if any two automorphisms of G coincide over the elements of S. The determining number of a graph G, is the size of a smallest determining set. The Mycielskian construction, denoted μ(G), takes a finite simple graph G to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted μt (G), takes G to a larger graph with the same chromatic number but with larger odd girth. In this chapter, the authors study the determining number, distinguishing number, and cost of distinguishing in graphs obtained as μ(G) and μt (G) where G is a finite graph G without loops or multiple edges, μ(G) denotes the (traditional) Mycielskian of G and μt (G) is the generalized Mycielskian of G. Chapter 6: Reconfiguration Graphs for Dominating Sets Kira Adaricheva, Chassidy Bozeman, Nancy E. Clarke, Ruth Haas, Margaret-Ellen Messinger, Karen Seyffarth, and Heather C. Smith Given a problem and a set of feasible solutions, the associated reconfiguration problem consists of determining whether or not one feasible solution can be transformed to another by following a predetermined set of rules. Many problems can be framed as reconfiguration problems. This chapter focuses on the study of reconfiguration for the classic graph theory problem of domination. The dominating graph of a graph G is defined as a graph whose vertices are all the dominating sets of G, and there is an edge between two dominating sets if one can be obtained from the other by adding or deleting a single vertex of G. This chapter provides a brief introduction to the study of reconfiguration of dominating sets and the dominating graph of a graph G, including the necessary background on the area and highlights of some previous work and known results. The authors also present new properties of reconfiguration graphs for dominating sets. In particular, this chapter contains

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Foreword

new results about the existence of a Hamilton path in the dominating graph of a graph G. Each chapter in this book features a succinct introduction to the topic of research so that readers can quickly advance and immerse themselves into current research and open problems. The emphasis of the material presented is on recent research developments and open problems on the topics of each chapter. This volume is intended to motivate the reader to become engaged in research in graph theory and its applications. San Marcos, TX, USA Ames, IA, USA Brooklyn, NY, USA Blacksburg, VA, USA March 2021

Daniela Ferrero Leslie Hogben Sandra R. Kingan Gretchen L. Matthews

Preface

The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and its Applications (IMA), University of Minnesota– Minneapolis, on August 19–23, 2019. The main objective of the Workshop for Women in Graph Theory and Applications was to promote the creation of a strong and lasting research collaboration network among female researchers working in graph theory and applications. Research collaborations are a central component of a successful research career and have been proven successful in increasing the number of women doing research in a field, raising the impact of their research findings, and, consequently, contributing to heightening the profile of female researchers in the specific area. For this reason, the workshop was proposed and organized as a working meeting in which participants devoted most of their time performing collaborative research in groups. During this workshop, 42 participants performed collaborative research in 6 teams, each focused on a different area of graph theory and its applications. Teams included experts in each area, who helped participants identify some open problems to work on. Due to the scarcity of women in graph theory and applications, the workshop also offered a unique opportunity for mentoring, and for this reason, the teams were formed with the purpose of promoting interaction among researchers at different stages in their career. The work of each team was led by two established researchers in the specific area of research. The organizers of the workshop were Daniela Ferrero, Sandra R. Kingan, and Linda Lesniak. The following are the research areas and the participants of each team: 1. Edge conditions in multipartite graphs Participants: Gabriela Araujo-Pardo, Zhanar Berikkyzy, Jill Faudree (mentor), Kirsten Hogenson, Rachel Kirsch, Linda Lesniak (mentor), and Jessica McDonald 2. Graph searching Participants: Sarah Anderson, Karen Collins, Daniela Ferrero (mentor), Leslie Hogben (mentor), Carolyn Mayer, Ann Trenk, and Shanise Walker

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Preface

3. Data storage, protection, and accessibility Participants: Katherine Benson, Jessalyn Bolkema, Katie Haymaker, Christine Kelley (mentor), Sandra R. Kingan, Gretchen L. Matthews (mentor), and Esmeralda N˘astase 4. The metric dimension of a graph Participants: Nadia Benakli, Shonda Dueck, Linda Eroh (mentor), Pamela Harris, Beth Novick, and Ortud Oellermann (mentor) 5. Creating chaos by breaking graph symmetries Participants: Debra Boutin (mentor), Minnie Catral, Sally Cockburn (mentor), Lauren Keough, Sarah Loeb, Katherine Perry, and Michaela Rombach 6. Reconfiguration problems Participants: Kira Adaricheva, Chassidy Bozeman, Nancy Clarke, MargaretEllen Messinger, Ruth Haas (mentor), Karen Seyffarth (mentor), and Heather Smith Prior to the workshop, team leaders provided their respective teams with a list of bibliographical references, carefully selected to facilitate their joint research during the week they collaborated at IMA. During the workshop, each team worked with their leaders to identify relevant and meaningful open problems in their area and to research one or more of these problems. In the months following the workshop, all six teams have continued collaborating and have made significant contributions to open problems in their respective areas. All teams were able to advance their research projects during the week of the workshop to the level that permitted to continue collaborating remotely in subsequent months. The Workshop for Women in Graph Theory and Applications held at IMA also led to the creation of the Women in Graph Theory and Applications Research Collaboration Network (WIGA), which provided the framework for the publication of this book.

Participants and Affiliations at the Time of the Workshop • Kira Adaricheva Hofstra University, Hempstead, NY, USA • Sarah Anderson University of St. Thomas, St. Paul, MN, USA • Gabriela Araujo-Pardo Universidad Nacional Autónoma de México, Juriquilla, Querétaro, México • Nadia Benakli New York City College of Technology, Brooklyn, NY, USA • Katherine Benson University of Wisconsin-Stout, Menomonie, WI, USA • Zhanar Berikkyzy University of California - Riverside, Riverside, CA, USA

Preface

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• Jessalyn Bolkema Harvey Mudd College, Claremont, CA, USA • Novi Bong University of Delaware, Newark, DE, USA • Debra Boutin Hamilton College, Clinton, NY, USA • Chassidy Bozeman Mount Holyoke College, South Hadley, MA, USA • Minnie Catral Xavier University, Cincinnati, OH, USA • Fan Chung Graham University of California- San Diego, La Jolla, CA, USA • Nancy Clarke Acadia University, Wolfville, Nova Scotia, Canada • Sally Cockburn Hamilton College, Clinton, NY, USA • Karen Collins Wesleyan University, Middletown, CT, USA • Shonda Dueck The University of Winnipeg, Winnipeg, Manitoba, Canada • Linda Eroh University of Wisconsin at Oshkosh, Oshkosh, WI, USA • Jill Faudree University of Alaska-Fairbanks, Fairbanks AK, USA • Daniela Ferrero Texas State University, San Marcos, TX, USA • Ruth Haas University of Hawaii at Manoa, Honolulu, HI, USA • Pamela Harris Williams College, Williamstown, MA, USA • Katie Haymaker Villanova University, Villanova, PA, USA • Leslie Hogben Iowa State University, Ames, IA, USA and American Institute of Mathematics, San Jose, CA, USA • Kirsten Hogenson Colorado College, Colorado Springs, CO, USA • Christine Kelley University of Nebraska-Lincoln, Lincoln, NE, USA • Lauren Keough Grand Valley State University, Allendale Charter Township, MI, USA • Sandra R. Kingan Brooklyn College, City University of New York, Brooklyn, NY, USA

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• Rachel Kirsch London School of Economics and Political Science, London, England • Linda Lesniak Western Michigan University, Kalamazoo, MI, USA • Sarah Loeb Hampden-Sydney College, Hampden-Sydney, VA, USA • Gretchen L. Matthews Virginia Polytechnic Institute and State University, Blacksburg, VA, USA • Carolyn Mayer Worcester Polytechnic Institute, Worcester, MA, USA • Jessica McDonald Auburn University, Auburn, AL, USA • Margaret-Ellen Messinger Mount Allison University, Sackville, New Brunswick, Canada • Esmeralda N˘astase Xavier University, Cincinnati, OH, USA • Beth Novick Clemson University, Clemson, SC, USA • Ortud Oellermann The University of Winnipeg, Winnipeg, Manitoba, Canada • Katherine Perry University of Denver, Denver, CO, USA • Michaela Rombach University of Vermont, Burlington, VT, USA • Karen Seyffarth University of Calgary, Calgary, Alberta, Canada • Heather Smith Davidson College, Davidson, NC, USA • Ann Trenk Wellesley College, Wellesley, MA, USA • Shanise Walker University of Wisconsin-Eau Claire, Eau Claire, WI, USA

Workshop Website https://www.ima.umn.edu/2018-2019/SW8.19-23.19 San Marcos, TX, USA Ames, IA, USA New York, NY, USA Blacksburg, VA, USA March 2021

Daniela Ferrero Leslie Hogben Sandra R. Kingan Gretchen L. Matthews

Acknowledgments

The editors express their gratitude to the authors of each of the chapters of this book and to the reviewers who carefully and thoroughly examined them and provided feedback to the authors. The diligent work of authors and reviewers was essential to have all chapters in this book peer-reviewed, and consequently, to ensure this book upholds the high standards characteristic of the AWM Springer Series. The continued support provided by the AWM ADVANCE Research Communities Program (funded by NSF-HDR-1500481, Career Advancement for Women Through Research-Focused Networks) since the early preparatory stages of the Workshop for Women in Graph Theory and Applications, is gratefully acknowledged, as is the support and the hospitality received from the Institute for Mathematics and its Applications in the organization and during the Workshop for Women in Graph Theory and Applications. Special thanks to Kristin Lauter, editor of the AWM Springer Series, for her encouragement and support to produce this volume. The editors are extremely grateful for the support received from Springer at every stage of the work in this book. Especially, many thanks to Elizabeth Loew and Dahlia Fisch who worked with the editors during the proposal and approval process of this book and to Jeffrey Taub and Saveetha Balasundaram who offered support for the production of this book.

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Contents

1

Finding Long Cycles in Balanced Tripartite Graphs: A First Step . . . . Gabriela Araujo-Pardo, Zhanar Berikkyzy, Jill Faudree, Kirsten Hogenson, Rachel Kirsch, Linda Lesniak, and Jessica McDonald

1

2

Product Thottling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sarah E. Anderson, Karen L. Collins, Daniela Ferrero, Leslie Hogben, Carolyn Mayer, Ann N. Trenk, and Shanise Walker

11

3

Analysis of Termatiko Sets in Measurement Matrices . . . . . . . . . . . . . . . . . . . Katherine F. Benson, Jessalyn Bolkema, Kathryn Haymaker, Christine Kelley, Sandra R. Kingan, Gretchen L. Matthews, and Esmeralda L. N˘astase

51

4

The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nadia Benakli, Novi H. Bong, Shonda Dueck (Gosselin), Beth Novick, and Ortrud R. Oellermann

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5

Symmetry Parameters for Mycielskian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . Debra Boutin, Sally Cockburn, Lauren Keough, Sarah Loeb, K. E. Perry, and Puck Rombach

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6

Reconfiguration Graphs for Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Kira Adaricheva, Chassidy Bozeman, Nancy E. Clarke, Ruth Haas, Margaret-Ellen Messinger, Karen Seyffarth, and Heather C. Smith

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Contributors

Kira Adaricheva Hofstra University, Hempstead, NY, USA Sarah E. Anderson University of St. Thomas, St. Paul, MN, USA Gabriela Araujo-Pardo Universidad Nacional Autónoma de México, Juriquilla, Querétaro, Mexico Nadia Benakli New York City College of Technology, Brooklyn, NY, USA Katherine F. Benson University of Wisconsin-Stout, Menomonie, WI, USA Zhanar Berikkyzy Fairfield University, Fairfield, CT, USA Jessalyn Bolkema California State University, Dominguez Hills, Carson, CA, USA Novi H. Bong University of Delaware, Newark, DE, USA Debra Boutin Hamilton College, Clinton, NY, USA Chassidy Bozeman Mount Holyoke College, South Hadley, MA, USA Minnie Catral Xavier University, Cincinnati, OH, USA Nancy E. Clarke Acadia University, Wolfville, NS, Canada Sally Cockburn Hamilton College, Clinton, NY, USA Karen L. Collins Wesleyan University, Middletown, CT, USA Shonda Dueck (Gosselin) The University of Winnipeg, Winnipeg, MB, Canada Jill Faudree University of Alaska-Fairbanks, Fairbanks, AK, USA Daniela Ferrero Texas State University, San Marcos, TX, USA Ruth Haas University of Hawaii at Manoa, Honolulu, HI, USA Kathryn Haymaker Villanova University, Villanova, PA, USA

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Contributors

Leslie Hogben Iowa State University, Ames, IA, USA American Institute of Mathematics, San Jose, CA, USA Kirsten Hogenson Skidmore College, Saratoga Springs, NY, USA Christine Kelley University of Nebraska-Lincoln, Lincoln, NE, USA Lauren Keough Grand Valley State University, Allendale, MI, USA Sandra R. Kingan Brooklyn College and the Graduate Center, City University of New York, New York, NY, USA Rachel Kirsch Iowa State University, Ames, IA, USA Linda Lesniak Western Michigan University, Kalamazoo, MI, USA Sarah Loeb Hampden-Sydney College, Hampden-Sydney, VA, USA Gretchen L. Matthews Virginia Polytechnic Institute and State University, Blacksburg, VA, USA Carolyn Mayer Worcester Polytechnic Institute, Worcester, MA, USA Sandia National Laboratories, Albuquerque, NM, USA Jessica McDonald Auburn University, Auburn, AL, USA Margaret-Ellen Messinger Mount Allison University, Sackville, NB, Canada Esmeralda L. N˘astase Xavier University, Cincinnati, OH, USA Beth Novick Clemson University, Clemson, SC, USA Ortud R. Oellermann The University of Winnipeg, Winnipeg, MB, Canada K. E. Perry Soka University of America, Aliso Viejo, CA, USA Puck Rombach University of Vermont, Burlington, VT, USA Karen Seyffarth University of Calgary, Calgary, AB, Canada Heather C. Smith Davidson College, Davidson, NC, USA Ann N. Trenk Wellesley College, Wellesley, MA, USA Shanise Walker University of Wisconsin-Eau Claire, Eau Claire, WI, USA

Chapter 1

Finding Long Cycles in Balanced Tripartite Graphs: A First Step Gabriela Araujo-Pardo, Zhanar Berikkyzy, Jill Faudree, Kirsten Hogenson, Rachel Kirsch, Linda Lesniak, and Jessica McDonald

1.1 Introduction In this chapter, all graphs are simple. For undefined terms, see [6]. The lack of loops or parallel edges means that if a graph has enough edges, then it must contain long cycles. We use e(G) to denote the number of edges in the graph G. In this chapter,

G. Araujo-Pardo Instituto de Matematicas-Campus Juriquilla, Universidad Nacional Autonoma de Mexico, Juriquilla, Mexico e-mail: [email protected] Z. Berikkyzy Department of Mathematics, Fairfield University, Fairfield, CT, USA e-mail: [email protected] J. Faudree Department of Mathematics and Statistics, University of Alaska-Fairbanks, Fairbanks, AK, USA e-mail: [email protected] K. Hogenson Department of Mathematics and Statistics, Skidmore College, Saratoga Springs, NY, USA e-mail: [email protected] R. Kirsch Department of Mathematics, Iowa State University, Ames, IA, USA e-mail: [email protected] L. Lesniak () Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] J. McDonald Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_1

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G. Araujo-Pardo et al.

vertices

1 vertices

1 vertices

Fig. 1.1 This graph is a Kn,n−1,n−1 with two additional degree-2 vertices, x and y, with a common neighbor in the largest partite set. It has 3n2 −4n+4 edges, and its longest cycle has 3n−2 vertices, thus demonstrating that Theorem 1.1 is sharp

we specifically consider the problem of finding long cycles in balanced tripartite graphs, and prove the following result. Theorem 1.1 Let G be a balanced tripartite graph on 3n vertices such that n = 1, 2 or n ≥ 14. If e(G) ≥ 3n2 − 4n + 5, then G contains a cycle of length at least 3n − 1. The edge bound in Theorem 1.1 is tight; see Fig. 1.1. While the cases 3 ≤ n ≤ 13 are not addressed in Theorem 1.1, we suspect that the theorem also holds for these values of n; the current proof just does not work for them. In the next section we survey degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and k-partite results where they exist. This survey provides context for Theorem 1.1 as the first step in the larger problem of determining edge conditions for long cycles in tripartite and k-partite graphs. The proof of Theorem 1.1 appears in Sect. 1.3 of this chapter. As a matter of notation, the variable p will denote the number of vertices of a graph, and n will denote the number of vertices in one part of a (typically balanced) multipartite graph.

1.2 Background Given an arbitrary graph, the problem of finding a Hamiltonian cycle, i.e. a cycle containing every vertex of the graph, is known to be NP-complete [19]. However, much study has been devoted to finding sufficient conditions for Hamiltonicity (i.e., for a graph to contain a Hamiltonian cycle).

1 Finding Long Cycles in Balanced Tripartite Graphs: A First Step

3

1.2.1 Degree Conditions The most famous of these results involve degree conditions.

1.2.1.1

Degree Conditions and Hamiltonicity

In 1952, Dirac gave the following simple minimum degree condition for Hamiltonicity. Theorem 1.2 ([10]) For p ≥ 3, if a p-vertex graph G has δ(G) ≥ p/2, then G is Hamiltonian. In 1960, Ore proved a variation on Dirac’s famous theorem. Theorem 1.3 ([14]) Let G be a p-vertex graph with p ≥ 3 such that for any pair of nonadjacent vertices u and v, deg(u) + deg(v) ≥ p. Then G is Hamiltonian. Ore’s result led to the definition of a new graph parameter: σ2 (G) = min{deg(u) + deg(v) | u, v ∈ V (G), u = v, uv ∈ E(G)}, for G = Kn , and σ2 (Kn ) = ∞. Results which involve a bound on this parameter are called σ2 results or Ore-type results. In 1963, Moon and Moser considered balanced bipartite graphs and were able to lower Dirac’s minimum degree bound by a factor of two. Theorem 1.4 ([13]) If G is a balanced bipartite graph on p vertices with p ≥ 4 and δ(G) > p/4, then G is Hamiltonian. They also proved the following Ore-type result with an improved lower σ2 bound. Theorem 1.5 ([13]) If G is a balanced bipartite graph on p vertices with p ≥ 4 and σ2 (G) > p/2, then G is Hamiltonian. Chen, Faudree, Gould, Jacobson, and Lesniak extended the Dirac-type Hamiltonicity results to balanced k-partite graphs. Theorem 1.6 ([8]) If G is a balanced k-partite graph on kn vertices and  δ(G) >

( k2 − ( k2

−

1 k+1 )n 2 k+2 )n

if k is odd if k is even,

then G is Hamiltonian. Chen and Jacobson proved an Ore-type result for Hamiltonicity in balanced k-partite graphs in 1997. Note this result does not use the σ2 notation because the Chen-Jacobson degree-sum condition is restricted to nonadjacent vertices in different partite sets; thus it implies a σ2 condition.

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Theorem 1.7 ([7]) Let G be a balanced k-partite graph of order kn with k ≥ 2. If ⎧ ⎨ k− deg(u) + deg(v) >  ⎩ k−



2 k+1  n 4 k+2 n

if k is odd if k is even

for every pair of nonadjacent vertices u and v which are in different partite sets, then G is Hamiltonian. In 2019, DeBiasio et al. [9] addressed k-partite graphs that are not necessarily balanced. Assuming a k-partite graph has the necessary condition that no part contains more than half the vertices, they gave a minimum degree condition that implies Hamiltonicity. This minimum degree condition, although somewhat cumbersome to describe, asymptotically implies the result of Chen et al. (above) for balanced k-partite graphs. A different type of degree condition for Hamiltonicity was given by Pósa in 1962. Theorem 1.8 ([17]) Let G be a graph of order p. If, for every integer r with 1 ≤ r < p2 , the number of vertices of degree at most r is less than r, then G is Hamiltonian. Moon and Moser gave a similar result for balanced bipartite graphs in 1963. Theorem 1.9 ([13]) Let G be a balanced bipartite graph of order 2n ≥ 4. If, for each r with 1 ≤ r ≤ n/2, the number of vertices of degree at most r is less than r, then G is Hamiltonian.

1.2.1.2

Degree Conditions and Long Cycles

The circumference of a graph, denoted c(G), is the length of a longest cycle in G. All of the results we have discussed so far concern Hamiltonian graphs, which always have c(G) = |V (G)|. However, minimum degree, σ2 , and Pósatype conditions can also imply the existence of more general long cycles in graphs. Dirac’s 1952 paper states a result of this type. Theorem 1.10 ([10]) Suppose G is a 2-connected graph on p vertices and 3 ≤ c ≤ p. If δ(G) ≥ c/2, then G has a cycle of length at least c. In 1975, Bermond published a generalization of Ore’s theorem. Theorem 1.11 ([5]) Suppose G is a 2-connected graph on p vertices and 3 ≤ c ≤ p. If σ2 (G) ≥ c, then G has a cycle of length at least c. Long cycle variations of bipartite Hamiltonicity results have also been proved. For example, in 2009 J. Adamus and L. Adamus proved an Ore-type result for long cycles in balanced bipartite graphs.

1 Finding Long Cycles in Balanced Tripartite Graphs: A First Step

5

Theorem 1.12 ([2]) Let G be a 2-connected balanced bipartite graph on 2n ≥ 4 vertices such that σ2 (G) ≥ n. Then G contains an even cycle of length at least 2n − 2. In 1963, Pósa generalized his own Hamiltonicity result with the following theorem that gives a degree condition for the existence of long cycles. We will use this theorem to prove our main result in Sect. 1.3. Theorem 1.13 ([16]) Suppose G is a 2-connected graph on p vertices, and let 3 ≤ c ≤ p. If |{v : deg(v) ≤ j }| < j for all 1 ≤ j ≤ c−1 2 , then G has a cycle of length at least c.

1.2.2 Edge Conditions In addition to the degree conditions, it is also possible to guarantee Hamiltonicity or long cycles in a graph using a global edge count. The seminal result on edge conditions for Hamiltonicity was given by Ore in 1961. Theorem 1.14 ([15]) If G is a simple graph of order p and e(G) ≥ 12 (p − 1)(p − 2) + 2, then G is Hamiltonian. A natural problem inspired by Ore’s result is the following: For any pair of nonnegative integers p and k < p2 − 1, find the minimum integer g(p, k) such that every graph of order p and size at least g(p, k) contains a cycle of length p − k. In 1972, Woodall proved the following theorem. Theorem 1.15 ([20]) Let G be a graph of order p ≥ 2k + 3, k ∈ N, and   p−k−1 k+2 + + 1. e(G) ≥ g(p, k) = 2 2 Then G contains a cycle of length  for each  such that 3 ≤  ≤ p − k. This result is best possible. An extremal graph for this problem is the graph obtained by identifying one vertex of a Kp−k−1 with one vertex of a Kk+2 . Note that this graph has exactly g(p, k) − 1 edges and cannot contain any cycle of length p − k. Analogous edge conditions have also been found for multipartite graphs. In 1985, Mitchem and Schmeichel proved the following result about balanced bipartite graphs. A bipartite graph is bipancyclic if it contains even cycles of all possible lengths. Thus, a bipancyclic balanced bipartite graph is also Hamiltonian. Theorem 1.16 ([12]) Let G be a balanced bipartite graph of order 2n. If e(G) ≥ n2 − n + 2, then G is bipancyclic. In 1999, Bagga and Varma proved a variation of this result for bipartite graphs that may not be balanced.

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Theorem 1.17 ([4]) Let G be a bipartite graph with partite sets X and Y , |X| = m ≤ n = |Y |. If e(G) ≥ n(m−1)+2, then G contains cycles C2 for all 2 ≤  ≤ m. In 2009, Adamus generalized the previous two results with the following theorem. Theorem 1.18 ([3]) Let G be a bipartite graph with partite sets X and Y , |X| = m ≤ n = |Y |, where m ≥ 12 k 2 + 32 k + 4, k ∈ N. If e(G) ≥ n(m − k − 1) + k + 1, then either G contains a cycle of length 2m−2k, or else e(G) = n(m−k −1)+k +1 and G is isomorphic to a graph in the family Gm,n,k (see below). In both cases, G contains C2 for all 2 ≤  ≤ m − k − 1. The family Gm,n,k is defined as the set of bipartite graphs with partite sets X and Y , |X| = m, |Y | = n, such that X = A ∪ B where |A| = m − k − 1, |B| = k + 1, and deg(v) = n for every v ∈ A, and deg(w) = 1 for every w ∈ B. Researchers have also found edge conditions for Hamiltonicity and large circumference in k-partite graphs with k ≥ 3. Adamus proved a result about Hamiltonicity of balanced tripartite graphs. Theorem 1.19 ([1]) Let G be a balanced tripartite graph of order 3n, n ≥ 2. If G has at least 3n2 − 2n + 2 edges, then G is Hamiltonian. Ferrero and Lesniak [11] later generalized this result to tripartite graphs which may not be balanced. The following theorem, which we will use in proving our main result, is a combination of their Theorem 18, Corollary 21, Theorem 22, and Theorem 23. Theorem

1.20 ([11]) Let k ≥ 3 and let n1 ≥ n2 ≥ · · · ≥ nk be positive integers with p = ki=1 ni . Let G be a k-partite graph with part sizes n1 , n2 , . . . , nk . Then G is Hamiltonian provided that e(G) ≥ e(Kn1 ,n2 ,...,nk )−(p−n1 −2) and n1 ≤ p/2.

1.3 Main Result In order to prove the main result, we will use Theorems 1.7, 1.13, and 1.20 (above) and a classic extremal result by Turán. For ease of reference the theorem is restated here. Theorem 1.21 ([18]) Let G be a graph on n vertices that does not contain Kr+1  2 r the complete r-partite graph such as a subgraph. Then e(G) ≤ nr 2 . Moreover,    that all partite sets have cardinality nr or nr is the unique graph that contains the maximum number of edges for a given n and r.

1 Finding Long Cycles in Balanced Tripartite Graphs: A First Step

7

Theorem 1.1 Let G be a balanced tripartite graph on 3n vertices such that n = 1, 2 or n ≥ 14. If e(G) ≥ 3n2 − 4n + 5, then G contains a cycle of length at least 3n − 1. Proof For n = 1 the theorem holds vacuously. Let n = 2 or n ≥ 14. Let G be a balanced tripartite graph on 3n vertices such that e(G) ≥ 3n2 − 4n + 5. Observe that if G is viewed as a subgraph of Kn,n,n , then its construction involves deleting at most 4n − 5 edges from Kn,n,n . Thus, subgraphs of Kn,n,n that would require the deletion of more than 4n − 5 edges from Kn,n,n are impossible. This idea will be used frequently in the proof. Claim If δ(G) ≤ 2, then G contains a cycle of length 3n − 1. Suppose G has a vertex v with deg(v) ≤ 2. Observe that G − v is a tripartite graph with part sizes n, n, n − 1 such that a largest partite set contains at most half of the vertices. Moreover, e(G − v) ≥ e(G) − 2 ≥ (3n2 − 4n + 5) − 2 = 3n2 − 4n + 3 and e(Kn,n,n−1 ) − (3n − 1 − n − 2) = 3n2 − 4n + 3. Thus, Theorem 1.20 implies G − v is Hamiltonian. Thus, G contains a cycle of length 3n − 1 and the claim holds.  Now we can assume δ(G) ≥ 3. If n = 2, then Theorem 1.2 implies G is Hamiltonian, so we can assume that n ≥ 3. Claim G is 2-connected. Suppose G has at least two components. Let H be a smallest component of G with partite sets H1 , H2 , and H3 . Let J1 , J2 , and J3 be the partite sets of G − H such that Hi ∪ Ji is a partite set of G for each i. Observe that no tripartite graph on 4 or fewer vertices can have minimum degree 3; thus, we can assume |H | ≥ 5. Observe that G is missing all the edges between Hi and Ji+1 ∪ Ji+2 , where indices are taken modulo 3. Let h = |H | and hi = |Hi |. Thus, construction of G from Kn,n,n would require deleting h1 (2n − h2 − h3 ) + h2 (2n − h1 − h3 ) + h3 (2n − h1 − h2 ) = 2nh − 2e(Kh1 ,h2 ,h3 ) ≥ 2nh − 2

2h2 3

(1.1) (1.2)

edges. Define f (h) = 2nh − 2h3 . Note that the inequality follows from an application of Turán’s theorem (Theorem 1.21). Since H is a smallest component, it follows that 5 ≤ h ≤ 3n/2 which implies that n ≥ 4. Observe that f (h) ≥ 0 on [5, 3n], and f (h) is increasing for h ≤

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3n/2. Thus, f (h) is minimized when h = 5 and, consequently, the number of edges deleted from Kn,n,n to construct G is at least f (5) = 10n − 50/3, which is greater than 4n − 5 for all n ≥ 3. Thus, G must be connected. Suppose there exists a cut-vertex v in G. Let H be a smallest component of G − v, and label the partite sets of H and G − H as in the previous case. Without loss of generality, we can assume v ∈ J1 . We will apply the same argument from the previous disconnected case with slightly different counts resulting from possible adjacencies to v. As before, the observation that no tripartite graph on 4 or fewer vertices can have minimum degree 3 implies that the component H along with the cut vertex v must consist of at least 5 vertices. Thus, |H | ≥ 4 which implies that n ≥ 3. If |H | = 4, then e(G) < 8 + (n − 1)2 + (n − 2)(2n − 2) = 3n2 − 8n + 13, which is too few edges when n ≥ 3. Thus, we can assume that |H | ≥ 5. As before, we know that G is missing all the edges between Hi and Ji+1 ∪ Ji+2 except possibly those to v, where indices are taken modulo 3. Thus, construction of G from Kn,n,n would require deleting at least h1 (2n − h2 − h3 ) + h2 (2n − h1 − 1 − h3 ) + h3 (2n − h1 − 1 − h2 ) = (2n − 1)h − 2e(Kh1 ,h2 ,h3 ) ≥ (2n − 1)h −

2h2 3

(1.3) (1.4)

2

edges. Define g(h) = (2n − 1)h − 2h3 . Again, the function g(h) is minimized when h = 5. Thus the minimum number of edges deleted from Kn,n,n to construct G is g(5) = 10n − 65/3, which is larger than 4n − 5 for all n ≥ 3. Thus, G must be 2-connected and the claim is proven.  Now that G is 2-connected, Pósa’s Theorem (Theorem 1.13) can be applied. If its hypotheses are satisfied for all c such that c ≤ 3n − 1, then G contains a cycle of length at least 3n − 1. If the hypotheses of Pósa’s Theorem fail, then there exists a value of j such that 3 ≤ j ≤ 3n−2 2 and |{v : deg(v) ≤ j }| ≥ j. Let r be such a value of j . Claim r > 3n−5 2 . Since G has at least r vertices of degree at most r, in constructing G from Kn,n,n , we must delete at least 2n − r edges from at least r vertices. Furthermore, at most 3(r/3)2 of the deleted edges may have both endpoints among the r low-degree vertices. Therefore, 4 4n − 5 ≥ e(Kn,n,n ) − e(G) ≥ r(2n − r) − 3(r/3)2 = 2nr − r 2 . 3 Solving for n, we obtain

1 Finding Long Cycles in Balanced Tripartite Graphs: A First Step

9

2 4 1 (4/3)r 2 − 5 = r + + (r − 2)−1 . 2r − 4 3 3 6

n≤ Since 3 ≤ r, we obtain

n≤

2 4 1 4r + 9 r+ + = . 3 3 6 6

Solving for r, we obtain r≥

3n − 5 1 + 2 4 

and the claim has been proven.

At this point, we jump over values n = 3, 4, . . . , 13 and assume n ≥ 14. Observe that if G is Hamiltonian then we have the desired long cycle. If G is not Hamiltonian then by Theorem 1.7, there must exist two nonadjacent vertices in different partite sets such that their degree sum is at most 5n/2. Thus, the two vertices of smallest degree must have degree sum at most 5n/2. We will again use Theorem 1.13 and recall that if the hypotheses of this Theorem fail, they must do so for a value of r such that r > (3n − 5)/2. Since n ≥ 14, it follows that r > (3 · 14 − 5)/2 = 17.5 > 10. Thus, if the hypotheses of Theorem 1.13 fail, there must exist at least 10 vertices all with degrees bounded above by (3n − 2)/2. Let R = {v1 , . . . , v10 } be the ten lowest-degree vertices of G, each of which necessarily has degree at most (3n − 2)/2. Let v1 and v2 be two vertices of R of smallest degree. Considering only degrees of vertices of R, constructing G from Kn,n,n requires deleting at least  4n −

5n 2



 3n − 2 + 8 2n − − 33 2

(1.5)

  3n−2 where 4n − 5n is the is the number of edges deleted from v and v , 8 2n − 1 2 2 2 number of edges deleted from the remaining 8 vertices of R, and at most 33 edges were counted twice in the previous two terms. Simplifying Eq. 1.5, we conclude that failing the hypotheses of Theorem 1.13 results in the deletion of 5.5n − 25 edges, which is more than 4n − 5 provided n ≥ 14. Thus, Theorem 1.13 holds for c = 3n − 1, and it follows that G contains a cycle of length at least 3n − 1.  Acknowledgments The work described in this article is a result of a collaboration made possible by the IMA’s Workshop for Women in Graph Theory and Applications. Research of the fifth author was supported in part by NSF grant DMS-1839918.

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References 1. J. Adamus. Edge condition for Hamiltonicity in balanced tripartite graphs. Opuscula Mathematica 29 337–343 (2009). 2. J. Adamus and L. Adamus. Ore and Erdös type conditions for long cycles in balanced bipartite graphs. Discrete Mathematics and Theoretical Computer Science 11 57–69 (2009). 3. L. Adamus. Edge condition for long cycles in bipartite graphs. Discrete Mathematics and Theoretical Computer Science 11 25–32 (2009). 4. J. S. Bagga and B.N, Varma. Hamiltonian properties in bipartite graphs. Bulletin of the Institute of Combinatorics and its Applications 26 71–85 (1999). 5. J.-C. Bermond. On Hamiltonian walks. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aerdeen, 1975), Congressus Numerantium XV, 41–51 (1976). 6. G. Chartrand, L. Lesniak and P. Zhang. Graphs & Digraphs. 6th.ed., CRC Press, Taylor & Francis Group, Boca Raton, FL (2016). 7. G. Chen and M.S. Jacobson. Degree sum conditions for Hamiltonicity on k-partite graphs. Graphs and Combinatorics 13 325–343 (1997). 8. G. Chen, R. J. Faudree, R. Gould, M.S. Jacobson and L. Lesniak. Hamiltonicity in balanced k-partite graphs. Graphs and Combinatorics 11 221–231 (1995). 9. L. DeBiasio, R. A. Krueger, D. Pritikin and E. Thompson. Hamiltonian cycles in k-partite graphs. Journal of Graph Theory 94 92-112 (2020). 10. G. A. Dirac. Some theorems on abstract graphs. Proceedings of the London Mathematical Society 3 69–81 (1952). 11. D. Ferrero and L. Lesniak. Chorded pancyclicity in k-partite graphs. Graphs and Combinatorics 34 1565–1580 (2018). 12. J. Mitchem and E. Schmeichel, Pancyclic and bipancyclic graphs—a survey. In: Graphs and applications (Boulder, Colorado, 1982) Wiley-Interscience. 271–278 (1985). 13. J. Moon and L. Moser. On Hamiltonian bipartite graphs. Israel Journal of Mathematics 1 163– 165 (1963). 14. O. Ore. Note on Hamilton circuits. American Mathematical Monthly 67 p. 55 (1960). 15. O. Ore. Arc coverings of graphs. Annali di Matematica Pura ed Applicata. Serie Quarta 55 315–321 (1961). 16. L. Pósa. On the circuits of finite graphs. A Magyar Tudomanyos Akademia Matematikai Kutató Intezetenek Közleményei 8 55–361 (1964). 17. L. Pósa. A theorem concerning Hamilton lines. A Magyar Tudomanyos Akademia Matematikai Kutató Intezetenek Közleményei 7 225–226 (1962). 18. P. Turán. On an extremal problem in graph theory. Matematikai és Fizikai Lapok 48 436–452 (1941). 19. D. B. West. Introduction to Graph Theory. 2nd. ed. Prentice Hall, Upper Saddle River, NJ (2001). 20. D. R. Woodall. Sufficient conditions for circuits in graphs. Proceedings of the London Mathematical Society third series 24 739–755 (1972).

Chapter 2

Product Thottling Sarah E. Anderson, Karen L. Collins, Daniela Ferrero Carolyn Mayer, Ann N. Trenk, and Shanise Walker

, Leslie Hogben

,

2.1 Introduction Throttling addresses the question of minimizing the sum or the product of the resources used to accomplish a task and the time needed to complete that task for various graph searching processes. Graph parameters of interest include various types of zero forcing, power domination, and Cops and Robbers. The resources used to accomplish a task can be blue vertices in zero forcing, Phasor Measurement Units (PMUs) in power domination, or cops in Cops and Robbers. The time is the number of rounds needed to complete the process (the propagation time or capture time). We begin by defining the graph parameters for which we will discuss product throttling. Our focus is on connected graphs of order at least two (unless otherwise

S. E. Anderson Department of Mathematics, University of St. Thomas, St. Paul, MN, USA e-mail: [email protected] K. L. Collins Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA e-mail: [email protected] D. Ferrero Department of Mathematics, Texas State University, San Marcos, TX, USA e-mail: [email protected] L. Hogben () Department of Mathematics, Iowa State University, Ames, IA, USA American Institute of Mathematics, San Jose, CA, USA e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_2

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stated). Zero forcing is a coloring game on a graph, where the goal is to color all the vertices blue (starting with each vertex colored blue or white). White vertices are then colored blue by applying a color change rule; the type of zero forcing is determined by the color change rule. Standard zero forcing uses the standard color change rule: If w is the unique white neighbor of a blue vertex v, then change the color of w to blue. Positive semidefinite (PSD) zero forcing uses the PSD color change rule: Let B be the set of (currently) blue vertices and let W1 , . . . , Wk be the sets of vertices of the components of G − B. If v ∈ B, w ∈ Wi , and w is the only white neighbor of v in G[Wi ∪ B], then change the color of w to blue. Note that it is possible that there is only one component of G − B, and in that case the effect of the PSD color change rule is the same as that of the standard color change rule. A nonempty set S ⊆ V (G) defines an initial set of blue vertices (with all vertices not in S colored white); this is called an initial coloring of G. Given an initial coloring S of G, the final coloring of S is the set of blue vertices obtained by applying the color change rule until no more changes are possible (other names for the final coloring include the derived set and the closure of S). A set S is a standard zero forcing set (respectively, PSD zero forcing set) of G if the final coloring of S is V (G) using the standard (respectively, PSD color change rule). The standard zero forcing number (respectively, PSD zero forcing number), denoted by Z(G) (respectively, Z+ (G)) is the minimum cardinality of a standard zero forcing set (respectively, a PSD zero forcing set). Hereafter, we will use the term forcing set to mean standard or PSD zero forcing set. If v is used to change the color of w by a color change rule, we say v forces w and write v → w. For a given forcing set S, we construct the final coloring, recording the forces. Depending on context, the symbol F is used to denote the set of forces that produces the final coloring, or an ordered list of forces (in the order they were performed), called a chronological list of forces. For a given set S, there are often choices as to which vertex forces a particular vertex, so a set of forces, or

C. Mayer Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, USA Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] A. N. Trenk Department of Mathematics, Wellesley College, Wellesley, MA, USA e-mail: [email protected] S. Walker Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI, USA e-mail: [email protected]

2 Product Thottling

13

a chronological list of forces, is usually not unique. However, the final coloring is unique for standard and PSD zero forcing [2, 27]. We can also approach (standard or PSD) zero forcing not as an individual sequence of forces but via rounds, where in each round we perform all possible forces that can be done independently of each other (rounds are also called time steps in the literature). Starting with S ⊆ V (G), we define two sequences of sets, the set S (i) of vertices that turn blue in round i and the set S [i] of vertices that are blue after round i. Thus S [0] = S (0) = S is the initial set of blue vertices. Assume S (i) and S [i] have been constructed and S (i) = ∅. Then S (i+1) = {w : w can be forced by some v (given S [i] blue)} and S [i+1] = S [i] ∪ S (i+1) .

Let p denote the greatest integer such that S (p) = ∅. Since S (i) = ∅ implies S (i+1) = ∅, S is a forcing set of G if and only if S [p] = V (G). When S is a forcing set, this p is called the propagation time of S in G, denoted pt(G; S) or pt+ (G; S) for standard and PSD zero forcing set, respectively; if S is not a forcing set, then pt(G; S) = ∞ or pt+ (G; S) = ∞. For k ∈ Z+ , pt(G, k) = min|S|=k pt(G; S) and pt+ (G, k) = min|S|=k pt+ (G; S). The standard propagation time of G (respectively, PSD propagation time of G) is pt(G) = pt(G, Z(G)) (respectively, pt+ (G) = pt+ (G, Z+ (G))). For each v ∈ V (G), define the round function by rd(v) = k for v ∈ S (k) . A propagating set of forces is one in which rd(u) < rd(v) implies u is forced before v in the associated chronological list of forces, and this is the only kind of forcing set we are concerned with. The round function will also be used for power domination and Cops and Robbers, but the meaning will be clear from the context or a subscript will be added to identify the parameter. The name zero forcing comes from the fact that the process describes forcing zeros in the null vector of a symmetric matrix using only the pattern of off-diagonal nonzero entries of the matrix (a graph describes the nonzero off-diagonal pattern of a symmetric matrix). The zero forcing number was introduced in [2] as an upper bound for the maximum nullity, or equivalently, maximum multiplicity of an eigenvalue, among real symmetric matrices having this graph. Zero forcing was introduced independently in mathematical physics in the study of control of quantum systems [12], and later reintroduced as fast mixed graph searching [33]. Arguably its first appearance was as part of the power domination process, which we describe next. Power domination models the observations that can be made by PMUs and was studied using graphs by Haynes et al. in [23]; Brueni and Heath [11] showed that a simplified version of the propagation rules is equivalent to the original version in [23], and we use their propagation rules. For a nonempty set S of vertices of G, N[S] denotes the closed neighborhood of S. A set S is a dominating set of G if N[S] = V (G), and the minimum cardinality of a dominating set is the domination number of G, denoted by γ (G). Given S ⊆ V (G), define the sequences of sets P (i) (S) and P [i] (S) by the following recursive rules: 1. P [0] (S) = P (0) (S) = S, P [1] (S) = N[S] and P (1) (S) = N [S] \ S.

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2. For i ≥ 1,   P (i+1) (S) = w ∈ V (G) \ P [i] (S) : ∃u ∈ P [i] (S), NG (u) \ P [i] (S) = {w} , P [i+1] (S) = P [i] (S) ∪ P (i+1) (S). Step (1) is called the domination step, because it results in P [1] (S) = N[S]. Step (2) is called the zero forcing step, because P (i+1) (S) = N[S](i) for i ≥ 1. For v ∈ P (i) (S), we say v is observed in round i or rd(v) = i (if necessary to distinguish from zero forcing, we write rdpd (v)). If every vertex is observed in some round, i.e., there is an i such that P [i] (S) = V (G), then S is a power dominating set of G; S is a power dominating set of G if and only if N[S] is a zero forcing set of G. The power domination number of G, denoted by γP (G), is the minimum cardinality of a power dominating set. When S is a power dominating set, the least positive integer p with the property that P [p] (S) = V (G) is the power propagation time of S in G, denoted by ptpd (G; S); if S is not a power dominating set, then ptpd (G; S) = ∞. Observe that ptpd (G; S) = pt(G; N[S]) + 1. For k ∈ Z+ , ptpd (G, k) = min|S|=k ptpd (G; S) and the power propagation time of G is ptpd (G) = ptpd (G, γP (G)). Cops and Robbers is a two-player game played on a graph. One player places and moves a collection of cops and the other places and moves a single robber. The goal for the cops is to capture the robber by having a cop occupy the same vertex the robber occupies. The goal of the robber is to avoid capture. After an initial placement of the cops on a multiset of vertices (meaning more than one cop can occupy a single vertex), followed by the placement of the robber, the game is played in a sequence of rounds during which the players take turns, both playing in a single round: The team of cops takes a turn by allowing each cop to move to an adjacent vertex or stay in place. Similarly, the robber takes a turn by moving to an adjacent vertex or staying in place. The cops win the game if after some finite number of rounds, a cop captures the robber. If the robber has a strategy to evade the cops indefinitely, the robber wins. The cop number c(G) of a graph G is the minimum number of cops required to capture the robber playing on G [1]. The capture time, denoted capt(G), is the number of rounds it takes for c(G) cops to capture the robber on the graph G (assuming all players follow optimal strategies) [8], and for any k ≥ c(G), the k-capture time of G, denoted by captk (G), is the minimum number of rounds it takes for k cops to capture the robber on G (assuming that all players follow optimal strategies) [9]. If k < c(G), then captk (G) = ∞. Throttling originated with a question of Richard Brualdi to Michael Young in a talk about zero forcing and propagation time at the 2011 International Linear Algebra Society Conference in Braunschweig, Germany. This led Butler and Young to initiate the study of sum throttling for (standard) zero forcing in [13]. Sum throttling has been studied for numerous parameters including standard zero forcing, PSD zero forcing, and their minor monotone floors; power domination; Cops and Robbers (see [27, Chapter 10] for a survey). Here we define sum throttling for the four graph games we discuss, i.e., standard zero forcing, PSD zero forcing, power domination, and Cops and Robbers.

2 Product Thottling

15

The standard throttling number of S in G is th(G; S) = |S| + pt(G; S), and the standard k-throttling number is th(G, k) = k + pt(G, k). The standard throttling number of G is th(G) = min th(G; S) = S⊆V (G)

min

Z(G)≤k≤n

th(G, k).

The PSD throttling number of S in G is th+ (G; S) = |S| + pt+ (G; S), and the PSD k-throttling number is th+ (G, k) = k + pt+ (G, k). The PSD throttling number of G is th+ (G) = min th+ (G; S) = S⊆V (G)

min

Z+ (G)≤k≤n

th+ (G, k).

The power domination throttling number of S in G is thpd (G; S) = |S|+ptpd (G; S), and the power domination k-throttling number is thpd (G, k) = k + ptpd (G, k). The power domination throttling number of G is thpd (G) = min thpd (G; S) = S⊆V (G)

min

γP (G)≤k≤n

thpd (G, k).

The cop throttling number of S in G is thc (G; S) = |S| + capt(G; S), and the cop k-throttling number is thc (G, k) = k + captk (G). The cop throttling number of G is thc (G) = min thc (G; S) = S⊆V (G)

min

c(G)≤k≤n

thc (G, k).

Product throttling minimizes a product of the number of vertices and the propagation time. In order to make product throttling interesting, the case of a zero product (where each vertex has a cop/PMU/blue color and the propagation time is zero) must be excluded. This can be done by requiring that the cost of positioning cops/PMUs/blue vertices be considered (e.g., by adding one to the number of rounds before multiplying by the number of vertices used), and we describe this as product throttling with initial cost. Alternatively, a requirement that at least one round be performed must be added or the initial set consisting of all vertices must be excluded. The study of product throttling was initiated in [7] for Cops and Robbers. It was assumed there is a time cost to placing cops and the product throttling number was defined as the number of cops times one more than the propagation time. In contrast, PMUs remain in place but it is natural to assume that the domination step always occurs. In the study of product throttling for power domination in [4], at least one round was required, and the product throttling number was defined as the product of the number of PMUs and the power propagation time. Formal versions of these definitions are given in Sect. 2.2 for Cops and Robbers and Sect. 2.3 for power domination. Requiring at least one round and excluding the zero round case are effectively the same for connected graphs of order at least two, but it is more convenient to

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exclude the zero round case by requiring that the number of vertices used is less than the order; this avoids having two different definitions of propagation time. We refer to this as product throttling with no initial cost. This is the approach taken in Sect. 2.4, where formal versions of the two definitions in universal notation are presented and discussed further, and in subsequent sections. Product throttling for the standard zero forcing number is not interesting for the first definition and not really a throttling question for the second definition (see Sect. 2.5). This fact (or at least the need to exclude the zero solution and that first definition results in product throttling number n for every graph of order n) may have delayed the introduction of product throttling. Using the universal perspective, we examine both definitions of product throttling for Cops and Robbers in Sect. 2.6, for power domination in Sect. 2.7, and for PSD zero forcing in Sect. 2.8. Section 2.9 compares product throttling for Cops and Robbers, power domination, and PSD zero forcing. We need some additional notation. The path, cycle, and complete graph on n vertices are denoted by Pn , Cn , and Kn , respectively, and Kr,n−r denotes a complete bipartite graph. For a graph G, α(G) denotes the independence number of G. For all the graph parameters discussed, the number of rounds is at least the maximum distance of any vertex to the initial set S, and this plays an important role in the analysis of throttling. Let S be a set of vertices of G. For v ∈ V (G), the distance from v to S is dist(S, v) = minx∈S dist(x, v). The eccentricity of S is defined by ecc(S) = maxv∈S dist(S, v). The k-radius of G is radk (G) = minS⊆V (G),|S|=k ecc(S).

2.2 Initial Cost Product Throttling for Cops and Robbers In this section, we follow the convention in [7] and do not assume G is connected or that its order n is at least two unless stated otherwise. The product cop throttling number with initial cost defined in [7] is th× c (G) =

min {k(1 + captk (G))}

c(G)≤k≤n

× × or equivalently, th× c (G) = minS⊆V (G) thc (G; S) where thc (G; S) = |S|(1 + capt(G; S)). This choice of definition for Cops and Robbers reflects the fact that there is a time cost to getting the cops in position. In this section we summarize results from [7].

Observation 2.1 ([7]) There are two immediate upper bounds: (1) th× c (G) ≤ c(G)(1 + capt(G)). (2) th× c (G) ≤ 2γ (G). Remark 2.2 ([7]) Let G be a graph of order n and suppose S ⊆ V (G) with |S| ≥ c(G). Then

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th× c (G; S)=|S|(1+ capt(G; S))=|S|+|S| capt(G; S) ≥ |S|+ capt(G; S)= thc (G; S), so th× c (G) ≥ thc (G). Equality occurs exactly when |S| = 1 or capt(G; S) = 0, so th× (G) = thc (G) if and only if thc (G) = thc (G; S) with |S| = 1 or |S| = n, i.e., c when the cop throttling number can be realized with a single cop or a cop on every vertex. The product cop throttling number can be determined for low values by factoring the proposed value. Proposition 2.3 ([7]) Let G be a graph of order n. (1) th× c (G) = 1 if and only if thc (G) = 1 if and only if G = K1 . (2) th× c (G) = 2 if and only if thc (G) = 2 if and only if either G = 2K1 or γ (G) = 1. (3) th× c (G) = 3 if and only if G satisfies at least one of the following conditions: ˙ K2 . (a) G = 3K1 or G = K1 ∪ (b) γ (G) ≥ 2 and there exists z ∈ V (G) such that (i) for all v ∈ V (G), dist(z, v) ≤ 2, and (ii) for all w ∈ V (G) \ N[z], there is a vertex u ∈ N [z] such that N[w] ⊂ N [u]. (4) th× c (G) = 4 if and only if G satisfies at least one of the following conditions: ˙ K2 . (a) G = 4K1 or G = 2K1 ∪ (b) γ (G) = 2 and capt1 (G) ≥ 3. (c) c(G) = 1 and capt(G) = 3. × It is immediate that th× c (Kn ) = 2 and thc (K1,n−1 ) = 2 for n ≥ 2. A graph G is a chordal graph if it has no induced cycle of length greater than 3. The next result is less elementary than the previous ones.

Theorem 2.4 ([7]) Let H be a chordal graph. Then captk (H ) = radk (H ). Furthermore, th× c (H ) = 1 + rad(H ) = c(H ) + capt(H ).   n−1 , and more generally, th× From Theorem 2.4, th× c (Pn ) = 1 + c (T ) = 2 1 + rad(T ) for any tree T . Theorem 2.4 provides many examples of graphs G with th× c (G) = c(G)(1 + capt(G)), thus achieving equality in the first upper bound in Observation 2.1. It can also be the case that th× c (G) is realized by small capture time and a larger number of cops, e.g., by using γ (G) cops. One example of this is provided by a graph in the family H (n) defined in [8]; it is shown there that c(H (n)) = 1 and capt(H (n)) = n − 4. It was observed in [7] that for H (11) (see Fig. 2.1), capt(H (11)) = 7, but vertices 5 and 7 dominate the graph, so th× c (H (11)) = 2(1 + 1) = 4.

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Fig. 2.1 The graph H (11)

9

5

10

6

2

1

4

8

3 7

11 Fig. 2.2 The graph M(3)

However, the next example provides a family of graphs G for which both × × × th× c (G, c(G)) > thc (G) and thc (G, γ (G)) > thc (G) for sufficiently large order.  Fix a positive integer r, define M (r) to be the graph that is the union of C4 and three disjoint copies of Pr+1 where one of the end points of each of the paths is on a distinct vertex of C4 , and define M(r) = M  (r) ◦ K1 . The graph M(3) is shown in Fig. 2.2; the order of M(r) is 6r + 8. Proposition 2.5 ([7]) For r ≥ 7, th× c (M(r)) < c(M(r))(1 + capt(M(r))) and th× c (M(r)) < 2γ (M(r)).

2.3 Product Throttling for Power Domination with No Initial Cost Let S be a power dominating set. In the papers that studied power propagation time and in this chapter, the power propagation time of S in G is defined to be the least nonnegative integer p such that P [p] (S) = V (G) and is denoted by ptpd (G; S). Thus ptpd (G; V (G)) = 0. When product throttling for power domination was introduced in [4], the perspective was that the domination step always takes place, so the power propagation time of S is at least one even if S = V (G). That is, the definition of power propagation time was modified to require p to be positive. Observe that ptpd (G; S) ≥ 1 for all S = V (G). For power domination (and all

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other parameters discussed), it is immediate that ptpd (G, n − 1) = 1 when G is a connected graph and has order n ≥ 2. Thus the restriction that ptpd (G, k) be positive can be achieved by allowing the value zero for ptpd (G, k), excluding k = |V (G)| from the definition of product power throttling, and requiring that G be connected and have order at least two. For a connected graph G of order n ≥ 2, the product power throttling number with no initial cost1 is th∗pd (G) = min1≤k 1. (ii) Y (G) = 1 and ptY (G, 1) = 2.

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(f) th∗Y (G) = 3 if and only if G satisfies exactly one of the following conditions: (i) Y (G) ≤ 3, ptY (G, 3) = 1, ptY (G, 2) > 1, and ptY (G, 1) > 3. (ii) Y (G) = 1, ptY (G, 1) = 3, and ptY (G, 2) > 1. (g) th× Y (G) = 4 if and only if G satisfies exactly one of the following conditions: (i) n = 4, Y (G) > 1, and ptY (G, 2) > 1. (ii) Y (G) ≤ 2, ptY (G, 2) = 1, and ptY (G, 1) > 3. (iii) Y (G) = 1 and ptY (G, 1) = 3. × For (c), note that th× Y (G) = thY (G, 2) implies ptY (G, 2) = 0 and thus G = K2 , which is covered by Y (G) = 1 and ptY (G) = 1. For (e)(ii) and (g)(ii), note that ptY (G, 1) ≥ 2 implies n ≥ 3, so ptY (G, 2) ≥ 1 and th× Y (G, 2) ≥ 4.

Observation 2.30 Let p be a prime number. (1) If th× Y (G) = p, then Y (G) = 1 and ptY (G) = p − 1, or |V (G)| = p. (2) If th∗Y (G) = p, then Y (G) = 1 and ptY (G) = p, or ptY (G, p) = 1.

2.5 Product Throttling for Standard Zero Forcing For any set S ⊆ V (G), at most |S| forces can be performed in each round, so n−|S| |S| ≤ pt(G; S). This implies for 1 ≤ k ≤ n, n−k ≤ pt(G, k). k

(2.1)

This fundamental bound is the reason that the initial cost version of product throttling for standard zero forcing, defined by th× (G) := minS⊆V (G) |S|(1 + pt(G; S)), is not interesting. Remark 2.31 For any graph G of order n, minS⊆V (G) |S|(1 + pt(G; S)) = n, which is achieved by coloring all vertices blue, because  n−k =n k(1 + pt(G, k)) ≥ k 1 + k by (2.1).

2.5.1 Characterization of th∗ (G) Next we consider the version of product throttling for standard zero forcing that has no initial cost and show that th∗ (G) := minZ(G)≤k t − i, so j < i. By the induction hypothesis, u ∈ (rev(S))[j ] ⊆ (rev(S))[i−1] . Now consider a neighbor v = w of u. Since v does not force w and w → u in round t − j , either v does not force or v → x in round t − j  with j  < j . In either case, v ∈ (rev(S))[i−1] . So if w ∈ (rev(S))[i−1] , then u can force w in the ith round of forcing starting with rev(S). Thus w ∈ (rev(S))[i] .  As in Sect. 2.4.1, define k(G, p) = min{|S| : pt(G; S) = p}. Theorem 2.33 For any graph G, th∗ (G) is the least k such that pt(G, k) = 1, i.e., th∗ (G) = k(G, 1). Necessarily k(G, 1) ≥ n2 . Proof Let k ≥ Z(G), let t = pt(G, k) and let S  V (G) be such that pt(G; S) = t. If pt(G, k) = 1, there is nothing to prove for this k, so suppose t ≥ 2. Define Sˆ = S ∪ rev(S) for some reversal of S. Then for i = 1, . . . , t, S [i] ∪

i 

S (t−j ) ⊆ S [i] ∪ (rev(S))[i] ⊆ Sˆ [i]

j =0

by Lemma 2.32. In particular (since t ≥ 2), V (G) =

t  j =0



S

(j )

⊆ Sˆ

t−1 2



.

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27

  ≤ 2t and (2k) pt(g, 2k) ≤ k pt(G, k). Apply this Thus pt(G, 2k) ≤ t−1 2 repeatedly as needed to show that minZ(G)≤k 1, which is a contradiction.    The characterization of all graphs G of odd order n such that th∗ (G) = n2 is given in terms of the a graph operation we recall next. If G is a graph of order n ≥ 2 and v is a vertex of G, then the graph G − v is defined by V (G − v) = V (G) \ {v} and E(G − v) = E(G) \ {uv : u ∈ NG (v)}. That is, G − v, the graph obtained by removing the vertex v in all edges incident with v in G.   Theorem 2.43 A connected graph G of odd order n satisfies th∗ (G) = n2 if and only if there exists v ∈ V (G) such that G − v is a matched-sum graph.

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Proof Suppose v ∈ V (G) and G − v is a matched-sum graph. Then by Theorem 2.42, there is a set S ⊂ V (G − v) such that |S| = n−1 1. 2 and pt(G − v; S) =    | = th∗ (G; S) ≥ th∗ (G), and th∗ (G) = n Define S  = S ∪ {v}. Then n+1 = |S 2 2 by Theorem 2.33. Let G be a graph of odd order n = 2r + 1 such that th∗ (G) = r + 1. By Observation 2.35, there exists S ⊂ V (G) with |S| = r + 1 such that S is a zero forcing set of G and pt(G; S) = 1. Let F be a propagating set of forces and let v ∈ S be the one vertex that does not perform a force. Then S  = S \ {v} is zero forcing set for G − v and pt(G − v; S  ) = 1. Thus th∗ (G − v) = r = |V (G−v)| , so 2 G − v is a matched-sum graph by Theorem 2.42.  Matched-sum graphs include several interesting graph families, and for these graphs the product zero forcing throttling number is obtained by applying Theorem 2.42. For example, the d-dimensional hypercube Qd = K2  · · ·  K2 (with d copies of K2 ) is a matched-sum graph for any d ≥ 2, so Theorem 2.42 yields th∗ (Qd ) = 2d−1 . The generalized Petersen graph G(r, s) is also a matched-sum graph, so th∗ (G(r, s)) = r.

2.5.3 High Values of th∗ (G) The maximum value of th∗ (G) over connected graphs G of order n ≥ 2 is n − 1 (see Observation 2.26) and this is realized by Kn . We can use Carlson and Kritschgau’s characterization of graphs having th(G) = n in [16] to characterize graphs having th∗ (G) = n − 1. Theorem 2.44 ([16]) For a connected graph G of order n, th(G) = n if and only if G does not have a P4 , C4 , or bowtie graph as an induced subgraph. Corollary 2.45 For a connected graph G of order n ≥ 2, th∗ (G) = n − 1 if and only if G does not have a P4 , C4 , or bowtie graph (see Fig. 2.4) as an induced subgraph. Proof Since th∗ (G) ≤ n − 1, it suffices to establish that th∗ (G) ≤ n − 2 if and only if G has a P4 , C4 , or bowtie graph as an induced subgraph. Let G be one of P4 , C4 , or the bowtie graph and let n be the order of G . Observe that pt(G , n − 2) = 1, so th∗ (G ) ≤ n − 2. Suppose that a connected graph G of order n contains G as an induced subgraph. Then by Remark 2.27, th∗ (G) ≤ n − n + (n − 2) = n − 2. Fig. 2.4 The bowtie graph

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Now suppose G is a connected graph of order n ≥ 2 and th∗ (G) ≤ n − 2. Then th(G) ≤ th∗ (G) + 1 = n − 1 by Proposition 2.28, so G has a P4 , C4 , or bowtie graph as an induced subgraph by Theorem 2.44. 

2.6 Product Throttling for Cops and Robbers Revisited In Sect. 2.2, we summarized known results on the product cop throttling number with initial cost, th× c (G). In this section, we study product cop throttling number with no initial cost. Let G be a connected graph of order at least two. For a set S ⊆ V (G) with c(G) ≤ |S| ≤ γ (G), th∗c (G; S) = |S| capt(G; S). Define th∗c (G, k) = k captk (G). The no initial cost product cop throttling number of G is th∗c (G) =

min

c(G)≤k≤γ (G)

k captk (G) =

min

c(G)≤k≤γ (G)

th∗c (G, k).

While initial cost product throttling seems more realistic if considering actual police officers, no initial cost product throttling is useful in other searching applications. This parameter has been studied as work wk = k captk (G) and speedup between using j > i cops, defined as wi /wj [28–30]. The product cop power throttling number with no initial cost extends this idea by considering the number of cops that yields the largest possible speed-up. We give bounds for th∗c (G) and determine this number exactly for certain families of graphs, including paths, cycles, complete graphs, complete bipartite graphs, full t-ary trees and unit interval graphs. We also establish a few additional results for th× c (G).

2.6.1 General Observations Observation 2.46 Let G be a connected graph of order at least two. There are several immediate upper bounds for th∗c (G): (1) th∗c (G) ≤ γ (G). (2) th∗c (G) ≤ c(G) capt(G). (3) th∗c (G) ≤ th× c (G) − c(G). There are several immediate lower bounds for th∗c (G): (4) th∗c (G) ≥ c(G). (5) th∗c (G) ≥ thc (G) − 1. (6) th∗c (G) ≥ minc(G)≤k≤γ (G) k radk (G). Furthermore, th∗c (G) = c(G) if and only capt(G) = 1.

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Remark 2.47 The following values for Kn and Kr,n−r follow immediately from Observation 2.46. Capture time agrees with power propagation time for paths and cycles, so the values of these graphs follow from Observation 2.9. (1) th∗c (Kn ) = 1 = γ (Kn ). (2) For 2 ≤ r ≤ n − 2, th∗c (Kr,n−r ) = 2 = γ (Kr,n−r ) and th∗c (K1,n−1 ) = 1 = γ (K1,n−1 ).  (3) th∗c (Pn ) = n3 = γ (Pn ).   (4) th∗c (Cn ) = n3 = γ (Cn ) for n ≥ 4. The next result is immediate from Remark 2.29 and Proposition 2.3. Remark 2.48 Let G be a connected graph of order at least two. (1) th∗c (G) = 1 if and only if γ (G) = 1. (2) th∗c (G) = 2 if and only if G satisfies at least one of the following conditions: (a) γ (G) = 2. (b) γ (G) ≥ 3 and there exists z ∈ V (G) such that (i) for all v ∈ V (G), dist(z, v) ≤ 2, and (ii) for all w ∈ V (G) \ N[z], there is a vertex u ∈ N[z] such that N[w] ⊂ N [u]. (3) th∗c (G) = 3 if and only if G satisfies at least one of the following conditions: (a) γ (G) = 3 and th∗c (G) = 2. (b) c(G) = 1, capt(G, 1) = 3, and capt(G, 2) ≥ 2. Note that whether or not capt(G, 2) ≥ 2 can be determined by a polynomial time algorithm (see [21] or [10, Algorithm 2]). Let G1 and G2 be graphs such that G1 ∩ G2 = Km for some m, and G1 , G2 = Km . Then G1 ∪ G2 is the clique sum of G1 and G2 . Next we state bounds for the (sum) cop throttling number for clique sums and establish analogous bounds for the product cop throttling numbers of clique sums. Theorem 2.49 ([10]) Let G be a clique sum of G1 and G2 . Let k1 and k2 be numbers such that thc (Gi ) = thc (Gi , ki ) for i ∈ {1, 2}. Then max{thc (G1 , k1 ), thc (G2 , k2 )} ≤ thc (G) ≤ k1 + k2 + max{captk1 (G1 ), captk2 (G2 )}. Proposition 2.50 Let G be a connected non-trivial clique sum of G1 and G2 . Let k1 , k2 , 1 , 2 be such that for i ∈ {1, 2}, th∗c (Gi ) = ki · captki (Gi ) and th× c (Gi ) = i · (capti (Gi ) + 1). Then max{th∗c (G1 ), th∗c (G2 )} ≤ th∗c (G) ≤ (k1 + k2 ) max{captk1 (G1 ), captk2 (G2 )}, and × × max{th× c (G1 ), thc (G2 )} ≤ thc (G) ≤ (1 + 2 )(max{capt1 (G1 ), capt2 (G2 )} + 1).

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Proof Let k = k1 + k2 . As in [10], note that G1 and G2 are retracts of G, so captk (G) ≤ max{captk1 (G1 ), captk2 (G2 )} (see [9]). Therefore th∗c (G) ≤ (k1 + k2 ) · max{captk1 (G1 ), captk2 (G2 )}. For the lower bound, as in [10], note that for i ∈ {1, 2}, if a robber’s movement within G is restricted to V (Gi ), then for ki cops, there is no benefit to the cops starting outside V (Gi ). The ki cops then catch the robber in time captki Gi . The proof for th×  c (G) is similar.

2.6.2 Chordal Graphs Recall that a graph G is a chordal graph if it has no induced cycle of length greater than 3. The next result is immediate from Theorem 2.4 and Observation 2.46. Remark 2.51 Let H be a connected chordal graph of order n ≥ 2. Then th∗c (H ) =

min

1≤k≤γ (H )

k radk (H ) ≤ min{rad(H ), γ (H )}.

Each of the upper bounds rad(G) and γ (G) is tight, but the inequality in Remark 2.51 cannot be changed to an equality. In Table 2.1, we provide examples of graphs such that th∗c (G) = rad(G) and th∗c (G) = γ (G) as well as an example of a graph where th∗c (G) < min{rad(G), γ (G)}. We can find th∗c (G) exactly for certain families of chordal graphs, including split graphs, full t-ary trees, and unit interval graphs. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. Table 2.1 Examples illustrating relationships among rad(G), γ (G), and th∗c (G) G

rad(G) 3

γ (G) 5

th∗c (G) 3 = rad(G) < γ (G)

4 7

3 9

3 = γ (G) < rad(G) 6 < min{rad(G), γ (G)}

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Fig. 2.5 The full ternary tree of height two T3,2

Fig. 2.6 An interval graph T with th∗c (T ) < γ (T )

Remark 2.52 Let G be a connected split graph of order two or more. If γ (G) = 1, then th∗c (G) = 1 by Remark 2.48. If γ (G) > 1, then th∗c (G) = 2 by Remark 2.51 since rad(G) = 2. A full r-ary tree of height h, denoted by Tr,h , is a rooted tree in which each node has r children unless it is at distance h from the root, and the distance between a vertex and the root is at most h; T3,2 is shown in Fig. 2.5. Proposition 2.53 Let Tr,h be the full r-ary tree of height h with h, r ≥ 2. Then th∗c (Tr,h ) = h. Proof Consider a set S ⊆ V (Tr,h ) of cardinality k ≥ 2 such that capt(Tr,h ; S) = capt(Tr,h , k). If k capt(Tr,h ; S) ≤ h, then each of the leaves must be within distance h h h− hk 2 k of a vertex in S. This requires at least r  ≥ r vertices; it is most efficiently h done by taking every vertex at depth h − k , so h

k capt(Tr,h ; S) ≥ 2 · 2 2 > h.  As shown in [4], the graph T in Fig. 2.6 is an interval graph (and chordal). As with power domination, th∗c (T ) = 2 < 3 = γ (T ). We show that th∗pd (G) = γ (G) for a unit interval graph G; some of the ideas come from the proof that th∗pd (G) = γ (G) in [4], but there are substantial differences. Instead of partitioning the vertices by the round in which they are observed, we partition the vertices of G by their distance from S ⊂ V (G): Define S (k) = {v : dist(S, v) = k}. Then V (G) = ˙ S (1) ∪˙ . . . ∪ ˙ S (ecc(S)) . For a unit interval graph G, fix a unit representation of G S∪ with induced order x is handled by R k (x).  Theorem 2.55 If G is a unit interval graph, then th∗c (G) = γ (G). Proof Let G be a connected unit interval graph of order at least two with a fixed unit representation and induced order. It suffices to show γ (G) ≤ th× c (G, k) for 1 ≤ k < γ (G). Let t = radk (G), and choose S ⊂ V (G) such that |S| = k and ecc(S) = t; note that t ≥ 2. Define T k (S) = ∪x∈S {Lk (x), R k (x)}. We consider two cases, t is even and t is odd. Assume first that t is even. Then Sˆ = T 1 (S) ∪ T 3 (S) ∪ · · · ∪ T t−1 (S) dominates ˆ ≤ |S|2 t = kt = V (G) = S ∪ S (1) ∪ · · · ∪ S (t) by Lemma 2.54. Then γ (G) ≤ |S| 2 × thc (G, k). Now assume that t is odd. Let Sˆ = S ∪ S (2) ∪ S (4) ∪ · · · ∪ S (t−1) . Then Sˆ = S ∪ T 2 (S) ∪ T 4 (S) ∪ · · · ∪ T t−1 (S) dominates V (G) = S ∪ S (1) ∪ · · · ∪ S (t) by Lemma 2.54 and since the vertices in S (1) are dominated by S by definition. Then   t−1 ˆ ≤ |S| 1 + 2 = kt = th×  γ (G) ≤ |S| c (G, k). 2

2.7 Product Throttling for Power Domination Revisited In this section we determine th∗pd (G), the product power throttling number with no initial cost, for additional families of graphs G and explore the definition of product throttling with initial cost, th× pd (G).

2.7.1 Determination of th∗pd (G) for Additional Families of Graphs The values of th∗pd (G) for some families of graphs were established in [4] and several families of graphs for which th∗pd (G) = γ (G) were presented (see Sect. 2.3). In this section, we establish th∗pd (G) for some additional families of graphs. We also construct infinite families of graphs where th∗pd (G) = γ (G). For these families, th∗pd (G) < min{γ (G), γP (G) ptpd (G)}.

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We need the following definitions. For j ≥ 2 and d ≥ 4, construct the j, dgeneralized necklace Nj,d by connecting j copies of Kd − e arranged cyclically to create a d − 1 regular graph; N3,5 is shown in Fig. 2.7. The results in the next remark follow from results stated in Sect. 2.3. Remark 2.56 Let n ≥ 2. (1) th∗pd (Qd ) = γ (Qd ) = 22

 −−1

for d = 2 − 1 using Corollary 2.8 because

the order of Qd is 2d , (Qd ) = d, and for d = 2 − 1, γ (Qd ) = 22 −−1 by Harary and Livingston   [22].   (2) th∗pd (Pr ◦ K1 ) = 2 3r by Theorem 2.18 and γ (Pn ) = n3 .     (3) th∗pd (Cr ◦ K1 ) = 2 3r by Theorem 2.18 and γ (Cn ) = n3 . (4) th∗pd (Nj,d ) = j by Corollary 2.8 because the order of Nj,d is j d, (Nj,d ) =   jd = j. d − 1, and γ (Nj,d ) = (d−1)+1 

We now construct a family G(n, s, m) of 2-connected graphs for which th∗pd (G(n, s, m)) is less than both γ (G(n, s, m)) and γP (G(n, s, m)) ptpd (G(n, s, m)). These examples lead to a family of r-connected graphs with the same properties, for any integer r ≥ 2. Let Kn be the complete graph on vertices {u1 , u2 , . . . , un }. Replace each edge ui uj of Kn with s ≥ 1 disjoint paths of length m ≥ 1 between ui and uj , for 1 ≤ i < j ≤ n. Call the resulting graph G(n, s, m); Fig. 2.8 shows G(3, 3, 4). Proposition 2.57 Let n ≥ 2, s ≥ 3, and m ≥ 4. Then Fig. 2.7 The 3,5-generalized necklace N3,5

Fig. 2.8 G(3, 3, 4) is formed by replacing each edge of K3 by 3 paths of length 4

2

1

3

2 Product Thottling

37

m−3 (1) γ (G(n, s, m)) ≥ s n(n−1) 2  3 . (2) γP (G(n, s, m)) = n − 1, ptpd (G(n, s, m)) = m, and γP (G(n, s, m)) · ptpd (G(n, s, m)) = (n − 1)m.   (3) th∗pd (G(n, s, m)) ≤ n m−1 . 2

For n, m ≥ 5, γ (G(n, s, m)) th∗pd (G(n, s, m)).

≥

γP (G(n, s, m)) · ptpd (G(n, s, m))

>

Proof (1) Since the domination number of a path is the ceiling of its number of vertices divided by 3 and m + 1 − 4 vertices are not accessible from the original vertices of the Kn , any dominating set of G(n, s, m) will have to contain at least  m−3 3  vertices from each of the s paths between ui and uj for 1 ≤ i < j ≤ n. (2) Note that {u1 , u2 , . . . , un−1 } is a power dominating set of G(n, s, m), because each path between vertices ui and uj will have a degree 2 neighbor of a blue endpoint turn blue in the first round. To show that γP (G(n, s, m)) ≥ n − 1, if S is a minimum power dominating set that contains neither ui nor uj , then S must contain at least s − 1 ≥ 2 vertices on the paths between them, and replacing these vertices by ui results in a power dominating set S  with |S  | < |S|, contradicting the minimality of S. Thus the only minimum power dominating sets are {u1 , u2 , . . . , un } \ {ui } for i = 1, . . . , n. To evaluate ptpd (G(n, s, m); {u1 , . . . , un−1 }), note that the last vertex to be observed will be un in round m.   m−1 , so th∗pd (G(n, s, m); (3) Note that ptpd (G(n, s, m); {u1 , . . . , un }) = 2   . {u1 , . . . , un }) = n m−1 2 Let n, m ≥ 5. Then nm ≥ 2m + 3n, so n(n − 1)(m − 3) ≥ 2(n − 1)m. m−3 Since s ≥ 3, s n(n−1) ≥ (n − 1)m. Also nm > n + 2m implies (n − 1)m > 2   3  n(m+1) m−1 . = n 2 + 1 > n m−1 2 2  We now construct a family of (r + 2)-connected graphs for any positive integer r ≥ 1 for which th∗pd (G) < min{γ (G), γP (G) ptpd (G)}. Theorem 2.58 Let r ≥ 1 and let s ≥ 2r+1 + 1. Define Hr = G(3, s, 4)Qr . Then th∗pd (Hr ) < γ (Hr ), th∗pd (Hr ) < γP (Hr ) ptpd (Hr ), and Hr is (r + 2)-connected. Proof Clearly Hr contains 2r copies of G(3, s, 4). By Proposition 2.57(1), γ (G(3, s, 4)) ≥ 3s ≥ 3(2r+1 + 1). Let S be a power dominating set of Hr . Assume first that |S| ≤ 2r+1 − 1. Then there is a copy of G(3, s, 4) in which at most one of the vertices from the original K3 are chosen. Hence there are at least two vertices in the original K3 of that copy, say u1 and u2 , that are not in S. For at least s −1 of the paths between u1 and u2 , there must be a vertex in S that is adjacent to a vertex on the path. No vertex in a different copy of G(3, s, 4) is adjacent to two of these paths, so there are at least s − 1 = 2r+1 vertices needed, contradicting our

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assumption that |S| ≤ 2r+1 − 1. Thus any power dominating set of Hr must contain at least 2r+1 vertices. Now suppose S is a set that has the same two vertices in each copy of G(3, s, 4), so |S| = 2 · 2r . Since power domination will occur simultaneously in each copy, S is a power dominating set of Hr and ptpd (Hr ; S) = ptpd (G(3, s, 4); S) = ptpd (G(3, s, 4)). Thus γP (Hr ) = 2r+1 . Since any minimum power dominating set must have this form, ptpd (Hr ) = ptpd (G(3, s, 4)). By Proposition 2.57(2), ptpd (G(3, s, 4)) = 4 and therefore γP (Hr ) ptpd (Hr ) = 2r+1 · 4 = 2r+3 . Finally, suppose S is the set of three original vertices in each copy of G(3, s, 4). Then |S| = 3 · 2r . As in the proof of Proposition 2.57(3), the number of rounds needed to complete the power domination simultaneously in each copy is 2. Thus, th∗pd (Hr ) ≤ 3 · 2r · 2 = 3 · 2r+1 < 3(2r+1 + 1) < 4 · 2r+1 = 2r+3 . If G is s-connected and H is r-connected, then GH is (s + r)-connected [31].  Thus Qr is r-connected and G(3, s, 4)  Qr is (2 + r)-connected.

2.7.2 The Initial Cost Definition th× pd (G) In this section we summarize some basic results about th× pd (G), and we prove that × thpd (G) < |V (G)| when G is a connected graph of order at least three. In contrast, 6 we present examples where th× pd (G) = 7 |V (G)|. The graphs G = H ◦K1 have high

3 domination number, but we show that th× pd (H ◦ K1 ) ≤ 4 |V (G)| if H is connected and nontrivial. We compare the results found in [4] about th∗pd (Pn Pm ) to new upper bounds on th× pd (Pn Pm ) that show that the best ways to power dominate are different in each case. For a graph G of order n, recall that th× pd (G, k) = k(1 + ptpd (G, k)) and

th× pd (G) =

min

γP (G)≤k≤n

k(1 + ptpd (G, k)) =

min

γP (G)≤k≤n

th× pd (G, k).

The next result follows from Observations 2.22 and 2.23. Observation 2.59 For every graph G of order n: (1) (2) (3) (4)

th× pd (G) ≤ γP (G)(1 + ptpd (G)). th× pd (G) ≤ 2γ (G). × γP (G) ≤ th× pd (G) ≤ n and γP (G) + 1 ≤ thpd (G) if G is connected and n ≥ 2. If γP (G) ≥ n2 , then th× pd (G) = n.

The next result follows from Remark 2.29 in the case G is connected, and the analysis of the disconnected case is straightforward.

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39

Remark 2.60 (1) th× pd (G) = 1 if and only if G = K1 . (2) th× pd (G) = 2 if and only if γ (G) = 1 or G = 2K1 . ˙ (3) th× pd (G) = 3 if and only if ptpd (G, 1) = 2 or G = 3K1 or G = K2 ∪ K1 . A description of a construction for a connected graph G with ptpd (G, 1) = 2, which is equivalent to γP (G) = 1 and ptpd (G) = 2, appears in [4]. In particular, th× pd (C4 ) = 3. Remark 2.61 Power domination on trees behaves like Cops and Robbers, and power domination on Cn behaves like power domination on Pn . The remaining parts of the next result follow from Observation 2.59 and Remark 2.60. (1) If T is a tree, then th× rad T . pd (T ) = 1 +   × n−1 (2) thpd (Pn ) = 1 + rad(Pn ) = 1 + 2 .   n−1 (3) th× . pd (Cn ) = 1 + rad(Pn ) = 1 + 2 (4) (5) (6) (7)

th× pd (Kn ) = 2. th× pd (K1,n−1 ) = 2. For n ≥ 4, th× pd (K2,n−2 ) = 3. For p, q ≥ 3, th× pd (Kp,q ) = 4.

In Theorem 2.63 we show that th× pd (G) < |V (G)| for a connected graph of order at least three, using the next result. Theorem 2.62 ([24, Theorem 2.2]) A connected graph G of order n ≥ 2 has γ (G) = n2 if and only if G = H ◦ K1 for some connected graph H or G = C4 . Theorem 2.63 Let G be a connected graph of order n ≥ 3. Then th× pd (G) < n. Furthermore, if G = H ◦ K1 for a connected graph H of order at least two, then × 3n th× pd (G) = 3γ (H ) and thpd (G) = 4 . Proof By Theorem 2.17, γ (G) ≤ n2 . If γ (G) < n2 , then by Observation 2.59(2), n th× pd (G) ≤ 2γ (G) < n. So assume γ (G) = 2 . By Theorem 2.62, either G = C4 or G consists of a connected graph H with a leaf attached to each vertex. If G = C4 , then th× pd (C4 ) = 3 < n by Remark 2.60(3). So suppose G = H ◦ K1 for a connected graph H . Since n ≥ 3, H has at least )| 3n 2 vertices, so by Theorem 2.17, γ (H ) ≤ |V (H 2 , and this proves that 3γ (H ) ≤ 4 . Let S be a dominating set of H with |S| = γ (H ). Then S is a power dominating set 3n of G with propagation time two. Thus th× pd (G) ≤ 3γ (H ) ≤ 4 < n. To show that th× pd (G) = 3γ (H ) for G = H ◦ K1 , we choose S ⊂ V (G) such × × that thpd (G; S) = th× pd (G) and show that thpd (G; S) ≥ 3γ (H ). Note that S is not a dominating set of G since γ (G) = n2 and th× pd (G) < n. Without loss of generality, we may assume that S ⊆ V (H ) since we can always replace a leaf of G by its neighbor in H . The set S must be a dominating set of H in order to be a

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1

2

Fig. 2.9 A graph G with |V (G)| = 7r and th× pd (G) = gray vertices may be included

6|V (G)| . 7

Any subset of edges between the

power dominating set of G, so |S| ≥ γ (H ) and the propagation time for S is two. Therefore th×  pd (G; S) = 3|S| ≥ 3γ (H ). The next result is immediate from Theorem 2.63. m Corollary 2.64 For m ≥ 2, th× pd (Pm ◦ K1 ) = 3 3 . Next we present a family of connected graphs that satisfy th× pd (G) =

6|V (G)| . 7

Example 2.65 Let G1 consist of a P4 and a P3 with an edge connecting them between a vertex of degree 2 on each, as shown in Fig. 2.9. The vertices of degree 3 are colored gray in the figure. Then γ (G1 ) = 3, hence th× pd (G1 ) ≤ 3 · 2 = 6. However, any power dominating set of G1 that is not a dominating set must contain a vertex from each path and have propagation time at least 2. Thus th× pd (G1 ) ≥

6|V (G1 )| 2 · 3 = 6. Since |V (G1 )| = 7, th× . pd (G1 ) = 7 Now let G consist of r disjoint copies of G1 , say G1 , G2 , . . . , Gr , with any subset of edges between the gray vertices in each copy of G1 . Any dominating set of G must contain at least 3 vertices from each Gi , 1 ≤ i ≤ r, in order to dominate the leaves. Any power dominating set of G must contain at least two vertices from 6|V (G)| each Gi , 1 ≤ i ≤ r. Hence th× . pd (G) = 7

Let G be j copies of P3 and i copies of P4 (all disjoint, so the order of G is th× (G)

3j + 4i). We show 3jpd+4i ≥ 67 if and only if j = i. Note that γ (G) = j + 2i, γP (G) = j + i, and ptpd (G) = 2. Then th× pd (G) = min{2(j + 2i), 3(j + i)} because ptpd (G, k) = 2 for γP (G) ≤ k < γ (G). Thus i = j implies th× pd (G) = th× pd (G)

6 7.

By

(G) 2(j +2i) 2j +4i choosing a dominating set, 3j +4i ≤ 2γ 3j +4i = 3j +4k = 3j +4i . If i < j , then 2j +4i 14j + 28i < 18j + 24i, which implies 3j +4i < 67 . By choosing a minimum power th× (G) +i) 3j +3i dominating set, 3jpd+4i ≤ 3(j 3j +4i = 3j +4i . If j < i, then 21j + 21i < 18j + 24i, th× +3i pd (G) 6 6 < . Hence i =  j implies which implies 3j 3j +4i 7 3j +4i < 7 . If G is connected and not K1 or K2 , then Theorem 2.63 shows that the ratio th× pd (G) 6 |V (G)| is less than 1, and Example 2.65 shows that it can be as large as 7 .

2 Product Thottling

Question 2.66 Is 67 the largest possible value of connected graphs of arbitrarily large order?

41 th× pd (G) |V (G)|

that is achieved for

Next we examine grid graphs, which are natural to consider in PMU placement problems. It is interesting to compare the value of th× pd (Pm ◦ K1 ) in Corollary 2.64 × with the value of thpd (Pm P2 ) in Theorem 2.69. n m Proposition 2.67 For n, m ≥ 2, th× pd (Pn Pm ) ≤ min{ 3 (n + 1), 3 (m + 1)}. Proof Arrange Pn Pm with  mn rows and m columns. By symmetry, we need only show that th× (P P ) ≤ m pd n 3 (n + 1). Let S be a minimum dominating set of the top row of Pn Pm , so |S| = γ (Pm ). After the first round, each vertex in the top row is observed and has at most one unobserved neighbor in the second row. Thus zero forcing can proceed row by row, so ptpd (Pn Pm ; S) ≤ n and th× pd (Pn Pm ) ≤ m  3 (1 + n). In [4], it was shown that th∗pd (Pn Pm ) = γ (Pn Pm ) for all m, n (see Theorem 2.15). As for all graphs, th× 2γ (Pn Pm ). For example, pd (Pn Pm) ≤    m+2 3m+4 and γ (P3 Pm ) = (see [3]), and therefore γ (P2 Pm ) = 2 4   m+2 th× ≤ m + 2 and th× pd (P2 Pm ) ≤ 2γ (P2 Pm ) = 2 pd (P3 Pm ) ≤ 2   . However, these bounds are not tight, as shown in the 2γ (P3 Pm ) = 2 3m+4 4 next remark.   Remark 2.68 In [20], it is shown that γ (Pn Pm ) = (m+2)(n+2) − 4 for m, n ≥ 5    m (m+2)(n+2) − 4 , th× 16. Since 3 (n + 1) < 2 pd (Pn Pm ) < 2γ (Pn Pm ) for 5 m, n ≥ 16. To establish the exact value of th× pd (P2 Pm ) in Theorem 2.69, we need some definitions. We use the notation u → v or u forces v to mean u observes v (this may involve a choice among several vertices that can observe v). For a power dominating set S ⊆ V (G), create a propagating power domination set of forces F of S as follows: Initially, F = ∅. For each w ∈ N[S] \ S, choose x ∈ S ∩ N(w) and add x → w to F . Then choose a propagating set of forces for N[S] (using the standard color change rule for zero forcing) and add that to F . Suppose S is a power dominating set of G and F is a propagating power domination set of forces of S. For a vertex x ∈ S, define Sx to be the set of all vertices w such that there is a sequence of forces x = v0 → v1 → · · · → vk = w in F ; the empty sequence of forces is permitted, i.e., x ∈ Sx . Theorem 2.69 For m ≥ 2, th× pd (P2 Pm ) = m if m ≡ 0 mod 3 and × thpd (P2 Pm ) = m + 1 if m ≡ 0 mod 3.

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  2 Proof Let G = P2 Pm . By Proposition 2.67, th× (G) ≤ pd 3 (m + 1) = m + 1.   × m m If m is divisible by 3, then thpd (G) ≤ 3 3 = 3 · 3 = m. This proves the upper bound. Next we prove the lower bound.   by [3], so th× For any k such that ptpd (G, k) = 1, k ≥ γ (G) = m+2 pd (G, k) ≥ 2   m+2 ≥ m + 1. Thus we need consider only sets S such that ptpd (G; S) ≥ 2. 2 2

Arrange G with 2 rows and m columns. Choose S such that th× pd (G) = × thpd (G; S) = |S|(1 + p) where p = ptpd (G; S). Create a propagating power domination set of forces for S by choosing the forcing vertex in the same row whenever there is a choice (row-forcing is preferred). For x ∈ S, we show that |Sx | ≤ 2(1 + p) and |Sx | < 2(1 + p) except under the additional conditions that |P (1) (S) ∩ Sx | = 3, |P (2) (S) ∩ Sx | = 2, and p = 2. By the rules of power domination, |P (i) (S)∩Sx | ≤ |P (1) (S)∩Sx | for i ≥ 2. Thus (1) |P (S)∩Sx | ≤ 2 implies |Sx | ≤ 1+2p < 2(1+p). So assume |P (1) (S)∩Sx | = 3 and x is in the top row. Denote the east, south, and west neighbors of x by xE , xS and xW , and name additional vertices similarly, according to their direction from x. Since row-forcing is preferred, xSE , xSW ∈ S. Let r ≥ 2 be the first round in which any of xW , xS , xE performs a force. We analyze the situation based on which force(s) occur in round r. To obtain a contradiction, suppose that xW → xW W in round r, which requires xSW ∈ P [r−1] (S) and xSW is not forced by xS . If xSW ∈ P [1] (S), then xW SW ∈ S and so xW W ∈ P (1) (S), which is a contradiction. Otherwise, xSW ∈ P (i) (S) with 1 < i < r and xW SW → xSW in round i requires that xW W must already be observed before round i, which is a contradiction. Therefore, xW → xW W cannot happen in round r. Similarly, xE → xEE cannot happen in round r. Now suppose that xS → xSW in round r. This requires xSE ∈ P [r−1] (S), which in turn requires xESE , xEE ∈ P [r−1] (S). So xE can never force. Then |P (r) (S) ∩ Sx | = 1 since xW cannot force in round r, |P (i) (S) ∩ Sx | ≤ 2 for i ≥ r + 1, and |P (i) (S) ∩ Sx | = 0 for 1 < i < r. Thus |Sx | ≤

p 

|P (i) (S) ∩ Sx | ≤ 1 + 3 + 0 + · · · + 0 + 1 + 2(p − r) < 2(1 + p).

i=0

The case xS → xSE in round r is similar. It remains to consider the case when xS does not force in round r, and in this case, |P (r) (S) ∩ Sx | ≤ 2. By definition of r, one of xW , xE must force in round r, and we have shown that it cannot force along a row. Without loss of generality, let xW → xSW in round r. Necessarily, xW W ∈ P [r−1] (S) or xW and xSW are the leftmost vertices in G. We show that xW SW ∈ P [r] (S) if xW W ∈ P [r−1] (S), and so in either case xSW never performs a force. If xW W ∈ S, then xW SW ∈ P [1] (S) ⊂ P [r] (S). If xW W W → xW W in round r − 1, then xW W → xW SW in round r. Note that xW SW cannot force xW W in round r − 1 because its neighbor xSW is also unobserved until round r. If xE → xSE in round r, a similar proof shows that xSE can never force. Thus |P (i) (S) ∩ Sx | ≤ 2 for i ≥ r + 1, and

2 Product Thottling

|Sx | ≤

p 

43

|P (i) (S) ∩ Sx | ≤ 1 + 3 + 0 + · · · + 0 + 2 + 2(p − r) ≤ 2(1 + p).

i=0

If r ≥ 3, then |P (2) (S) ∩ Sx | = 0, and |Sx | < 2(1 + p). If r = 2 and xW → xSW or xE → xSE , then xW or xE cannot force in round 2, and |Sx | ≤ 1+3+1+2(p−2) < 2(1 + p). If r = 2 and xW → xSW and xE → xSE in round 2, then as described above, none of xSW , xSE , or xS can ever force, so |P (i) (S) ∩ Sx | = 0 for i = 0, 1, 2 and |Sx | < 2(1 + p) if p > 2. Thus we have shown that |Sx | ≤ 2(1 + p) in all cases, and |Sx | < 2(1 + p) unless |P (1) (S) ∩ Sx | =

3, |P (2) (S) ∩ Sx | = 2, and p = 2. Since |Sx | ≤ 2(1 + p) in all cases, 2m = x∈S |Sx | ≤ |S|2(1 + p) = 2 th× pd (G) × × (1) and thpd (G) ≥ m for all G. Furthermore thpd (G) > m unless |P (S) ∩ Sx | = 3, |P (2) (S) ∩ Sx | = 2, and p =

2 for all x ∈ S. In this case |Sx | = 6 = 2(1 + p) for all x ∈ S and hence 2m = x∈S |Sx | = 6|S| and m = 3|S|, and m is divisible by 3. 

2.8 Product Throttling for PSD Zero Forcing The next result implies that product throttling for Z+ is nontrivial for th× + (G) and th∗+ (G), based on results from Cops and Robbers. As in Sect. 2.4, define k+ (G, p) = min{|S| : pt+ (G; S) = p}. Theorem 2.70 ([10]) Let S ⊆ V (G) be a PSD zero forcing set. Then |S| ≥ c(G), capt(G; S) ≤ pt+ (G; S), and c(G) ≤ Z+ (G). If T is a tree and S ⊆ V (T ), then capt(T ; S) = pt+ (T ; S) = ecc(S) for S ⊆ V (T ). Thus captk (T ) = pt+ (T , k) = radk (T ). Note that c(G) and Z+ (G) can be substantially different, resulting in very different product throttling numbers. For example, Z+ (Kn ) = n − 1 whereas c(Kn ) = 1.

2.8.1 Initial Cost Definition th× + (G) Let G be a graph of order n. Define th× + (G, k) = k(1 + pt+ (G, k)) and th× + (G) =

min

Z+ (G)≤k≤n

th× + (G, k) =

min

Z+ (G)≤k≤n

k(1 + pt+ (G, k)).

The results in the next remark follow immediately from the universal forms of these results in Sect. 2.4.1. Remark 2.71 Let G be a graph of order n.

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(1) (2) (3) (4) (5)

× Z+ (G) ≤ th× + (G) ≤ n. If G is connected and n ≥ 2, then Z+ (G)+1 ≤ th+ (G). × th+ (G) ≤ Z+ (G)(1 + pt+ (G)). th× + (G) ≤ 2k+ (G, 1). th× + (G) ≥ minZ+ (G)≤k≤n k(1 + radk (G)). × If Z+ (G) ≥ n2 , then th× + (G) = n. Examples include th+ (Kn ) = n and × th+ (Qd ) = 2d .

The next result follows from Theorems 2.4 and 2.70 since a tree is chordal. × Corollary 2.72 Let T be a tree. Then th× th× + (T , k) =  c (T , k) and th+ (T ) = × n−1 . th× c (T ) = 1 + rad T . In particular, th+ (Pn ) = 1 + 2 × For any graph G, th× c (G, k) ≤ th+ (G, k) for c(G) ≤ k ≤ n.

The next result follows from Remark 2.29 in the case G is connected, and the analysis of the disconnected case is straightforward (for n = 4 see [25]). Remark 2.73 Let G be a graph. (1) th× + (G) = 1 if and only if G = K1 . (2) th× + (G) = 2 if and only if Z+ (G) = 1 and pt+ (G) = 1 ( i.e., G = K1,n−1 ) or G = 2K1 . (3) th× + (G) = 3 if and only if G satisfies exactly one of the following conditions: ˙ K1 , or G = 3K1 . (a) G = K3 , G = K2 ∪ (b) Z+ (G) = 1 and pt+ (G, 1) = 2 (i.e., G is a tree and rad G = 2). (4) th× + (G) = 4 if and only if G satisfies at least one of the following conditions: ˙ K1 , G = K2 ∪˙ 2K1 , or G = 4K1 . (a) G = K4 , G = K3 ∪ (b) pt+ (G, 2) = 1 and pt+ (G, 1) > 3. (c) Z+ (G) = 1 and pt+ (G, 1) = 3.

2.8.2 No Initial Cost Definition th∗+ (G) Let G be a connected graph of order n ≥ 2. Define th∗+ (G, k) = k pt+ (G, k) and th∗+ (G) =

min

Z+ (G)≤k 2 for r ≥ 3. Example 2.86 The generalized wheel GW (k, r) with k ≥ 4 and r ≥ 1 is the graph of order kr + 1 constructed from Ck  Pr by adding a new vertex adjacent to every vertex in one end copy of Ck ; Fig. 2.12 shows GW (6, 2). Let k ≥ 5, so γ (GW (k, 2)) ≥ 3. It is immediate that th× pd (GW (k, 2)) = 3 and th∗pd (GW (k, 2)) = 2, whereas c(GW (k, 2)) = 2 and capt(GW (k, 2)) ≥ 2, so × ∗ ∗ th× + (GW (k, 2)) ≥ thc (GW (k, 2)) ≥ 6 and th+ (GW (k, 2)) ≥ thc (GW (k, 2)) ≥ 3. From Theorems 2.33, 2.17, and Observation 2.6, it follows that for any graph G, th∗ (G) ≥

n ≥ γ (G) ≥ th∗pd (G). 2

Moreover, there exist graphs G such that th∗ (G) = n2 = γ (G) = th∗pd (G). For example, a straightforward verification shows th∗ (C4 ) = 2, γ (G) = 2 and th∗pd (G) = 2. We now combine results from previous sections to obtain the following characterization of graphs G such that th∗ (G) = n2 = γ (G) = th∗pd (G). Proposition 2.87 Let G be a connected graph of order n such that th∗pd (G) = Then th∗ (G) = n2 = γ (G).

n 2.

Proof By Theorem 2.19, th∗pd (G) = n2 if and only if G = (H ◦ K1 ) ◦ K1 for some connected graph H , G = C4 ◦ K1 , or G = C4 . To conclude the proof we show that G is a matched-sum graph, and apply Theorem 2.42 to conclude th∗ (G) = n2 . If G = C4 , then G = H1 M + H2 where H1 = K2 , H2 = K2 and M is any matching between V (H1 ) and V (H2 ). Now suppose G = H1 ◦ K1 for some connected graph H1 . Let V (H2 ) = V (G) \ V (H1 ). For each v ∈ V (H2 ), v has a unique neighbor uv in G; note that uv ∈ V (H1 ). Let M = {vuv : v ∈ V (H2 )}. Then G is a matchedsum graph, specifically, G = H1 M + H2 .  Corollary 2.88 A connected graph G satisfies th∗ (G) = n2 = γ (G) = th∗pd (G) if and only if G = (H ◦ K1 ) ◦ K1 for some connected graph H , G = C4 ◦ K1 , or G = C4 .

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Acknowledgments The work of A. Trenk was partially supported by a grant from the Simons Foundation (#426725). The work of C. Mayer was partially supported by Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

References 1. M. Aigner and M. Fromme. A game of Cops and Robbers. Discrete Applied Mathematics 8 1–11 (1984). 2. AIM Minimum Rank – Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S.M. Cioaba, D. Cvetkovi´c, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi´c, H. van der Holst, K. Vander Meulen and A. Wangsness Wehe). Zero forcing sets and the minimum rank of graphs. Linear Algebra and its Applications 428 1628–1648 (2008). 3. S. Alanko, S. Crevals, A. Isopoussu, P. Östergård and V. Pettersson. Computing the Domination Number of Grid Graphs. Electronic Journal of Combinatorics 8 Paper 141 (2011). 4. S.E. Anderson, K.L. Collins, D. Ferrero, L. Hogben, C. Mayer, A.N. Trenk and S. Walker. Product throttling for power domination. Under review. arXiv:2010.16315 (2020). 5. B. Bagheri. (G1 , G2 )-permutation graphs. Discrete Mathematics, Algorithms and Applications 7 (2015). 6. C. Balbuena, P. García-Vázquez and X.Marcote. Connectivity measures in matched sum graphs. Discrete Applied Mathematics 308 1985–1993 (2008). 7. A. Bonato, J. Breen, B. Brimkov, J. Carlson, S. English, J. Geneson, L. Hogben, K.E. Perry and C. Reinhart. Optimizing the trade-off between number of cops and capture time in Cops and Robbers. To appear in Journal of Combinatorics. arXiv:903.10087 (2019). 8. A. Bonato, P. Golovach, G. Hahn and J. Kratochvíl. The capture time of a graph. Discrete Mathematics 309 5588–5595 (2009). 9. A. Bonato, X. Pérez-Giménez, P. Prałat and B. Reiniger. The game of overprescribed Cops and Robbers played on graphs. Graphs and Combinatorics 57 801–815 (2017). 10. J. Breen, B. Brimkov, J. Carlson, L. Hogben, K.E. Perry and C. Reinhart. Throttling for the game of Cops and Robbers on graphs. Discrete Mathematics 341 2418–2430 (2018). 11. D.J. Brueni and L.S. Heath. The PMU placement problem. SIAM Journal on Discrete Mathematics 19 744–761 (2005). 12. D. Burgarth and V. Giovannetti. Full control by locally induced relaxation. Physical Review Letters 99 100501 (2007). 13. S. Butler and M. Young. Throttling zero forcing propagation speed on graphs. Australasian Journal of Combinatorics 57 65–71 (2013). 14. J. Carlson. Throttling for Zero Forcing and Variants. Australasian Journal of Combinatorics 75 96–112 (2019). 15. J. Carlson, L. Hogben, J. Kritschgau, K. Lorenzen, M.S. Ross, S. Selken and V. Valle Martinez. Throttling positive semidefinite zero forcing propagation time on graphs. Discrete Applied Mathematics 254 33–46 (2019). 16. J. Carlson and J. Kritschgau. Various characterizations of throttling numbers. arXiv:1909.07952 (2019). 17. E. Conrad and L. Hogben. Product PSD throttling. In preparation.

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18. D. Ferrero, L. Hogben, F.H.J. Kenter and M. Young. Note on power propagation time and lower bounds for the power domination number. Journal of Combinatorial Optimization 34 736–741 (2017). 19. J.P. Georges and D.W. Mauro. On generalized Petersen graphs labeled with a condition at distance two. Discrete Mathematics 259 311–318, (2002). 20. D. Goncalves, A. Pinlou, M. Rao and S. Thomasse. The domination number of grids. SIAM Journal on Discrete Mathematics 25 1443–1453 (2011). 21. G. Hahn and G. MacGillivray. A note on k-cop, l-robber games on graphs. Discrete Mathematics 306 2492–2497 (2006). 22. F. Harary and M. Livingston. Independent domination in hypercubes. Applied Mathematics Letters 6 27–28 (1993). 23. T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi and M.A. Henning. Domination in graphs applied to electric power networks. SIAM Journal on Discrete Mathematics 15 519–529 (2002). 24. T.W. Haynes, S.T. Hedetniemi and P.J. Slater. Fundamentals of domination in graphs. Marcel Dekker, New York, 1998. 25. L. Hogben. Sage code for product throttling with examples and computations. https:// sage.math.iastate.edu/home/pub/136/. PDF available at https://aimath.org/~hogben/Product_ Throttling--Sage.pdf. 26. L. Hogben, M. Huynh, N. Kingsley, S. Meyer, S. Walker and M. Young. Propagation time for zero forcing on a graph. Discrete Applied Mathematics 160 1994–2005 (2012). 27. L. Hogben, J.C.-H. Lin and B. Shader. Inverse Problems and Zero Forcing for Graphs. To be published by the American Mathematical Society in the Mathematical Surveys and Monographs series in 2022. 28. F. Luccio and L. Pagli. More agents may decrease global work: A case in butterfly decontamination. Theoretical Computer Science 655 41–57 (2016). 29. F. Luccio and L. Pagli. Cops and robber on grids and tori. arXiv:1708.08255 (2017). 30. F. Luccio and L. Pagli. Capture on grids and tori with different numbers of cops. In: Malyshkin V. (eds) Parallel Computing Technologies. PaCT 2019. Lecture Notes in Computer Science, vol 11657, Springer, Cham, pp. 431–444 (2019). 31. G. Sabidussi. Graphs with given group and given graph-theoretical properties. Canadian Journal of Mathematics 9 515–525 (1957). 32. N. Warnberg. Positive semidefinite propagation time. Discrete Applied Mathematics 198 274– 290 (2016). 33. B. Yang. Fast-mixed searching and related problems on graphs. Theoretical Computer Science 507 100–113 (2013).

Chapter 3

Analysis of Termatiko Sets in Measurement Matrices Katherine F. Benson, Jessalyn Bolkema, Kathryn Haymaker, Christine Kelley, Sandra R. Kingan , Gretchen L. Matthews , and Esmeralda L. N˘astase

3.1 Introduction Compressed sensing is a novel technique used to recover a sparse signal by sampling at a rate much lower than the Nyquist rate. Researchers have shown that perfect recovery is achievable by effectively exploiting the sparsity of the signal in some basis known only at the receiver. While the signals can be of any nature, the results also apply to recovering sparse discrete-time signals that may be represented as sparse vectors. In particular, to recover a signal that is a k-sparse vector, meaning the number of its nonzero positions is at most k, Candes and Tao showed that using a small number of measurements and a linear programming (LP) algorithm over the 1 -norm was effective in recovering the signal perfectly and provided bounds on the number of measurements needed [2].

K. F. Benson Department of Mathematics, Statistics, and Computer Science, University of Wisconsin-Stout, Menomonie, WI, USA e-mail: [email protected] J. Bolkema Mathematics Department, California State University Dominguez Hills, Carson, CA, USA e-mail: [email protected] K. Haymaker Department of Mathematics and Statistics, Villanova University, Villanova, PA, USA e-mail: [email protected] C. Kelley Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, USA e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_3

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While linear programming is a practical technique in many applications, its implementation complexity is polynomial in the length of the signal. More efficient and almost linear-time algorithms are desirable for many practical applications. A close to linear-time algorithm is the Interval-Passing Algorithm (IPA) [3]. Typically, to recover a k-sparse vector x of length n, an m × n measurement matrix M is used. The measurement y is obtained as Mx. Random real-valued entries in the measurement matrix have been shown to be effective for the compressed sensing problem, and it was shown recently that sparse binary matrices can be as effective [5, 17]. When sparse measurement matrices are used, the IPA complexity is almost linear-time. In [6], it was shown that there are close connections between LP decoding of low-density parity-check (LDPC) codes over a communication channel [9] and LP reconstruction of sparse signals in compressed sensing. Specifically, using the same sparse binary parity-check matrices for LDPC codes as measurement matrices in compressed sensing, the authors showed that if the LP decoder could recover any k symbol errors, then the LP reconstruction could recover any k-sparse signal. LDPC codes are typically decoded using iterative graph-based message-passing algorithms, such as the Sum-Product Algorithm (SPA), which can be visualized as operating on the incidence graph, called the Tanner graph, of the corresponding parity-check matrix. Similarly, taking the incidence graph of the measurement matrix, one can visualize the IPA as a message-passing algorithm operating on this graph. The sparser the matrix, the lower the complexity of the algorithm. The efficiency of algorithms like the SPA and IPA makes them preferred in practice, despite the fact that they are suboptimal. Therefore, a main goal is to improve their performance by careful design of the corresponding matrices (equivalently, graphs). Failure of the sum-product algorithm operating on an LDPC code graph has been characterized by combinatorial substructures in the Tanner graph. Among these substructures are stopping sets (in the case of the binary erasure channel), trapping sets, absorbing sets [7, 8], and pseudocodewords [4, 7, 10, 11, 14]. Similarly, it was recently shown that certain graph structures characterize when the IPA fails to recover the signal [17]. Specifically, the IPA will fail if and only if the k nonzero positions of the signal correspond to the vertices in a so-called termatiko set.

S. R. Kingan Department of Mathematics, Brooklyn College and the Graduate Center, City University of New York, New York, NY, USA e-mail: [email protected] G. L. Matthews () Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA e-mail: [email protected] E. L. N˘astase Department of Mathematics, Xavier University, Cincinnati, OH, USA e-mail: [email protected]

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In this paper, we consider the relationship between termatiko sets and other combinatorial structures that characterize failure in related iterative decoding algorithms. This includes stopping, trapping, and absorbing sets. We also examine some classes of Tanner graphs including those based on finite geometries and those with degree regularity to obtain bounds on the cardinality of the smallest termatiko sets. These sets are those most likely to lead to failure in IPA. This paper is organized as follows. In Sect. 3.2, we provide the necessary background. In Sect. 3.3, we present some results characterizing the relationship between termatiko sets and stopping sets, and we give a lower bound on the size of termatiko sets in left-regular Tanner graphs. We also examine canonical examples of trapping and absorbing sets to determine whether or not they are also termatiko. In Sect. 3.4, we share a case study for whether redundancy in the measurement matrix affects the presence of termatiko sets. In Sect. 3.5, we give results on the sizes and types of termatiko sets that occur in finite-geometry based measurement matrices. Section 3.6 concludes the paper.

3.2 Preliminaries In this section, we provide any necessary definitions and notation used in this paper, including the background from coding theory as well as compressed sensing. Given a parity-check matrix H of a binary LDPC code C, the columns correspond to the coordinate positions of a codeword and the rows correspond to the paritycheck equations (or more general constraints). The corresponding Tanner graph G is a bipartite graph where one set of vertices V corresponds to the columns (called variable nodes) and one set of vertices W corresponds to the rows (called check nodes). An edge connects the ith variable node to the j th check node if and only if the (j, i)th (j, i)th entry of H is non-zero. We sometimes say the Tanner graph of C to mean the Tanner graph corresponding to a particular parity-check matrix of C. The Tanner graph of an m × n measurement matrix M in compressed sensing is defined similarly. The columns correspond to the coordinate positions of the signal and the rows correspond to the measurements. In the corresponding Tanner graph, the columns are represented by a set V of variable nodes and the rows are represented by a set W of measurement nodes. If the matrix is non-binary, the edges in the Tanner graph are labeled with the corresponding entry. However, for binary matrices, the presence of an edge implies a 1 in the matrix, and no edge label is needed. In this paper, we consider only binary matrices. Example 3.1 Figure 3.1 shows an example of a measurement matrix M and its corresponding Tanner graph. The variable nodes are represented by shaded circles, and the measurement nodes are represented by squares. For a subset X of vertices in a graph G, let N(X) denote the set of vertices that are adjacent to at least one vertex in X. In addition, N(u) := N({u}) for a vertex u of G.

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Fig. 3.1 A measurement matrix M on the left, with its corresponding Tanner graph representation on the right

It is common in coding theory to draw Tanner graphs in their left-right representation, meaning that the variable node set is on the left, and the check/measurement node set is on the right, as in Fig. 3.1. Moreover, when all variable nodes have degree j , the Tanner graph is said to be j -left regular, and when additionally, all check/measurement nodes have degree k, the Tanner graph is said to be (j, k)regular. The girth of a graph is the length of its shortest cycle. For bipartite simple graphs (meaning those with no multiple edges), the girth is even and at least four. Since small cycles are known to cause problems with iterative decoding [15], it is common to consider graphs with girth at least six. In [4], it was shown that iterative decoder failure on the Binary Erasure Channel (BEC) is characterized by the presence of stopping sets in the associated Tanner graph. Definition 3.2 A stopping set S ⊆ V is a subset of the variable nodes such that each check node in N(S) is adjacent to at least two vertices in S. Equivalently, S is a stopping set if there is no check node with exactly one edge adjacent to a vertex in S. The idea is that if all of the vertices in the stopping set correspond to positions in the codeword that are erased, then none of the check nodes will be able to recover any of the erased bits on the stopping set since each check neighbor of the set connects to it at least twice. Thus, the decoder fails exactly when a subset of the erased positions forms a stopping set. Consequently, one wants the smallest stopping sets to be as large as possible to minimize the probability of decoder failure. The size of a smallest nonempty stopping set in a Tanner graph is called the stopping distance and is denoted by smin .

3 Analysis of Termatiko Sets in Measurement Matrices Fig. 3.2 The set {v1 , v3 , v6 } is both a stopping set and a termatiko set in the Tanner graph

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Example 3.3 In the Tanner graph in Fig. 3.1, the vertices v1 , v3 , v6 form a stopping set S, since each vertex in the set of their neighbors {w1 , w2 , w3 } connects to the set S at least twice. The set S = {v1 , v3 , v6 } is highlighted in Fig. 3.2. On the other hand, the set {v1 , v6 } does not form a stopping set since w1 is only adjacent to v1 . The smallest stopping set in the graph in Fig. 3.2 is {v2 , v4 }, thus the stopping distance of the graph is smin = 2. In [17], it was shown that IPA failure is characterized by the presence of termatiko sets in the associated Tanner graph. Definition 3.4 Let G = (V , W ; E) be a Tanner graph corresponding to a measurement matrix M. Let T ⊆ V , let N(T ) ⊆ W be the set of check nodes neighbors of T and let U = {u ∈ V \T | N(u) ⊆ N(T )}. Then T is a termatiko set if and only if for each c ∈ N(T ) one of the two conditions holds: I. c is adjacent to a vertex u ∈ U . II. c is not adjacent to any vertex u ∈ U , but c is adjacent to at least two vertices v1 and v2 in T and every check node c ∈ N(v1 ) ∪ N(v2 ) is adjacent to at least two vertices in T . Throughout this paper, when we refer to Condition I and Condition II, we will mean the conditions in Definition 3.4. Moreover, references to U will mean the set U corresponding to a termatiko set T (from context) in Definition 3.4. Example 3.5 The set T = {v1 , v3 , v6 } in Fig. 3.2 is also a termatiko set. To see this, note that N(T ) = {w1 , w2 , w3 } and none of the vertices v2 , v4 , v5 , v7 have all of their neighbors in T . Thus, U = ∅. To be a termatiko set, each measurement node

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in N(T ) must satisfy Condition I or II. It is easy to verify that each of the nodes w1 , w2 , w3 satisfy Condition II. Notice that T = {v2 , v4 } is another termatiko set. Indeed, in this case, N(T ) = {w1 , w3 , w4 } and U = {v7 }. Both w1 and w4 are adjacent to v7 ∈ U ; hence, w1 and w4 satisfy Condition I. However, w3 fails to satisfy Condition I as it is not adjacent to v7 , the only element of U . Even so, w3 is adjacent to both v2 and v4 (which are vertices in T ). Moreover, investigating the vertices in N(v2 ) ∪ N(v4 ) = {w1 , w3 , w4 }, we can confirm that each of them is adjacent to both v2 and v4 . As a result, w3 satisfies Condition II. This verifies that T = {v2 , v4 } is a termatiko set and demonstrates the situation in which some elements of N(T ) satisfy Condition I while others satisfy Condition II. As with stopping sets, the IPA algorithm is improved when the smallest termatiko sets are as large as possible. The size of a smallest nonempty termatiko set is called the termatiko distance and is denoted by tmin .

3.3 Connections Between Termatiko Sets and Other Structures We start this section by examining the connection between termatiko sets and stopping sets. As noted in [17], every stopping set is a termatiko set. To see this, consider the following. If S ⊆ V is a stopping set, then by Definition 3.2, each c ∈ N (S) ⊆ W is adjacent to at least two vertices v1 and v2 in S. Since every check node c ∈ N (v1 ) ∪ N(v2 ) is also adjacent to at least two vertices in S, it follows that S is termatiko; indeed, each node in N(T ) satisfies Condition II. Therefore, for any Tanner graph, tmin ≤ smin . However, not every termatiko set is a stopping set. Example 3.6 For an example of a termatiko set that is not a stopping set, see Fig. 3.3. The set T = {v6 , v7 } is termatiko because all check node neighbors satisfy Condition I of the definition. In this case U = {v1 , v2 , v3 , v4 , v5 }, since N (T ) = {w1 , w2 , w3 , w4 }. T is not a stopping set since (for example) the check node w1 is adjacent to one vertex in T . In what follows, we will prove some results relating termatiko sets to stopping sets. Throughout we assume the set U is as defined in Definition 3.4. Although a termatiko set T is not necessarily a stopping set, we can show that T ∪ U is one. Proposition 3.7 For any termatiko set T in a Tanner graph G, T ∪ U is a stopping set in G. Proof Let T be a termatiko set, and U be as defined in Definition 3.4. Let c ∈ N (T ∪ U ) = N (T ). Suppose c ∈ N(U ). Thus, there exists u ∈ U that is adjacent

3 Analysis of Termatiko Sets in Measurement Matrices Fig. 3.3 The set T = {v6 , v7 } is a termatiko set in which every check node neighbor of vertices in T satisfies Condition I of the termatiko set definition

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to c. Also, since c ∈ N(T ), there exists a t ∈ T that is adjacent to c. Therefore, c is adjacent to at least two vertices in T ∪ U . Now suppose c ∈ N(T ) \ N(U ). By definition of a termatiko set, c must satisfy Condition II. In particular, this means that c is adjacent to at least two vertices in T and therefore c is adjacent to at least two vertices in T ∪ U . Since any c ∈ N(T ∪ U ) is always adjacent to at least two vertices in T ∪ U , the set T ∪ U is a stopping set.  The previous result shows that every termatiko set T is contained in a stopping set. In particular, when U = ∅, T is a stopping set. Note that this is not a necessary condition, since T may be a stopping set with U = ∅. Proposition 3.8 Suppose T is a termatiko set that is not a stopping set. Let W  be the set of check nodes that are adjacent to exactly one vertex in T , and let U  = N (W  ) ∩ U . Then the set T ∪ U  is a stopping set. Proof Suppose T is a termatiko set that is not a stopping set, and assume W  ⊆ N (T ) is such that each c ∈ W  is adjacent to exactly one vertex in T . Since T is termatiko, each c ∈ W  is adjacent to at least one vertex in U . Let U  = N(W  ) ∩ U . Then T ∪ U  is a stopping set since each vertex c ∈ N(T )\W  is adjacent to at least two vertices in T , and each vertex w ∈ W  is adjacent to one vertex in T and at least one vertex in U  .  Sparse matrices that have j nonzero positions per column for j ≥ 3 have been heavily studied in coding theory and are good candidates for measurement matrices. They correspond to j -left regular Tanner graphs. We first look at the cardinality of the smallest termatiko sets that can exist in a j -left regular graph, assuming the girth of the graph is at least six.

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Theorem 3.9 Suppose G is a j -left regular Tanner graph with girth at least 6. Then any termatiko set T in G has size |T | ≥ j. Moreover, for every integer j ≥ 2, there exists a j -left regular Tanner graph with girth at least 6 and termatiko distance j . Proof We first treat the case j = 2. If G is 2-left regular, then it remains to show there cannot be a termatiko set of size 1. Observe that if T = {v1 } is a termatiko set, then without loss of generality, let N(v1 ) = {c1 , c2 }. If u ∈ U , then N(u) = {c1 , c2 } since u has degree 2 also. But then G has a 4-cycle which contradicts the assumption. Therefore U = ∅ and neither c1 nor c2 can satisfy Condition I. Observe that c1 and c2 also cannot satisfy Condition II since each has only one neighbor in T . Thus, the smallest termatiko set in G has size j ≥ 2. Let T be a termatiko set in G, and suppose that |T | < j . We first show that no check node in N (T ) satisfies Condition II. Suppose there exists a vertex c ∈ N(T ) such that c ∈ / N(U ) and c is adjacent to x, y ∈ T and every c ∈ N(x) ∪ N(y) is adjacent to some two vertices in T . Then any d ∈ N(x)\{c} is adjacent to x and at least one other vertex in T . If two vertices in N(x) were adjacent to the same vertex v = x ∈ T , then there would be a 4-cycle in G, e.g. (c, x, d, v, c). Thus, each of the j − 1 elements in N(x) \ {c} are adjacent to a different vertex v = x ∈ T . So we have j − 1 different vertices other than x in T that are adjacent to d ∈ N(x)\{c}. This means T has size greater than j − 1, which is a contradiction. Thus, Condition II cannot hold for any c ∈ N(T ). We can therefore assume that for all c ∈ N(T ), Condition I is satisfied, so c ∈ N (U ). Let c ∈ N (T ) and suppose c is adjacent to some v ∈ T . By assumption, c is adjacent to some u ∈ U. Since G is j -left regular, u has j − 1 neighbors other than c in N (T ). Moreover, no two vertices d1 , d2 ∈ N(u) are adjacent to the same vertex x ∈ T , since this would create a 4-cycle. This means N(u) is adjacent to j distinct vertices in T , which is a contradiction. Thus, |T | ≥ j . Given an integer j ≥ 2, we may construct a j -left regular Tanner graph G = (V , W ; E) with girth at least 6 and termatiko distance j as follows. Let V = T ∪ U , where T = {v1 , . . . , vj } and U = {u1 , . . . , uj }, and W = {c1 , c2 , . . . , cj 2 }. For i = 1, . . . , j , let N(vi ) = {cj (i−1)+1 , cj (i−1)+2 , . . . , cij }, and let N(ui ) = {ci , ci+j , . . . , ci+(j −1)j }. Then G is j -left regular, N(U ) = N(T ) = W, and each c ∈ W is adjacent to a vertex in U . Hence, Condition I is satisfied for each ci ∈ N (T ), and T is a termatiko set of size j . It remains to show that there are no 4cycles. First observe that the neighborhoods of any pair vi , vj ∈ T , i = j have empty intersection, as do the neighborhoods of any pair ui , uj ∈ U, i = j . Further, the neighborhoods of vk ∈ T and u ∈ U intersect only in check node cj (k−1)+ , and thus there are no 4-cycles in G. By the previous argument, G does not have any smaller termatiko set, and G has termatiko distance j .  While stopping sets characterize iterative decoder failure on the binary erasure channel, absorbing sets and trapping sets have been shown to characterize iterative decoder failure on other communication channels. In the rest of the section, we will look at specific types of absorbing sets and trapping sets that have been heavily studied and examine whether they are termatiko.

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Let GD = (D, N (D); ED ) denote the subgraph induced by D ⊆ V and N(D) ⊆ W in a Tanner graph G, where ED is the set of edges between D and N(D). Definition 3.10 An (a, b)-trapping set is a subset D ⊆ V of variable nodes in the Tanner graph G = (V , W ; E) of a code such that |D| = a and |O(D)| = b, where O(D) is the subset of check nodes of odd degree in the subgraph GD . Definition 3.11 An (a, b)-absorbing set is a subset D ⊆ V of variable nodes in the Tanner graph G = (V , W ; E) of a code such that |D| = a, |O(D)| = b, and each variable node in D has strictly fewer neighbors in O(D) than in W \ O(D), where O(D) is the subset of check nodes of odd degree in the subgraph GD . Figure 3.4 contains examples of the smallest trapping and absorbing sets that occur in 3-left regular Tanner graphs with girth at least 6. These are regarded as the most harmful trapping and absorbing sets with respect to some iterative algorithms. Note that the absorbing sets in Fig. 3.4c and d are the unique absorbing sets of the given sizes. We now show that in a 3-left regular Tanner graph with girth at least 6, if the (3, 3)-absorbing set (shown in Fig. 3.4c) occurs as a subgraph, it is not termatiko. A similar statement also holds for the (4, 2)-absorbing set (Fig. 3.4d). Proposition 3.12 Consider a 3-left regular Tanner graph G with girth g ≥ 6. Then a (3, 3)-absorbing set of G is not termatiko, and a (4, 2)-absorbing set is not termatiko. Proof Assume T is a (3,3)-absorbing set in a 3-left regular Tanner graph with girth g ≥ 6. As in Fig. 3.4c, let v1 , v2 , v3 denote the vertices in T and c1 , c2 , . . . , c6 denote the vertices of N(T ) where c4 , c5 , and c6 are each adjacent to exactly one vertex in T . Suppose by way of contradiction that T is termatiko. We first show that no vertices in N(T ) satisfy Condition II. Vertices c4 , c5 , c6 are not adjacent to two vertices in T so Condition II of the termatiko set definition is not

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satisfied for these vertices. Each of the vertices c1 , c2 , c3 are adjacent to exactly two vertices in the set {v1 , v2 , v3 }. Since each vertex in {v1 , v2 , v3 } is adjacent to one vertex in {c4 , c5 , c6 }, N(vi ) ∪ N(vj ) where i, j ∈ {1, 2, 3}, i = j , includes a vertex that is adjacent to only one vertex in T and therefore none of the vertices c1 , c2 , c3 satisfy Condition II. Since T is termatiko, each vertex in {c1 , c2 , . . . , c6 } must satisfy Condition I. This means each is adjacent to at least one vertex in a set U ⊂ V such that N (U ) = N (T ). Since the Tanner graph is 3-left regular, each vertex of U has degree 3. For i, j ∈ {1, 2, 3}, and i = j , ci and cj cannot be adjacent to the same vertex u∗ ∈ U because otherwise there is a 4-cycle in the graph. Thus, let u1 , u2 , u3 be vertices in U that are adjacent to c1 , c2 , c3 respectively. Since the Tanner graph is 3-left regular, each of u1 , u2 , and u3 must be adjacent to exactly two vertices from {c4 , c5 , c6 }. Consider u1 which is adjacent to c1 . If u1 and c4 are adjacent, then (u1 , c4 , v1 , c1 , u1 ) is a 4-cycle, a contradiction. If u1 and c5 are adjacent, then (u1 , c5 , v2 , c1 , u1 ) is a 4-cycle, a contradiction. Thus, u1 can only be adjacent to c1 and c6 , which means u1 does not have degree 3 which contradicts the 3-left regular assumption on the graph. Therefore, T is not termatiko. The proof that a (4, 2)-absorbing set, as in Fig. 3.4d, is not termatiko is similar to the previous result and is omitted. We show that while one measurement node in the absorbing set graph satisfies Condition II (assuming that the set is termatiko), it is not possible to get the rest of the measurement nodes to satisfy Condition I and degree regularity simultaneously.  Proposition 3.13 There exists a 3-left regular Tanner graph with girth at least 6 such that a (4, 4)-trapping set is a termatiko set. Moreover, there exists a 3left regular Tanner graph with girth at least 6 such that the (5,3)-trapping set is termatiko. Proof Consider the graph in Fig. 3.4a. Let T = {v1 , v2 , v3 , v4 }. We will show how to embed the graph into a 3-left regular Tanner graph with girth ≥ 6 so that T is termatiko. First, it is straightforward to show that none of the measurement nodes ci ∈ N (T ), for i = 1, . . . , 8 satisfy Condition II. Therefore, each ci must satisfy Condition I and be adjacent to least one element u ∈ U . Observe that it is not possible for two of the nodes in {c1 , c2 , c3 , c4 } to be adjacent to the same ui ∈ U for the graph to have both girth at least 6 and 3-left regularity. We will construct the set U as follows. Let U = {u1 , u2 , u3 , u4 }. Define the neighborhoods as follows: N (u1 ) = {c1 , c7 , c8 }, N(u2 ) = {c2 , c5 , c8 }, N(u3 ) = {c3 , c5 , c6 }, N (u4 ) = {c4 , c6 , c7 }. Note that each measurement node in N(T ) is adjacent to at least one vertex in U , each vertex in U has degree 3, N(U ) = N(T ), and the girth of the resulting graph is 6. Consider the (5, 3)-trapping set in Fig. 3.4b. Let T = {v1 , v2 , v3 , v4 , v5 }. As in the previous proof, we will show how to embed the graph into a 3-left regular Tanner graph with girth ≥ 6 so that T is termatiko. Since none of the measurement nodes in N (T ) satisfy Condition II, we define a set U = {u1 , u2 , u3 } as follows. Let N (u1 ) = {c1 , c3 , c9 }, N(u2 ) = {c2 , c5 , c8 }, and N(u3 ) = {c4 , c6 , c7 }. Note that

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each measurement node in N(T ) is adjacent to a vertex in U , each vertex in U has degree 3, N (U ) = N(T ), and the girth of the resulting graph is 6.  This result shows that whenever there is a (4, 4)-trapping set, it may coincide with a termatiko set of size 4. More importantly, some of the existing techniques to remove (4, 4)-trapping sets [16] can be used to remove these types of termatiko sets. Similarly, it shows that whenever there is a (5, 3)-trapping set, it may coincide with a termatiko set of size 5. Thus, one may consider methods for removing them.

3.4 Redundancy Redundancy in the parity-check matrix representations of LDPC codes has been shown to improve iterative decoder performance due to the fact that harmful structures like stopping sets, absorbing sets, and trapping sets tend to be removed when additional constraints are present [13]. This gain comes at the expense of iterative decoder complexity, so finding the optimal tradeoff is advantageous. In this section, we examine several measurement matrix representations to see the effect of redundancy on the presence of termatiko sets in the corresponding Tanner graphs. This case study gives insight to design rules for measurement matrices that can improve IPA reconstruction performance. Consider the measurement matrix ⎡

⎤ 1001011 M = ⎣0 1 0 1 1 1 0⎦. 0010111

Fig. 3.5 The Tanner graph associated to the measurement matrix M

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The corresponding Tanner graph is shown in Fig. 3.5. The following matrices have additional rows of redundancy. ⎡

1 ⎢0 MA = ⎢ ⎣0 0

⎡

1 ⎢0 ⎢ ⎢ MC = ⎢ 0 ⎢ ⎣0 1

0 1 0 1

0 1 0 1 1

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⎡ ⎤ 1 1001 ⎢0 1 0 1 0⎥ ⎥ , MB = ⎢ ⎣0 0 1 0 1⎦ 1 1110

1 1 0 1

0 1 1 0

1 1 0 1 0

1 ⎢ 011 ⎢0 ⎢0 1 1 0⎥ ⎢ ⎥ ⎢ ⎥ 1 1 1 ⎥ , MD = ⎢ 0 ⎢ ⎥ ⎢1 0 0 1⎦ ⎢ ⎣1 101 1 ⎤

⎡

0 1 0 1 1 0 1

0 0 1 1 0 1 1

1 1 0 1 0 1 0

0 1 1 0

1 1 1 1

0 1 1 0 1 1 0

1 1 1 0 0 0 1

⎤ 1 0⎥ ⎥, 1⎦ 0 ⎤ 1 0⎥ ⎥ 1⎥ ⎥ ⎥ 1⎥. ⎥ 1⎥ ⎥ 0⎦ 0

In particular, if Rowi S denotes the ith row of a matrix S, then Row4 MA = Row2 M + Row3 M; Row4 MB = Row1 M + Row2 M + Row3 M; Row4 MC = Row2 M + Row3 M and Row5 MC = Row1 M + Row2 M; and Row4 MD = Row2 M + Row3 M, Row5 MD = Row1 M + Row2 M, Row6 MD = Row1 M + Row3 M, and Row7 MC = Row1 M + Row2 M + Row3 M. Since the smallest stopping set in each of the representations is 3, tmin ≤ 3. The following table shows the numbers of termatiko sets of sizes 1 and 2 in the corresponding graphs. Representation # Size 1 Termatiko # Size 2 Termatiko M 4 15 MA 3 15 MB 1 12 MC 0 7 MD 0 0 This example demonstrates that redundancy in the parity-check matrix can result in fewer termatiko sets of small size. Moreover, the choice of redundant measurement node matters. Indeed, representations MA and MB each have only one redundant row but different number of small cardinality termatiko sets. One may also observe that representation MD attains the best termatiko distance possible in this scenario with tmin = 3.

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3.5 Measurement Matrices from Finite Geometries In this section, we focus on measurement matrices obtained from incidence structures in finite geometries. LDPC codes based on finite geometries have been shown to have nice properties [12]. Specifically, we consider the incidences of points and lines in the finite Euclidean and projective geometries and analyze the sizes of termatiko sets in the corresponding graphs. The Euclidean geometry constructions involve defining a subgeometry without the origin point and creating incidence matrices of points and lines for these families of subgeometries. We recall the basic properties of finite projective and Euclidean geometries, starting with the definitions of affine and projective space [1]. A linear space is a collection of points and lines such that any line has at least two points, and two points are on precisely one line. A hyperplane of a linear space is a maximal proper subspace. A projective plane is a linear space in which any two lines meet, and there exists a set of four points, no three of which are collinear. A projective plane has dimension 2. A projective space is a linear space in which any two-dimensional subspace is a projective plane. An affine space is a projective space with one hyperplane removed. Throughout, Fq denotes the field with q elements, where q is a power of a prime. Like the Euclidean space Rm , the set of points formed by m-tuples with entries from the finite field Fq forms an affine space, called a finite Euclidean geometry. A finite Euclidean geometry satisfies the axioms of affine space, and comprises one of the families of finite geometries that we will consider in this paper. In the case of m = 2, lines are sets of points (x, y) satisfying an equation y = hx + b or x = a, where h, b, a ∈ Fq . Definition 3.14 The m-dimensional finite Euclidean geometry EG0 (m, q), has the following parameters. • There are q m points. q m−1 (q m − 1) lines. • There are q −1 • Each line contains q points. qm − 1 lines. • Each point is on q −1 Any two points have exactly one line in common, and any two lines either have one point in common or are parallel (i.e., have no points in common). It is common to define a code from a modified version of EG0 (m, q) in which the origin point is removed and every line containing the origin is also deleted. By convention, the notation EG(m, q) is used to refer to the Euclidean geometry with the origin removed [12]. The finite Euclidean plane EG(2, 3) is shown in Fig. 3.6. We use EG0 (m, q) to distinguish the case when the origin and all lines containing it are included. The parameters for EG(2, q) and EG0 (2, q) are provided in Table 3.1.

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Table 3.1 Parameters of finite planes Number of points Number of lines Number of points on each line Number of lines that intersect at a point

EG0 (2, q) q2 q(q + 1) q q +1

EG(2, q) q2 − 1 q2 − 1 q q

PG(2, q) q2 + q + 1 q2 + q + 1 q +1 q +1

Definition 3.15 The m-dimensional finite projective geometry PG(m, q) has the following parameters. q m+1 − 1 points. q −1 (q m + · · · + q + 1)(q m−1 + · · · + q + 1) . • The number of lines is (q + 1) • Each line contains q + 1 points. qm − 1 lines. • Each point is on q −1

• There are

Any two points have exactly one line in common, and each pair of lines has exactly one point in common. An essential subclass of finite geometries are the finite Euclidean and projective planes. The parameters in these cases are provided in Table 3.1. Any Tanner graph of a PG(m, q) or an EG(m, q)-LDPC code with m > 1 has girth 6 [12]. One way to obtain a measurement matrix from a finite geometry is to create an incidence matrix from the points and lines of the geometry, and let that matrix be M. In this paper, the columns of the incidence matrix correspond to lines in the geometry, and the rows of the matrix correspond to points in the geometry. This choice allows for less redundancy in the resulting measurement matrix in the Euclidean geometry case than if the role of points and lines were to be reversed. Therefore, the lines in the geometry correspond to variable nodes in the Tanner graph, and the points in the geometry correspond to check nodes. This process of obtaining a measurement matrix M from a finite geometry is illustrated in the next example. Example 3.16 An incidence matrix for the points and lines of PG(2, 2) is given in Fig. 3.7, along with the corresponding Tanner graph. The geometry is commonly known as the Fano plane.  It is convenient to rephrase the definition of termatiko set in terms of points and lines, where the lines correspond to the variable nodes and the points correspond to the measurement nodes of the measurement matrix based on the underlying finite geometry. Let L be the set of all lines in EG(m, q). Let T ⊆ L, and let N(T ) be the set of points that lie on at least one line in T . Define U = { ∈ L\T | N() ⊆ N (T )}. Then T is a termatiko set if and only if for each p ∈ N(T ) one of the two conditions holds:

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Fig. 3.6 The finite Euclidean plane EG(2, 3)

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Fig. 3.7 The point-line incidence graph of PG(2, 2) is on the left, its corresponding measurement matrix in the middle, and its corresponding Tanner graph is on the right

I. p lies on a line in U . II. p is not on a line in U , and there are at least two lines 1 and 2 in T that contain p and every point in 1 ∪ 2 is on at least two lines from T . Proposition 3.17 In the line-point incidence graph of EG(2, q), a maximal set of parallel lines is a termatiko set, all of whose check nodes satisfy Condition I. The complement of a maximal set of parallel lines is also a termatiko set of this type. Furthermore, any set of parallel lines that is not maximal is not a termatiko set. Proof Let T be a maximal set of parallel lines in EG(2, q). Then T is incident to all points in the geometry. Thus, V = U ∪ T . Moreover, since each point is on q + 1 > 1 lines, each point is incident to a line in U , and therefore each point satisfies Condition I. The set of lines V \ T = U is also termatiko, since each point in EG(2, q) is incident to one line in T , which now plays the role of U in Condition I. Therefore, all points satisfy Condition I. We now prove that a set of parallel lines that is not maximal is not a termatiko set. Let T be a set of k ≤ q − 1 parallel lines in EG(2, q). We show that U = { ∈ V \T | N() ⊆ N (T )} = ∅. Suppose  ∈ V \T . From the geometry, it is known that  contains q points which

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means that in the line-point incidence graph of EG(2, q), |N()| = q. Since a pair of lines can have at most one point in common, N()∩N(T ) ≤ 1 for each line T ∈ T . Since there are k lines in T , this means |N() ∩ N(T )| ≤ k ≤ q − 1. It follows that there exists at least one point in N()\N(T ) and therefore N()  N(T ). Since U = ∅, there are no points in N(T ) that satisfy Condition I. Additionally, since T is a set of parallel lines, N(1 ) ∩ N(2 ) = ∅ for all 1 , 2 ∈ T . Therefore, there is no point p ∈ N(T ) that is contained in two lines of T and thus there are no points that satisfy Condition II. Therefore, T is not termatiko.  Remark 3.18 We state a few other observations for termatiko sets in EG(2, q). • If T is the union of a maximal set of parallel lines with one additional line, then U has parallel lines and one more line. Then T is a termatiko set and both Conditions I and II occur. • If T has q lines, none of which are parallel, then U = ∅, and T is a termatiko set with each point satisfying Condition II. • If T has three lines intersecting in a point, then T is not termatiko since there are some points that satisfy neither condition. Theorem 3.19 Let T be a subset of the lines in EG(2, q). Then T is a termatiko set of smallest size in the line-point incidence graph of EG(2, q) if and only if T is a maximal set of parallel lines. Proof Let T be a subset of lines in EG(2, q), and let tmin denote the smallest size of a termatiko set in the line-point incidence graph of EG(2, q). We first show that if T is a set of q lines and is not a maximal set of parallel lines, then T is not a termatiko set. Suppose T is a set of q lines, and assume T is not a bundle of parallel lines. Then there are at most q − 1 parallel lines and in particular, there is a line in T that intersects the q −1 (or fewer) parallel lines. Without loss of generality, suppose there are q−1 parallel lines in T , and label these such that for 0 ≤ i ≤ q−2, line i+1 ∈ T contains points N (i+1 ) = {piq+1 , . . . , p(i+1)q } ⊂ N(T ), and for 1 ≤ i ≤ q − 1, line q has points N(q ) = {p1 , pq+1 , . . . , piq+1 , . . . , p(q−1)q+1 } ⊂ N(T ). Thus, q intersects the parallel lines 1 , . . . , q−1 . Recall that U = { ∈ V \T | N() ⊆ N (T )}. Towards a contradiction, suppose one of the intersection points of q and a point of T , say p1 , is on line ∗ ∈ U . Since p1 is on ∗ , it follows that ∗ cannot contain points p2 , . . . , pq because otherwise, points p1 and some point of {p2 , . . . , pq } is on both 1 and ∗ . This means that the other q − 1 points of ∗ are in the sets N (2 ), . . . , N (q ). If ∗ has two points in the set N(i ), for some 2 ≤ i ≤ q, then the lines ∗ and i have two points in common, giving a contradiction. Thus, ∗ has a point pi∗ in N (i ) for each i = 2, . . . , q. Then, in particular, ∗ and q intersect in both points p1 and pq∗ , resulting in a contradiction. Thus, none of the special intersection points, like p1 , are incident to the lines of U . Thus, they fail Condition I in the definition of a termatiko set. In the case of fewer than q − 1 parallel lines, even more points in N(T ) are these special intersection points, and thus, these points still fail Condition I.

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We now need to show that one of these intersection points in N(T ) also fails Condition II. Without loss of generality, we once again consider p1 ∈ N(T ), since from the above argument we know p1 is not on a line of U . The only lines that contain p1 in T ∪ U are 1 and q . Now the points of N(1 ) ∪ N(q ) are {p1 , p2 , . . . , pq } ∪ {p1 , pq+1 , . . . , p(q−1)q+1 }, and we need all these points to be on at least two lines in T . But we know for example that p2 is only on 1 since the lines 1 , . . . , q−1 are parallel. Thus, we have shown that there are points in N (T ) that also fail Condition II. In the case of fewer than q − 1 parallel lines, it is easy to show that there is again a common intersection point that fails to satisfy Condition II, regardless of whether a subset of the lines intersecting the parallel set are themselves intersecting or parallel. Thus, if T is a set of q lines in EG(n, q), not all parallel, then T is not a termatiko set. Moreover, by Theorem 3.9, since the graph is q-left regular with girth at least 6, any set of fewer than q lines is not termatiko.  Corollary 3.20 The termatiko distance of EG(2, q) is q. Proof This follows immediately from Theorem 3.19.



Theorem 3.21 The termatiko distance of EG(m, q) is at most q. Proof A maximal set of parallel lines in EG(2, q) occurs as a subset of lines in EG(m, q), and still forms a termatiko set.  Proposition 3.22 In EG(m, q), for m ≥ 2, the set of all lines through a given point is not a termatiko set. Proof Let p∗ be a point in EG(m, q), and let T = {1 , . . . , nt } be the set of nt = (q m − 1)/(q − 1) lines1 through point p∗ . Furthermore, let N(T ) be the set of points that lie on at least one line in T , and let U = { ∈ V \T | N() ⊆ N(T )}. Then by choice of T , point p∗ does not lie on a line in U . Now consider any two lines in T that contain p∗ , say 1 and 2 . Then any point p ∈ N(1 ) ∪ N(2 ) with p = p∗ is on only one line in T since all lines of T contain p∗ and every two points in EG(m, q) have at most one line in common. Since this holds for any pair of lines of T , it follows that T is not termatiko.  We will now consider measurement matrices based on finite projective geometries. For shorthand, we use the notation tmin (PG(m, q)) to denote the size of a termatiko set of minimum size in the Tanner graph constructed as the incidence graph of lines and points in PG(m, q), where variable nodes correspond to lines in the geometry and measurement nodes correspond to points. Similar to Theorem 5.7, we now show the relationship between termatiko sets in PG(m, q) and PG(2, q), for m > 2. Theorem 3.23 For all integers m ≥ 2, tmin (PG(m, q)) ≤ tmin (PG(2, q)).

1 In

EG(m, q), each point is on nt = (q m − 1)/(q − 1) lines.

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Proof Let T be a termatiko set in PG(2, q). Recall that PG(2, q) is a subspace m+1 m+1 −q)(q m+1 −q 2 ) copies of PG(2, q) of PG(m, q); specifically, there are (q (q−1)(q 3 −1)(q 3 −q)(q 3 −q 2 ) appearing in PG(m, q). We claim that the termatiko set T of PG(2, q) is also a termatiko set in PG(m, q). Since PG(2, q) is a subspace of PG(m, q), all lines through the points in N(T ) are contained in PG(2, q). Therefore, the conditions satisfied by the points in N(T ) in PG(2, q) are also satisfied in PG(m, q). Since a minimum size termatiko set in PG(2, q) also exists in PG(m, q), we must have that the termatiko distance of PG(m, q) is no more than the termatiko distance of PG(2, q).  Conjecture 3.24 For m ≥ 2, tmin (PG(m, q)) = tmin (PG(2, q)). Based on the structure of PG(m, q), Conjecture 3.24 says that it is not possible to obtain a smaller termatiko set in a higher dimensional geometry. Next we give two examples of termatiko sets of minimum size in PG(2, q) for small q. Example 3.25 In this example we show that tmin (PG(2, 2)) = 4. The termatiko set in Fig. 3.8 has all lines in T , and U = ∅. Thus, T is also a stopping set. Next we will show that no set of three or fewer lines in PG(2, 2) can form a termatiko set. It is clear that a set of one or two lines in PG(2, 2) cannot be a termatiko set since there would be at least three points on these lines that do not satisfy either of the conditions on points in N(T ). For sets of three lines in PG(2, 2), there are two cases. 1. Suppose that a set of three lines in PG(2, 2) intersect pairwise at distinct points. In this case there would be three points, p1 , p2 , p3 where two of the lines in T intersect, and three other points, each lying on a single line in T . By inspection in Fig. 3.9, the points p1 , p2 , and p3 would fail to satisfy either condition on N(T ), so T cannot be a termatiko set. 2. Suppose that the set of three lines intersects at a single point, p. Then all other points p on the lines in T would have the property that p is on a line in U , since U = L \ T in this case. However, the point of intersection p is not in U , and it also fails the other condition on elements of N(T ). Thus, the set T is not a termatiko set.

Fig. 3.8 Minimum termatiko set in PG(2, 2), shown in thick gray lines

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Fig. 3.9 Two sets T of size three in PG(2, 2) that are not termatiko. The sets T are shown in thick dark gray, and the sets U are shown in thick light gray Fig. 3.10 A termatiko set of size 6 in PG(2, 3) is shown in gray

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Example 3.26 This example deals with the case of PG(2, 3). The set of six lines in Fig. 3.10 is a termatiko set since every point in N(T ) is on at least two lines in T , meaning that all points in N(T ) satisfy Condition II in the definition of termatiko set. This also makes T a stopping set, and U = ∅. Therefore, tmin (PG(2, 3)) ≤ 6. Next we give partial results on the sizes of neighborhoods of T , assuming that T is a termatiko set in a code from PG(2, q). Proposition 3.27 Let T be a termatiko set in PG(2, q). Then  |T | |T |(q + 1) − ≤ |N(T )| ≤ |T |q + 1. 2 Proof Given a termatiko set T in PG(2, q), we consider the smallest number of points lying on T . Notice that each line in T has q + 1 points, but each pair of lines must intersect at a point. We consider two extreme scenarios: if all lines in T intersect at a single point, the number of points lying on T could be as large as |T |q + 1. An equivalent way of counting this set of point is: |T |(q + 1) − (|T | − 1).

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On the other hand, if we assume that each pair of lines in T intersects at a distinct point, the number of points lying on T could be as small as |T |(q + 1) − |T2 | .  Remark 3.28 We conclude this section with a discussion about |U | in minimum size termatiko sets in PG(m, q). First, we note that in PG(2, 2), if |S| > 1, then |T | > 4, by case analysis. Therefore, any minimum-size termatiko set in PG(2, 2) has |U | = ∅. Likewise all examples of termatiko sets in PG(2, 3) with |U | > 0 that we constructed were not minimum size. This leads to the following conjecture. Conjecture 3.29 Termatiko sets of minimum size in PG(m, q) have U = ∅, and thus are stopping sets.

3.6 Conclusion In this paper, we considered the relationship between termatiko sets and stopping, trapping, and absorbing sets. We characterized the relationship between termatiko sets and stopping sets and provided a lower bound on the size of termatiko sets in left-regular Tanner graphs. We determined whether certain trapping and absorbing sets are also termatiko. It would be interesting to pursue a formal study of how redundancy in the measurement matrix affects the presence of termatiko sets, based on the case study provided here. While initial results on types of termatiko sets that occur in finite-geometry based measurement matrices have been described here, a number of questions regarding their sizes remain open. Acknowledgments The authors would like to thank the Institute for Mathematics and its Applications (IMA) and the organizers of the IMA conference “Workshop for Women in Graph Theory and Applications (WIGA)" where this collaboration began. The work of G.L. Matthews is partially supported by NSF DMS-1855136.

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6. A. G. Dimakis, R. Smarandache and P.O. Vontobel. LDPC Codes for Compressed Sensing. IEEE Transactions on Information Theory 58 (5), 3093–3114 (2012) https://doi.org/10.1109/ TIT.2011.2181819. 7. Lara Dolecek. On absorbing sets of structured sparse graph codes. In: 2010 Information Theory and Applications Workshop (ITA), IEEE, 1–5 (2010). 8. Lara Dolecek, Zhengya Zhang, Venkat Anantharam, Martin Wainwright and Borivoje Nikolic. Analysis of absorbing sets for array-based LDPC codes. 2007 IEEE International Conference on Communications, 6261–6268 (2007). 9. Robert Gallager. Low-density parity-check codes. IRE Transactions on Information Theory 8 (1), 21–28 (1962). 10. Christine A. Kelley, Deepak Sridhara and Joachim Rosenthal. Tree-based construction of LDPC codes having good pseudocodeword weights. IEEE Transactions on Information Theory 53 (4) 1460–1478 (2007). 11. Christine A. Kelley and Deepak Sridhara. Pseudocodewords of Tanner graphs. IEEE Transactions on Information Theory 53 (11) 4013–4038 (2007). 12. Y. Kou, S. Lin. and M. P. C. Fossorier. Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Transactions on Information Theory 47 (7), 2711–2736 (2001) https://doi.org/10.1109/18.959255. 13. Moshe Schwartz and Alexander Vardy. On the stopping distance and the stopping redundancy of codes. IEEE Transactions on Information Theory 52 (3), 922–932 (2006). 14. T. Richardson. Error floors of LDPC codes. Proceedings of the Annual Allerton Conference on Communication, Control, and Computing 41 (3), 1426–1435 (2003). 15. R. Tanner, A recursive approach to low complexity codes. IEEE Transactions on Information Theory 27 (5), 533–547 (1981). 16. B. Vasi´c and S. K. Chilappagari and D. V. Nguyen and S. K. Planjery. Trapping set ontology. 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, pp. 1–7 (2009) https://doi.org/10.1109/ALLERTON.2009.5394825. 17. Yauhen Yakimenka and Eirik Rosnes. Failure Analysis of the Interval-Passing Algorithm for Compressed Sensing. IEEE Transactions on Information Theory 66 (4), 2466–2486 (2020) https://doi.org/10.1109/TIT.2020.2969163.

Chapter 4

The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey Nadia Benakli, Novi H. Bong, Shonda Dueck (Gosselin), Beth Novick, and Ortrud R. Oellermann

4.1 Introduction The problem of uniquely determining the location of an intruder in a network, with distance detecting devices, motivated Slater [15] to initiate the study of the metric dimension of a graph. Let u and v be distinct vertices of a connected graph G. A vertex w of G is said to resolve u and v, if the distance dG (u, w) from u to w does not equal the distance dG (v, w) from v to w. (We omit the subscript if G is clear from context.) A set W of vertices of G resolves G if every pair of vertices in G is resolved by some vertex of W . A smallest resolving set of a graph is called a metric basis or simply a basis and its cardinality the metric dimension or simply the dimension of G, denoted by β(G). Thus, if W = {w1 , w2 , . . . , wk } is a resolving set for a graph G and u, v are any two vertices of G, then the vectors (d(u, w1 ), d(u, w2 ), . . . , d(u, wk )) and (d(v, w1 ), d(v, w2 ), . . . , d(v, wk )) are distinct since for some i ∈ {1, 2, . . . , k} we have d(u, wi ) = d(v, wi ). The vectors (d(x, w1 ), d(x, w2 ), . . . , d(x, wk )) for x ∈ V (G) are called the

N. Benakli Department of Mathematics, New York City College of Technology, Brooklyn, NY, USA e-mail: [email protected] N. H. Bong Department of Mathematical Sciences, University of Delaware, Newark, DE, USA e-mail: [email protected] S. Dueck (Gosselin) · O. R. Oellermann () Department of Mathematics and Statistics, The University of Winnipeg, Winnipeg, MB, Canada e-mail: [email protected]; [email protected] B. Novick School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_4

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distance vectors of G relative to the set W = {w1 , w2 , . . . , wk } and the ordering w1 , w2 , . . . , wk of its vertices. Thus the location of an intruder in a network can be uniquely determined if distance detecting devices are placed at each of the vertices in a resolving set W . Sebö and Tannier [14] considered the question of whether two graphs with the same vertex set and the same distance vectors relative to some resolving set are isomorphic. They observed that there are non-isomorphic graphs of the same order and with the same distance vectors relative to some ordering of the vertices of a resolving set. A pair of such graphs is shown in Fig. 4.1. The vertices are labeled with their distance vectors relative to the resolving set, which is comprised of the white vertices which are labeled (0, 3) and (3, 0). This observation motivated them to introduce a stronger version of the metric dimension of a graph for which the distance vectors, relative to a corresponding basis, uniquely determine all adjacencies of the graph. A vertex w is said to strongly resolve two vertices u and v of a graph G if there is either a shortest u-w path that contains v or a shortest v-w path that contains u or, equivalently, either the interval between u and w contains v or the interval between v and w contains u.1 If every pair of vertices of G is strongly resolved by a vertex in some set W of vertices of G, then W is a strong resolving set for G. A smallest strong resolving set is called a strong basis and its cardinality the strong dimension of G, denoted by βs (G). It is readily seen that a strong resolving set of a graph is also a resolving set. Hence β(G) ≤ βs (G) for every connected graph G. It was observed in [15] and [14] that paths are precisely the graphs with metric dimension 1 and strong dimension 1, respectively. Unlike some graph parameters, such as the chromatic number or connectivity that cannot decrease when a set of edges is added to a graph, or the independence number of a graph which cannot increase when a set of edges is added to a graph, both the dimension and the strong dimension can increase or decrease if sets of edges are added to a graph. For example, if Kn − e is the complete graph of order n ≥ 3 with one edge deleted, then both the metric dimension and strong dimension increase from n − 2 to n − 1 if the missing edge is added. Moreover, if we add an edge to the star K1,4 , then the metric dimension decreases from 3 to 2. However, Fig. 4.1 Non-isomorphic graphs of the same order and with the same distance vectors relative to the same resolving set

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adding a single edge to K1,4 does not decrease its strong dimension. But if we add two adjacent edges to K1,4 , then the strong dimension of the resulting graph (which is also called the dart graph) is 2 since the set consisting of the leaf of the dart and one vertex of degree 2 forms a strong resolving set for this graph. If we now reconsider the problem of installing a smallest number of distance detecting devices that will permit the unique identification of the location of an intruder in a network, it is natural to ask if the number of detecting devices that are required could be reduced if additional links between some pairs of non-adjacent nodes are added. From an applied perspective this could represent a cost savings if the cost of installing additional links is small in comparison to the cost of purchasing a distance detecting device. This prompts the question by how much the dimension of a graph can be reduced by adding edges. Relationships between the metric dimension of a graph and that of its subgraphs had previously been studied, for example, in [4] and [7]. In particular, Chartrand et al. [4] showed that for every  > 0 there is a graph H and a proper subgraph G of H such that β(H )/β(G) < , and Khuller et al. [7] established a lower bound for the metric dimension in terms of its clique number. The problem of determining the smallest metric dimension among all graphs having a given graph G as spanning subgraph was introduced by Mol et al. in [11]. This minimum is called the threshold dimension of G and is denoted by τ (G). Let U(G) denote that family of graphs having G as a spanning subgraph. If H ∈ U(G) is such that β(H ) = τ (G), then H is called a β-threshold graph of G. Graphs for which the threshold dimension equals the dimension are called β-irreducible. Graphs that are not β-irreducible are called β-reducible. The seminal work on β-irreducible graphs appears in [12]. The threshold strong dimension was introduced by Benakli et al. in [2]. For a graph G, the threshold strong dimension of G, denoted by τs (G), is defined as the smallest strong dimension among all graphs having G as a spanning subgraph. A graph H ∈ U(G) such that βs (H ) = τs (G) is called a βs -threshold graph. If G is a graph such that βs (G) = τs (G), then G is said to be βs -irreducible and G is βs -reducible otherwise. It is readily seen that the metric dimension (the strong dimension) of a path is 1 and that either end-vertex of the path forms a metric basis (strong basis). Indeed paths are the only graphs with dimension (strong dimension) 1, see [14, 15]. So paths are the only graphs with threshold dimension and threshold strong dimension equal to 1. Thus all graphs with dimension (strong dimension) 2 also have threshold dimension (threshold strong dimension, respectively) equal to 2 and are therefore β-irreducible (βs irreducible, respectively). In this paper, we survey results on the threshold dimension and threshold strong dimension of a graph. Section 4.6 also contains several new results for the threshold strong dimension that parallel known results for the threshold dimension.

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4.2 Background and Preliminaries We begin our survey by introducing pertinent background material on the metric dimension and strong dimension of a graph.

4.2.1 The Metric Dimension of a Graph Since the metric dimension of a graph was introduced by Slater [15] and independently by Harary and Melter [5], it has resurfaced many times as a result of its diverse applications. This has led to a substantial body of work on this parameter. It is well-known that the problem of finding the metric dimension of a connected graph is NP-hard. (For a proof see, for example, Khuller et al. [7].) A variety of papers that highlight applications and both the theoretical and computational aspects of this invariant are cited, for example, in the publications of Cáceres et al. [3] and Belmonte et al. [1]. The metric dimension of trees can be computed efficiently and this has been (re)discovered several times, see [4, 5, 15]. To state this result we introduce some additional notation. A vertex of degree at least 3 in a tree T is called a major vertex, and a leaf u is called a terminal vertex of a major vertex v if every interior vertex on the u–v path of T has degree 2. The number of terminal vertices of a major vertex v of T is called the terminal degree of v, denoted by ter(v). If u is a terminal vertex of a major vertex v, the maximal path of T containing u but not v, is called a limb at v. A major vertex with positive terminal degree is called an exterior major vertex. For trees that are not paths, the metric dimension and a basis can be determined as stated in the following result. Theorem 4.1 ([4, 5, 15]) Let T be a tree that is not isomorphic to a path and let S be the set of exterior major vertices of T . Then β(T ) =

 (ter(v) − 1). v∈S

Moreover, a basis for T can be constructed by selecting, for every exterior major vertex v of positive degree exactly one vertex from all but one of its limbs. For example, the tree of Fig. 4.2 has metric dimension 6 and the white vertices form a metric basis.

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Fig. 4.2 A tree having metric dimension 6

Fig. 4.3 A graph G and its strong resolving graph GSR

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4.2.2 The Strong Dimension It was established in [13] that the problem of finding the strong dimension of a graph is NP-hard. Moreover, a useful tool for working with the strong dimension of a graph was introduced in [13]. It was shown that the problem of finding the strong dimension of a connected graph can be polynomially transformed to the vertex covering problem. Let u and v be vertices of a connected graph G. Then v is maximally distant from u, denoted v MD u, if every neighbour of v is no further from u than v, i.e., d(u, x) ≤ d(u, v) for all x ∈ N(v). If u MD v and v MD u, then we say u and v are mutually maximally distant and denote this by u MMD v. The strong resolving graph GSR of G has as its vertex set V (G) and two vertices u, v of GSR are adjacent if and only if u MMD v. Thus if u, v are any two leaves of G, then u MMD v and hence uv ∈ E(GSR ). For any graph H , let α(H ) denote the vertex covering number of the graph H , i.e., the cardinality of a smallest set S of vertices of H such that every edge is incident with a vertex of S. The following reduction of the strong dimension problem to the vertex covering problem was given in [13]. Theorem 4.2 ([13]) If G is a connected graph, then βs (G) = α(GSR ). Moreover, a minimum vertex cover of GSR corresponds to a strong basis of G. Figure 4.3 shows a graph G and its strong resolving graph GSR . Since GSR has only one non-trivial component, which is isomorphic to a 5-cycle, we see that the vertex covering number of the graph GSR is 3. Thus the strong dimension of G is 3. By an observation made prior to Theorem 4.2, if T is a non-trivial tree with  leaves, then βs (T ) ≥  − 1. Sebö and Tannier [14] in fact showed that the following is true:

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Theorem 4.3 ([14]) If T is a non-trivial tree with  leaves, then βs (T ) =  − 1. Moreover, any set of  − 1 leaves of T form a strong resolving set for T . We observed earlier that for any connected graph G, β(G) ≤ βs (G). One may wonder if there is a constant c such that βs (G) ≤ cβ(G) for every graph G. We illustrate that no such c exists. To this end we define for every integer n ≥ 6 a comb, denoted by Tn , as follows: for even n ≥ 6 the comb Tn is the tree of order n obtained from a path of order n+2 2 by attaching a leaf to each of its vertices of degree 2, and for odd n ≥ 7 a comb is defined to be a tree obtained from Tn−1 by subdividing any one of its edges. Then, by Theorems 4.1 and 4.3, we see that β(Tn ) = 2 and n−1 βs (Tn ) =  n−1 2 . Thus βs (Tn )/β(Tn ) ≥ 4 → ∞ as n → ∞. Note that if IH is the set of isolated vertices of H , then the vertex covering number of H equals the vertex covering number of H − IH . So βs (G) = α(GSR − IGSR ). To simplify our discussion we adopt, in the sequel, the convention that any isolated vertices have been omitted from the strong resolving graphs. We now state several simple bounds for the metric dimension and strong dimension of a graph that will be used in this paper. The first of these is an extension of the results that no graph of metric dimension 2 has K5 as a subgraph [7, Theorem 3.2]. The authors of [7] pointed out that this result has the following straightforward generalization: Theorem   4.4 Let G be a connected graph with Kn as a subgraph. Then β(G) ≥ log2 n . Since βs (G) ≥ β(G) for all connected graphs G this result also gives a lower bound for the strong dimension. Corollary  4.5 Let G be a connected graph with Kn as a subgraph. Then βs (G) ≥  log2 n . The second useful tight lower bound on the metric dimension of a graph of order n and diameter 2 was proven independently in [4, 7]. In order to state this result we first introduce a function defined in [11]. Definition 4.6 We let g : (1, ∞) → N be the function defined as follows: g(x) is the smallest integer d such that 2d + d ≥ x, i.e., we have g(x) = d if x ∈ (2d−1 + d − 1, 2d + d]. Theorem 4.7 Let G be a graph of order n and diameter 2. Then β(G) ≥ g(n). Again, since βs (G) ≥ β(G) we have the following: Corollary 4.8 Let G be a graph of order n and diameter 2. Then βs (G) ≥ g(n).

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4.3 Geometric Interpretations for the Threshold Dimension and the Threshold Strong Dimension of a Graph In this section we describe characterizations for both the threshold dimension and threshold strong dimension of a graph, developed in [11] and [2], respectively, in terms of the smallest number of paths whose “strong product” admits certain types of embeddings of these graphs. Just as some geometric objects have distance preserving embeddings in R2 but not in R, as for example a square, or in R3 but not in R2 , as for example the cube or tetrahedron, we will see that graphs with threshold dimension (threshold strong dimension) t can be embedded in a certain way in the “strong product” of t paths but not in the strong product of fewer than t paths. Loosely speaking, in what follows, a path can be thought of as a discrete version of the real line. We begin by reminding the reader of the definition of the strong product of graphs. If G1 , G2 , . . . , Gk are graphs, then their strong product is the graph k

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with vertex set {(x1 , x2 , . . . , xk ) : xi ∈ V (Gi )}, and for which two distinct vertices x = (x1 , x2 , . . . , xk ) and y = (y1 , y2 , . . . , yk ) are adjacent if and only if for every 1 ≤ i ≤ k, either xi yi ∈ E(Gi ) or xi = yi . The distance between x and y in G1  G2  · · ·  Gk is given by max{dGi (xi , yi ) : 1 ≤ i ≤ k}. For a graph G, we use G,k to denote the strong product of k copies of G, i.e., k

G,k =  G. i=1

The work we will be discussing focuses on embeddings in strong products of paths. Figure 4.4 shows the strong product of two paths of order 5. Fig. 4.4 The strong product P5  P5 = P5,2

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Let G and H be graphs. A map ϕ : V (G) → V (H ) is called an embedding of G in H if it is injective and preserves the edge relation (i.e., if xy ∈ E(G), then ϕ(x)ϕ(y) ∈ E(H )). Thus an embedding is an injective homomorphism. Recall that U(G) denotes the set of all graphs that have G as a spanning subgraph. For a graph G and a subset X ⊆ V (G), we let G[X] denote the subgraph of G induced by X. For an embedding ϕ of G in H , we let ϕ(G) = H [ϕ(V (G))], i.e., ϕ(G) is the subgraph of H induced by the range of ϕ. Thus, the graph ϕ(G) is isomorphic to the graph G ∈ U(G) with vertex set V (G ) = V (G) and edge set E(G ) = {xy : ϕ(x)ϕ(y) ∈ E(ϕ(G))}. To describe the embeddings developed in [11] and [2], we let V (Pn ) = {0, . . . , n−1}. Figure 4.4 illustrates this labeling for the strong product of two copies of P5 . Hence, the vertices of Pn,k are k-tuples over the set {0, . . . , n − 1}. Using this notation for the vertex set of Pn , distances in Pn,k can easily be computed.   Fact 4.9 If x = (x1 , . . . , xk ) and y = (y1 , . . . , yk ) are in V Pn,k , then d(x, y) = max{|xi − yi | : 1 ≤ i ≤ k}. In particular, if x and y are distinct, then they are adjacent if and only if |xi −yi | ≤ 1 for every 1 ≤ i ≤ k. The significance of the vertex labels in V (Pn ) is that they correspond to distances, and the labels of the vertices of Pn,k will correspond to vectors of distances. Let G be a connected graph with resolving set W = {w1 , w2 , . . . , wk }. Then every vertex x ∈ V (G) is uniquely determined by its distance vectors relative to the given ordering of the vertices in W . It was shown in [11] that the map which takes every vertex x to this vector of distances to W is an embedding of G in P ,k for some path P . As a result it follows that if W = {w1 , w2 , . . . , wk } is a resolving set for some graph in U(G), then there is an embedding ϕ of G in P ,k for some path P , such that for every vertex x ∈ V (G), the label of ϕ(x) is exactly the vector of distances in ϕ(G) from ϕ(x) to the vertices of ϕ(W ). If D = diam(G) denotes the diameter of G, i.e., the maximum distance between a pair of vertices of G, and W ,k , since the cois a resolving set for G, then there is an embedding of G in PD+1 ordinates of the distance vectors relative to W is at most D. We now describe these embeddings more formally. A Geometric Interpretation for the Threshold Dimension Definition 4.10 Let G be a graph, let W = {w1 , w2 , . . . , wk } be a subset of V (G), and let P be a path. A W -resolved embedding of G in P ,k is an embedding ϕ of G in P ,k such that for every x ∈ V (G), we have  ϕ(x) = dϕ(G) (ϕ(x), ϕ(w1 )), . . . , dϕ(G) (ϕ(x), ϕ(wk )) ,

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i.e., for every 1 ≤ i ≤ k, the ith coordinate of ϕ(x) is exactly the distance between ϕ(wi ) and ϕ(x) in ϕ(G). Figure 4.5 illustrates such an embedding for the given graph G and one of its resolving sets W = {w1 , w2 }. The embedding shown in Fig. 4.6 is a W -resolved embedding of the given tree T where W is not a resolving set of T , but ϕ(W ) is a resolving set for the graph ϕ(T ) ∈ U(T ). The geometric interpretation of the threshold dimension of a graph given in [11] is summarized in the following two results. Theorem 4.11 ([11]) Let G be a connected graph of diameter D, and let W = {w1 , w2 , . . . , wk } ⊆ V (G). Then W is a resolving set for some graph H ∈ U(G) ,k if and only if there is a W -resolved embedding of G in PD+1 . A consequence of this theorem gives a geometric interpretation for the threshold dimension. 1

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Corollary 4.12 ([11]) Let G be a connected graph of diameter D. Then τ (G) is the minimum cardinality of a set W ⊆ V (G) such that there is a W -resolved embedding ,|W | of G in PD+1 . We observed in the Introduction that the threshold dimension and threshold strong dimension of the star K1,4 is 2 but that a β-threshold graph for K1,4 may not be a βs -threshold graph. Moreover, the embeddings ϕ(G) and ϕ(T ) shown in Figs. 4.5 and 4.6, respectively also demonstrate that the threshold strong dimension for the graphs G and T shown there is 2. To see this, observe that the strong resolving graph ϕ(G)SR for the graph ϕ(G) in Fig. 4.5 is isomorphic to P5 and the strong resolving graph ϕ(T )SR for the graph ϕ(T ) in Fig. 4.6 is isomorphic to the graph obtained from a P3 by attaching two leaves to each of its ends, both of which have vertex covering number 2. This prompts the question whether there are graphs for which the threshold dimension does not equal the threshold strong dimension. It was verified in [2] that the graph given in Fig. 4.7 (shown as a {w1 , w2 }-resolved embedding), has threshold dimension 2 and threshold strong dimension exceeding 2. With a computer aided search it was shown that the threshold strong dimension of the graph in Fig. 4.7 is 3. Moreover, the graph obtained from G1 by adding the edges in the set {c2 e3 , c3 e3 , c3 f3 } is a βs -threshold graph for G1 with strong basis {w1 , w2 , c1 }. Indeed it was conjectured in [2], that the difference τs (G) − τ (G) can be arbitrarily large. We describe next a family of graphs from [2] that have threshold dimension 2 and for which it is conjectured that the gap τs (G) − τ (G) can be made as large as desired. To this end let G1 be the graph shown in Fig. 4.7. Let G2 be the graph obtained from two copies G11 and G21 of the graph G1 by identifying the Fig. 4.7 The graph G1 with threshold dimension 2 and threshold strong dimension 3

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Fig. 4.8 The graph G2 shown in black with threshold dimension 2 and threshold strong dimension 4

vertices corresponding to w2 and f1 in G11 with the vertices w1 and a3 , respectively in G21 and adding the edge between the vertex f2 in G11 and the vertex b4 from G21 , as well as the edge between the vertex e1 in G11 and the vertex a2 in G21 . The graph G2 is shown in Fig. 4.8. In general for n ≥ 2, let Gn be the graph obtained from n copies G11 , G21 , . . . , Gn1 of G1 by identifying for each 1 ≤ i < n the vertices labeled w2 and f1 in Gi1 with the vertices labeled w1 and a3 in Gi+1 and then adding the 1 i+1 i edge between the vertex f2 in G1 and the vertex b4 in G1 , as well as adding the edge between the vertex e1 in Gi1 and the vertex a2 in Gi+1 1 . By Theorem 4.11 and Corollary 4.12, τ (Gn ) = 2. Using an exhaustive computer search it was shown that τs (G2 ) = 4. The following was conjectured in [2]: Conjecture 4.13 ([2]) For every positive integer k, there is a positive integer n such that τs (Gn ) ≥ τ (Gn ) + k. The family of graphs {Gn : n ∈ N}, described above, all have threshold dimension 2. For b ≥ 3 we have not yet been able to find graphs with threshold dimension b and threshold strong dimension exceeding b.

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A Geometric Interpretation for the Threshold Strong Dimension In Corollary 4.12 we presented a geometric characterization for the threshold dimension of a graph developed in [11]. These ideas were extended in [2] to provide a geometric characterization for the threshold strong dimension of a graph. Indeed it was shown in [2] that with one additional condition, the W -resolved embeddings, defined above, give rise to a geometric interpretation of the threshold strong dimension. The notion of a distance-preserving subgraph is key to this result. A subgraph G of a graph H , is an isometric subgraph of H if dG (u, v) = dH (u, v) for all vertices u, v ∈ V (G). Theorem 4.14 ([2]) Let G be a connected graph of diameter D, and let W = {w1 , w2 , . . . , wk } ⊆ V (G). Then W is a strong resolving set for some graph ,k H ∈ U(G) if and only if there is a W -resolved embedding ϕ of G in PD+1 such ,k that ϕ(G) is an isometric subgraph of PD+1 .

As a consequence of this theorem we have the following. Corollary 4.15 ([2]) Let G be a connected graph of diameter D. Then τs (G) is the minimum cardinality of a set W ⊆ V (G) for which there is a W -resolved embedding ,|W | ,|W | ϕ of G in PD+1 such that ϕ(G) is an isometric subgraph of PD+1 . Since the embeddings of the graphs G and T shown in Figs. 4.5 and 4.6, respectively, are W (= {w1 , w2 })-resolved embeddings of these graphs that are also isometric subgraphs of the strong product of the two paths in which they have been embedded, we see that their threshold strong dimension is 2. Remark 4.16 These embeddings inspired the constructions of graphs given in [2], that provide a solution to the realizability problem posed in [9], namely which graphs with covering number 2 can be realized as the strong resolving graphs of some graph. It turns out that K2,2 -free graphs with covering number 2 are precisely those that can be realized as the strong resolving graph of some graph. Since the graphs in these constructions all have strong dimension 2, they are βs -irreducible. More results on βs -irreducible graphs are discussed in Sect. 4.6.

4.4 Trees In this section we survey what is known about the threshold dimension and threshold strong dimension of trees with small dimension and small strong dimension, respectively as established in [11] and [2], respectively and highlight, for these cases, some similarities and differences between these two invariants. As it was pointed out in the Introduction, paths are the only graphs with metric dimension 1 and strong dimension 1. Therefore the threshold dimension (and threshold strong dimension) of trees with metric dimension (respectively, strong

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dimension) 2 is also 2. In the case of trees with metric dimension at least 3, it was shown in [11] that the metric dimension can be reduced by adding a single edge. Theorem 4.17 ([11]) If T is a tree with β(T ) ≥ 3, then there exits an edge e ∈ E(T ) such that β(T + e) < β(T ). As a consequence of this theorem, we have the following result for trees with metric dimension 3. Corollary 4.18 ([11]) If T is a tree such that β(T ) = 3, then τ (T ) = 2. Moreover, the β-threshold graph of T can be obtained from T by adding a single edge. The next result shows that trees with metric dimension 4 also have threshold dimension 2. In this case, only two edges need to be added to obtain a β-threshold graph of the tree. Theorem 4.19 ([11]) If T is a tree with β(T ) = 4, then there exists a set of two edges E = {e1 , e2 } ⊂ E(T ) such that β(T + E) = 2. Similarly to the threshold dimension, the following result for the threshold strong dimension was established in [2]. Theorem 4.20 ([2]) If T is a tree with βs (T ) ∈ {2, 3, 4}, then τs (T ) = 2. It appears as though, in general, more edges need to be added to a tree to produce a βs -threshold graph than a β-threshold graph. It was observed, in the Introduction, that the addition of one edge to K1,4 is sufficient to yield a β-threshold graph, whereas two edges need to be added to produce a βs -threshold graph for K1,4 . Similarly, only two edges need to be added to the star K1,5 to obtain a β-threshold graph while five edges need to be added to construct a βs -threshold graph. For the tree T shown in Fig. 4.9, β(T ) = 3 and βs (T ) = 4. From Theorem 4.17 and Corollary 4.18 we conclude that there is a single edge whose addition to T yields a β-threshold graph for T . On the other hand the addition of a single edge to this tree cannot produce a βs -threshold graph for T . In [2], the geometric interpretation of the threshold strong dimension was used to prove Theorem 4.20. For the tree T of Fig. 4.9 we illustrate the embedding used in the proof of this theorem to show that τs (T ) = 2. In particular, Fig. 4.9 shows a {y2 , u3 }-resolved embedding of T that is also an isometric subgraph of P7  P7 . The solid black edges correspond to the edges of T while the dashed black edges are the edges that are added to T to produce a βs -threshold graph of T . Observe that 11 edges were added to T to obtain the βs -threshold graph for T shown in Fig. 4.9. Given what we have learned from trees with small metric dimension (strong dimension), i.e., β ≤ 4 (respectively, βs ≤ 4), we can say that, even though the threshold dimension and the threshold strong dimension of a tree share some similar results, in fact they are quite different invariants. For trees with metric dimension (strong dimension) greater than 4, the question of determining the threshold dimension (respectively, threshold strong dimension) is still open. In the next section, we will discuss bounds on the threshold dimension and the threshold strong dimension for trees. Note that the threshold dimension

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Fig. 4.10 A {w1 , w2 }-resolved embedding ϕ of L18 in P7  P7 . (a) The tree L18 . (b) An embedding ϕ of L18 in P7  P7 . (c) The graph ϕ(L18 ) as an embedding of P7  P7

(threshold strong dimension) of a tree T can exceed 2 if β(T ) ≥ 5 (respectively, βs (T ) ≥ 5). Indeed, it was shown in [11] that τ (K1,6 ) = 3. It follows that τs (K1,6 ) ≥ 3. We recall that βs (K1,6 ) = β(K1,6 ) = 5. It is natural to ask by how much the dimension (strong dimension) of a graph can exceed the threshold dimension (respectively, the threshold strong dimension) of the graph. It was shown in [11] that there are trees with threshold dimension 2 and arbitrarily large metric dimension. It was pointed out in [2] that the same trees also have threshold strong dimension 2. Thus, this gives examples of trees with threshold strong dimension 2 and arbitrarily large strong dimension. We next describe these trees. Let n ≥ 2 be an integer and let L3n be the tree of order 3n obtained by adding two leaves to each vertex on a path of order n. Then β(L3n ) = n and βs (L3n ) = 2n − 1. If w1 and w2 are two leaves adjacent with a vertex of degree 3 in L3n , then there is a {w1 , w2 }-resolved embedding of L3n in Pn+1  Pn+1 . This embedding is illustrated for the case n = 6 in Fig. 4.10. Regarding the question of complexity, linear time algorithms to find the metric dimension (strong metric dimension) of a tree were given in [4, 5, 7, 15] (respectively, [10, 14]). We believe that it is unlikely that there are efficient algorithms for finding the threshold dimension or the threshold strong dimension of trees.

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4.5 Bounds It appears to be a difficult problem to determine the threshold dimension and the threshold strong dimension of a graph. In this section we describe bounds for these invariants for graphs in general, and for trees, as derived in [2, 11, 12].

4.5.1 Bounds on the Threshold Dimension of a Graph One way in which we can establish a bound on the threshold dimension of a given graph G is by finding a subset W of its vertex set such that it is possible to give every vertex of V (G) − W a unique W -neighbourhood, by adding some edges between nonadjacent vertices of G as necessary. Then W is a resolving set of the resulting supergraph H , and since H contains G as a spanning subgraph, it follows that τ (G) ≤ β(H ) ≤ |W |. The Assignment Algorithm, which first appeared in [11, 12], describes how one can assign each vertex of a set P of vertices, disjoint from W , a unique W -neighbourhood by adding edges, as long as |W | is large enough (i.e., |W | ≥ log2 |P |), and as long as no pair of vertices of P share the same nonempty W neighbourhood in G. Algorithm 4.21 (The Assignment Algorithm [11, 12]) Input a graph G of order n, and nonempty sets of vertices W, P ⊆ V (G), such that (i) W ∩ P = ∅, (ii) |W | ≥ log2 (|P |), and (iii) for each pair of vertices u, v ∈ P , we have NW (u) or NW (v) is empty or NW (u) = NW (v). Step 1: Assign an ordering x1 , x2 , . . . , xk to the vertices of W . Let {u1 , u2 , . . . , uj } be the set of all vertices in P with non-empty W neighbourhoods. Order the vertices of P as follows: u1 , u2 , . . . , uj , v1 , v2 , . . . , v . Step 2: Let N = {NW (ui ) : 1 ≤ i ≤ j }. Let S = P(W ) − N, sorted first by cardinality and then lexicographically. (i.e., via shortlex, where x1 < x2 < · · · < xk ). Step 3: In a one-to-one manner, assign to each vi (1 ≤ i ≤ ) the i-th element of S. (This is possible since |S| = 2|W | − j ≥ |P | − j = .) Let the set assigned to vi be denoted Svi . Step 4: Let Ei = {vi xj |xj ∈ Svi } for 1 ≤ i ≤ .

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Let E = ∪i=1 Ei . Output E. Algorithm 4.21 assigns to each vertex of P a set of edges which, if added to G, gives each vertex in P a unique W -neighbourhood. Also, by construction, some vertex is assigned the empty W -neighbourhood unless all W -neighbourhoods were initially non-empty. The shortlex ordering is chosen so that the ordering of S is well-defined and unique. Finally, unless some vertex of P is W -universal, the set W is the last set that could be assigned to a vertex of P , if it is assigned at all. Since no two distinct vertices share the same leaf-neighbourhood in a graph, properties (i), (ii) and (iii) of the Assignment Algorithm hold for the set W of leaves of a graph G, with P = V (G) − W , as long as there are enough leaves in G (i.e., at least log2 |V (G) − W | leaves). To guarantee that there are enough leaves in a graph so that we can use the Assignment Algorithm in this way, the function g : (1, ∞) → N defined by g(x) = d if x ∈ (2d−1 + d − 1, 2d + d], and given in Definition 4.6, was used in [11]. The Assignment Algorithm, together with this definition of g, yields the following upper bound on the threshold dimension of a graph, by taking W to be a set of g(n) leaves of G and letting P = V (G) − W . Lemma 4.22 ([11]) Let G be a graph of order n with  leaves. If  ≥ g(n), then τ (G) ≤ g(n). Theorem 4.1 gives an expression for the dimension of a tree. Moreover, this theorem states that a basis for a tree T (that is not a path) can be constructed by selecting for each exterior major vertex with positive terminal degree a vertex from all but one of its limbs. So if T has  leaves we see that β(T ) ≤ . Theorem 4.23 ([11]) For any tree T of order n > 1, we have τ (T ) ≤ g(n). The bound in Theorem 4.23 is tight for the stars K1,n−1 of order n ≥ 2. The upper bound follows from Theorem 4.23. For the lower bound, it was shown in [4] that if G is a graph of order n and diameter D, then n ≤ D β(G) + β(G). Since a star K1,n−1 of order n ≥ 2 has diameter 2, g(n) ≤ β(K1,n−1 ). Every graph G constructed by adding edges between nonadjacent vertices of K1,n−1 also has diameter 2, so g(n) ≤ β(G) for every supergraph G of K1,n−1 for which K1,n−1 is a spanning subgraph. Hence the lower bound g(n) ≤ τ (K1,n−1 ) also holds. So if a graph has a universal vertex τ (G) ≥ g(n). Now we will present several upper bounds on the threshold dimension of general graphs. The first was established in [12] and the proof employs a process similar to the Assignment Algorithm. Theorem 4.24 ([12]) Let G be a graph of order n. If there exists a set W ⊆ V (G) such that for all x ∈ V (G) − W , x is adjacent to at most  vertices in W , and |W | ≥ g(n) + , then τ (G) ≤ |W |. If we take diam(G) vertices along a diametral path of a graph G and apply Theorem 4.24 to these vertices, we obtain the following corollary.

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Corollary 4.25 ([12]) If G is a graph of order n and diam(G) ≥ g(n) + 3, then τ (G) ≤ diam(G). Corollary 4.25 shows that for a graph G of order n to be β-irreducible, it cannot have the property that β(G) > diam(G) ≥ g(n) + 3. Thus, although it was proved in [12] that β-irreducible graphs of any order and dimension exist, there do not exist β-irreducible graphs for some combinations of diameter, order and dimension. Now we will present bounds on the threshold dimension of a graph G in terms of its chromatic number χ (G). These bounds follow chiefly from the following observation first noted in [12], and the fact that every graph G is a spanning subgraph of a complete multipartite graph with χ (G) partite sets, since one can simply add all possible edges between vertices of distinct colour classes in a proper χ (G)colouring of G. Observation 4.26 Let G be a graph with chromatic number k and order n. Let the sets X1 , X2 , . . . , Xk be the colour classes induced by a proper k-colouring. Then τ (G) ≤ τ (K|X1 |,|X2 |,...,|Xk | ). By Observation 4.26, if we can determine the threshold dimension of the complete k-partite graphs, then we have upper bounds for the threshold dimension of graphs with chromatic number k. Before we establish the threshold dimension of complete k-partite graphs, we need some definitions. Definition 4.27 1. Define the function f : [1, ∞) → N ∪ {0} by f (x) = d if x ∈ [2d−1 + d − 1, 2d + d). 2. For a given integer d ≥ 1, the d lower limit is defined as d = 2d−1 + d − 1. The next result first appeared in [12] and gives the exact value of the threshold dimension of complete multipartite graphs Kx1 ,x2 ,...,xk . The proof involves applying the Assignment Algorithm k times, where each time we apply the algorithm to a subset Wi of each partite set Xi , with Pi = Xi − Wi , where |Wi | = f (xi ) − 1 for |Xi | = xi . Lemma 4.28 ([12]) Let K = Kx1 ,x2 ,...,xk be a complete k-partite graph. Then

τ (K) = ki=1 f (xi ) − c where  0 if no xi = f (xi ) c= 1 otherwise. Lemma 4.26 can be used to establish the following closed-form, sharp upper bound of logarithmic order on the threshold dimension of all k-chromatic graphs, which was first established in [12]. Theorem 4.29 ([12]) Let G be a graph of order n with χ (G) = k. Then τ (G) ≤ k(f (n/k) + 1) − 1.

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Moreover, this bound is sharp for all k. Note that f (n/k) will always be at most log2 (n/k). Thus, with k = 4, we have that k(f (n/k) + 1) − 1 ≤ 4(log2 (n/4) + 1) − 1 = 4(log2 n − 1) − 1. With this observation and the well known fact that planar graphs are 4-colourable, we obtain the following corollary which appeared in [12]. Corollary 4.30 ([12]) Let G be a planar graph of order n. Then τ (G) ≤ 4log2 n − 5.

4.5.2 Bounds on Threshold Strong Dimension of a Graph In this section we establish bounds on the threshold strong dimension for graphs in general and for trees, as derived in [2]. One way in which we can establish an upper bound on the threshold strong dimension of a given graph G is by adding edges to form a supergraph H of G for which βs (H ) ≤ B. We will use the fact that βs (H ) is equal to the minimum cardinality of a vertex covering of the strong resolving graph HSR , in which two vertices u, v are adjacent if and only if u MMD v in H . Similar to our approach for bounding the threshold dimension of a graph G, we start by finding a subset W of its vertex set such that it is possible to give every vertex of V (G) − W a unique W -neighbourhood, by adding some edges between nonadjacent vertices of G as necessary. In addition to these added edges, we also bring the diameter of the resulting supergraph down to 2, by adding edges to make V (G) − W a clique. Then the MMD pairs of vertices in the supergraph H we obtain are precisely the nonadjacent vertices. Thus the strong resolving graph HSR consists of a clique W plus some edges from vertices of W to vertices of V (G)−W , and so W is a vertex cover of HSR , which implies that β(H ) ≤ |W |, and hence τs (G) ≤ |W |. Now we apply the strategy that we just described to the colour classes V1 , V2 , . . . , Vk in a proper k-colouring of a graph G. We start by adding all additional edges between the distinct pairs of colour classes to form a complete k-partite supergraph KV1 ,V2 ,...,Vk of G, with partite sets V1 , V2 , . . . , Vk . For each colour class Vi for which |Vi | > 1, we choose a subset Wi ⊆ Vi such that |Wi | = log2 |Vi |, and add edges via the Assignment Algorithm to give each vertex of Vi − Wi a unique Wi -neighbourhood. Then we add edges to make Vi − Wi complete. Let the resulting supergraph be denoted by H . The set W consisting of all vertices in colour classes of cardinality 1, as well as all vertices of the sets Wi we chose from colour classes of cardinality at least 2, will form a vertex cover of HSR , and thus βs (H ) ≤ |W |, from which we conclude that τs (G) ≤ |W |. By this method we obtain Theorem 4.31 below. Note that if G is a complete graph of order n, then τs (G) = βs (G) = n − 1. Theorem 4.31 first appeared in [2], and gives an upper bound for τs (G) when G is not complete. Theorem 4.31 can be seen as a sort of an analogue to Lemma 4.28 for the threshold strong dimension.

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Theorem 4.31 ([2]) Let G be a non-complete graph with χ (G) = k and let V1 , V2 , . . . , Vk be the colour classes in a proper k-colouring of G, where |V1 | ≤ |V2 | ≤ · · · ≤ |Vk |. 1. If there is an  ≥ 1 such

that |Vi | = 1 for 1 ≤ i ≤  and |Vi | > 1 for  < i ≤ k, then τs (G) ≤  − 1 + ki=+1 log2 |Vi |.

2. If |V1 | ≥ 2, then τs (G) ≤ ki=1 log2 |Vi |. As an example, we establish a bound on the threshold strong dimension of the n-cube Qn , whose vertex set is the set of binary strings of length n, in which two n-bit strings are adjacent if they differ in exactly one position. Let V1 and V2 denote the sets of n-bit strings which contain an odd number of 1’s and an even number of 1’s, respectively. Then V1 and V2 are both independent sets, so V1 and V2 are the colour classes in a proper 2-colouring of Qn , where |V1 | = |V2 | = 2n−1 . Hence Theorem 4.31 implies that τs (Qn ) ≤ 2log2 2n−1  = 2(n − 1). For the sake of comparison, we note that by Theorem 4.29, we obtain an upper bound on the threshold dimension τ (Qn ) ≤ 2((n − 1) + 1) − 1 = 2n − 1. We now present an improved bound on the threshold strong dimension for trees. We obtain this result by letting W be a set of leaves of the tree T of order n. Let  denote the number of leaves in a tree T . If  < log2 n, then TSR ∼ = K , which has vertex covering number  − 1, which is an upper bound on τs (T ). If  ≥ log2 n, we can take a set W of log2 n leaves, and it is possible to apply the Assignment Algorithm to add edges to give each vertex of V (T ) − W a unique W -neighbourhood. Then we add edges to make V (T ) − W a clique. This brings the diameter of the resulting supergraph H down to 2, and so the MMD pairs of vertices are precisely the nonadjacent vertices of H . The vertices of W form a clique in the strong resolving graph HSR of H . All other edges of HSR join vertices of W with vertices in V − W . Thus W is a minimum vertex covering of HSR , and so βs (H ) = |W | = log2 n. Since T is a spanning subgraph of H , it follows that τs (T ) ≤ |W | = log2 n. Thus, no matter how many leaves a tree T of order n has, we have τs (T ) ≤ log2 n. This is stated formally in the following theorem, which may be seen as an analogue of Theorem 4.23 for the threshold strong dimension. Theorem 4.32 ([2]) If T is a tree with n ≥ 2 vertices, then τs (T ) ≤ log2 n.

4.6 Irreducible Structures Recall that a graph G is said to be β-irreducible (βs -irreducible) if β(G) = τ (G) (respectively, βs (G) = τs (G)). As mentioned in the Introduction, paths are the only graphs with threshold dimension and threshold strong dimension equal to 1, and thus they are irreducible. This result implies that all graphs with dimension (strong dimension) 2 are β-irreducible [11] (βs -irreducible [2], respectively). In general, it seems to be difficult to determine whether a graph is irreducible. In [11], it was mentioned that graphs of dimension n − 1 are β-irreducible. In fact,

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graphs with strong dimension n − 1 are precisely the complete graphs. Thus they are also βs -irreducible. Javaid et al. [6] showed that the dimension of the generalized Petersen graphs P (n, 2) is 3 for all n ≥ 5. Later, Mol et al. [12] proved that the graphs in this infinite family are β-irreducible of dimension 3. In the same paper, they also showed that the square of the cycle Cn2 , for every n ≥ 3, is also a β-irreducible graph of dimension 3. In the case of strong dimension, Kratica et al. [8] showed that  βs (P (n, 2)) =

4k + 2 n = 4k + 2, for all k 5k

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and βs (P (4k + 1, 2)) ≤ 5k + 5 for k ≥ 3. Unlike the threshold dimension, the generalized Petersen graph P (4k + 2, 2) is βs -reducible. Computational evidence suggests that the strong dimension of graphs P (4k + 2, 2) can be reduced to 2k + 1 by the addition of 2k + 1 edges: in particular, if we denote the vertices of the outer cycle by u1 , u2 , . . . , u4k+2 and add edges in the set {ui ui+2 : 1 ≤ i ≤ 4k + 2} with subscripts expressed modulo 4k + 2, then it appears that the strong dimension of the resulting graph is 2k + 1. Such a supergraph for P (6, 2) is shown in Fig. 4.11. Here the additional edges (shown in gray) reduce the strong dimension from 6 to 3. Since τ (P (6, 2)) = 3 and τs (P (6, 2)) ≥ τ (P (6, 2)), we see that this graph is a βs -threshold graph for P (6, 2). The Petersen graph, which is known to be β-irreducible of dimension 3, see [16], has strong dimension 8, see [8]. The graph shown on the left in Fig. 4.12 is obtained by adding edges to the Petersen graph, and its strong resolving graph, shown on the right in Fig. 4.12, has vertex covering number 3. Thus τs (P (5, 2)) ≤ 3. Since 3 = τ (P (5, 2)) ≤ τs (P (5, 2)) ≤ 3, we have τs (P (5, 2)) = 3.

4.6.1 Irreducible Graphs of Given Order and Dimension In this section, we describe the construction of an infinite family of β-irreducible (βs -irreducible) graphs from a given β-irreducible (respectively, βs -irreducible) Fig. 4.11 A βs -threshold graph of P (6, 2)

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graph of diameter at most 2. Let G ∨ H denote the join of graphs G and H . The following result was established in [12]. Lemma 4.33 ([12]) If diam(G) ≤ 2, then 1. β(G ∨ K2 ) = β(G) + 1; and 2. β(G ∨ K2 ) ∈ {β(G) + 1, β(G) + 2}. We now prove an analogous result for the strong dimension. Lemma 4.34 If diam(G) ≤ 2, then 1. βs (G ∨ K2 ) = βs (G) + 1; and 2. βs (G ∨ K2 ) ∈ {βs (G) + 1, βs (G) + 2}. Proof Observation: In a graph of diameter at most 2, the MMD pairs are precisely one of the following: • a pair of universal vertices, • a pair of vertices distance 2 apart, or • a pair of twin vertices. Proof of 1 Let H = G ∨ K2 and let x1 , x2 be the pair of vertices joined to every vertex of G. Then H has diameter 2. Let W be a strong basis of H . Since diam(G) = 2, we have dH (u, v) = dG (u, v) for all vertices u, v ∈ V (G). Since x1 and x2 are twins, they form an MMD pair of H , by the above observation. Let u ∈ V (G). Then neither x1 (nor x2 ) is MMD with u in H since x2 (respectively, x1 ) is a neighbour of u that is further in H from x1 (x2 , respectively). Moreover, any pair of vertices that is MMD in G remains MMD in H . Thus HSR = GSR ∪ K2 . Therefore βs (G ∨ K2 ) = βs (G) + 1. Proof of 2 Let H = G ∨ K2 and let x1 , x2 be the two vertices joined to every vertex of G. Then x1 and x2 are universal vertices of H . Since G and H have diameter at most 2 we have dH (u, v) = dG (u, v) for all vertices u, v ∈ V (G). Thus, by the above Observation, GSR ⊆ HSR .

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Case 1 G does not contain a universal vertex. Then x1 and x2 are the only universal vertices of H . The vertex x1 cannot be MMD with any vertex in G since, for each vertex u of G, there exists another vertex v of G such that dH (u, v) = 2 (as u is not universal in G). Since dH (v, x1 ) = 1 we see that x1 has a neighbour which is further away from u. Thus HSR = GSR ∪ K2 . Therefore βs (G ∨ K2 ) = βs (G) + 1. Case 2 If G contains exactly n universal vertices, then GSR has Kn as a component. Since the vertices x1 , x2 in K2 are universal in H , they are both joined, in HSR , to one another and every vertex in the component of GSR isomorphic to Kn . Thus the vertex covering number of HSR exceeds that of GSR by 2. The result now follows.  The result in Lemma 4.34 does not hold if the diameter of the graph G is more than 2. For example, if the graph G is formed by subdividing the edges in the star K1,3 , then the strong dimension βs (G) = 2, but βs (G ∨ K2 ) = 6. Lemma 4.33 was used in [12] to describe the construction of an infinite family of β-irreducible graphs from a β-irreducible graph of diameter at most 2. Theorem 4.35 ([12]) Let G be a β-irreducible graph with diam(G) ≤ 2. Then G ∨ K2 is also β-irreducible, with τ (G ∨ K2 ) = β(G) + 1. Lemma 4.34 can now be used to establish an analogous result for the strong dimension. Theorem 4.36 Let G be a βs -irreducible graph with diam(G) ≤ 2. Then G ∨ K2 is also βs -irreducible, with τs (G ∨ K2 ) = βs (G) + 1. Proof Let H = G ∨ K2 . Since G is βs -irreducible, no edges can be added to G to reduce its strong dimension. The only edge which remains to be considered is the edge between the two vertices in K2 . If this edge is added, it is readily seen that the strong dimension of the resulting graph will either remain the same (if G has no universal vertex) or increase by 1 (if G has a universal vertex). Hence, H is βs -irreducible.  For example, since the path P3 is a βs -irreducible graph of diameter 2 with strong dimension 1, it follows from Theorem 4.36, that the wheel graph W4 is a βs -irreducible graph of strong dimension 2. If G is a β-irreducible graph of order n ≥ 2 and dimension b, then b ∈ {1, 2, . . . , n − 1}. The next result, established in [11], states that for every such pair n, b there is a β-irreducible graph. Theorem 4.37 ([11]) For every integer n ≥ 2, and every integer b ∈ {1, . . . , n−1}, there exists a connected β-irreducible graph of order n and dimension b. We now explain why the constructions used in [11] to prove this result also show that the analogous result holds for the strong dimension. We begin by describing two infinite families of β-irreducible graphs that were used in [11] to establish Theorem 4.37. We show that these graphs are also βs irreducible. For the first of these, we use the function g given in Definition 4.6.

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For every n ≥ 2, let An = Kg(n) and let Bn = Kn−g(n) . Apply the Assignment Algorithm (using the Reverse Shortlex ordering rather than the Shortlex ordering in Step 2) to the disjoint union An ∪ Bn with W = V (An ) and P = V (Bn ). Let E be the edges output by the algorithm. Define Sn = (An ∪ Bn ) + E. Since the Reverse Shortlex ordering is used in the Assignment Algorithm, some vertex of An is assigned the entire set V (Bn ). This vertex is universal in Sn , and we therefore conclude that Sn has diameter at most 2. Then each vertex of An is MMD with some vertex of Bn , since it either has a twin in Bn or it is distance 2 from some vertex of Bn . Also no two vertices from Bn are MMD with one another. Hence Sn has order n, and the vertices of An form a strong resolving set for Sn . By Corollary 4.8 we see that the vertices of An form a strong basis. Corollary 4.8 also implies that no graph of order n can have strong dimension less than g(n). Hence Sn is a βs -irreducible graph. Figure 4.13a shows the graph S8 , with the vertices of A8 coloured white. For the second family of βs -irreducible graphs, described in [12], let b > 1 and s ≥ 1 be integers and let Fb,s be the graph obtained from the disjoint union K2b ∪ Kb ∪ Ps by joining a leaf x1 of the path Ps to a single vertex v of Kb . Let x2 be the other leaf of Ps . Apply the Shortlex Assignment Algorithm to Fb,s with W = V Kb and P = V (K2b ), and let the output be E. Define Sb,s = Fb,s + E.  Note that Sb,s has order 2b + b + s, and that the set V Kb − {v} ∪ {x2 } strongly resolves Sb,s . Finally, by Corollary 4.5, Sb,s is irreducible. The graph S2,3 is shown in Fig. 4.13b, with the vertices of a strong resolving set coloured white. Finally, for a given integer k ≥ 1 and a given graph G, the graph k ∗ G is defined recursively as follows: 1. 1 ∗ G = G. 2. Suppose that (k − 1) ∗ G has been defined for some k > 1. Then k ∗ G = ((k − 1) ∗ G) ∨ G. Thus, for example, the graph k ∗ K2 is the complete k-partite graph with each partite set of size 2.

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Fig. 4.13 The graphs S8 and S2,3 . The vertices of a strong basis for each graph are shown in white. (a) The graph S8 . (b) The graph S2,3

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We now show that the constructions from [11], which illustrate that for every n ≥ 2 and every b ∈ {1, . . . , n − 1} there is a β-irreducible graph of order n with dimension b, are also βs -irreducible with strong dimension b. If b = 1, then Pn is a connected irreducible graph of order n and strong dimension b. From now on, assume that b > 1. We consider three different cases. Case 1: n > 2b + b. Let s = n − 2b + b. Then Sb,s is a connected βs -irreducible graph of dimension b and order n. Case 2: 2b < n ≤ 2b + b. Let k be the smallest non-negative integer such that 2b−k−1 + b − k ≤ n − 2k ≤ 2 b−k + b − k. It was shown in [11] that such a k always exists. Let G = Sn−2k ∨ k ∗ K2 . Note first that G has order n. Further, since 2b−k−1 + b − k ≤ n − 2k ≤ 2b−k + b − k we have g(n − 2k) = b − k. By repeated application of Theorem 4.36, we have τs (G) = βs (G) = βs (Sn−2k ) + k = b. Thus, we have shown that G is a βs -irreducible graph with strong dimension b and order n. Case 3: n ≤ 2b. Let k = n − 1 − b. Since b ≤ n − 1, we must have k ≥ 0. Since n ≤ 2b, we also have k ≤ n − 1 − n/2 ≤ n/2 − 1. Let G = Kn−2k ∨ k ∗ K2 . Note that G can also be obtained from Kn by deleting a matching of size k from the complete graph of order n. Thus GSR ∼ = Kn−2k ∪ kK2 . Since the vertex covering number of this graph is n − 2k − 1 + k = n − k − 1 = b, we see that G has strong dimension b. By repeated application of Theorem 4.36, we conclude that G is in fact a βs -irreducible graph of order n and strong dimension βs (Kn−2k ) + k = n − 2k − 1 + k = b. We have thus proven the following result: Theorem 4.38 ([11]) For every integer n ≥ 2, and every integer b ∈ {1, . . . , n−1}, there exists a connected βs -irreducible graph of order n and strong dimension b.

4.7 Concluding Remarks and Open Problems In this article we surveyed results on the threshold and threshold strong dimension of a graph. We saw that the geometric interpretations for the threshold dimension and threshold strong dimension are useful tools for evaluating these parameters for some graphs. Open Problem 1 Is the problem of finding the threshold dimension (threshold strong dimension) NP-hard? We described an infinite family of trees that contains trees of arbitrarily large dimension and threshold dimension 2. This observation suggests the following problem:

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Open Problem 2 For a given integer b ≥ 2, does there exist an infinite family of graphs that contains graphs of arbitrarily large dimension (strong dimension) and threshold dimension (respectively, threshold strong dimension) b? We described an infinite family of graphs with threshold dimension 2 that we conjecture to contain graphs of arbitrarily large threshold strong dimension. More generally: Open Problem 3 Given integers b ≥ 2 and k ≥ 1, is there a graph G with τ (G) = b and τs (G) ≥ b + k? Even though we observed that β-threshold graphs for trees need not be βs threshold graphs, we do not know of any tree for which the threshold dimension and threshold strong dimension are not equal. Open Problem 4 Are there trees for which the threshold dimension and threshold strong dimension are not the same? Open Problem 5 Are there conditions on a graph G that guarantee τ (G) = τs (G)? Acknowledgments The work of O. R. Oellermann was partially supported by an NSERC Grant CANADA, Grant number RGPIN-2016-05237.

References 1. R. Belmonte, F. V. Fomin, P. A. Golovach and M. S. Ramanujan. Metric dimension of bounded width graphs. In: Mathematical Foundations of Computer Science (Eds., G. Italiano, G. Pighizzini,and D. Sannella) Lecture Notes in Computer Science, Vol. 9235. Springer, Berlin, Heidelberg, 115–126 (2015). 2. N. Benakli, N.H. Bong, S. M. Dueck, L. Eroh, B. Novick, and O. R. Oellermann. The threshold strong dimension of a graph. Discrete Mathematics 344(7) 112402 (2021). 3. J. Cáceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C. Seara and D. R. Wood. On the metric dimension of Cartesian products of graphs. SIAM Journal on Discrete Mathematics 21(2) 423–441 (2007) . 4. G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann. Resolvability in graphs and the metric dimension of a graph. Discrete Applied Mathematics 105 99–113 (2000) . 5. F. Harary and R. Melter. The metric dimension of a graph. Ars Combinatoria 2 191–195 (1976). 6. I. Javaid, M.T. Rahim and K. Ali. Families of regular graphs with constant metric dimension. Utilitas Mathematica 65 21–33 (2008). 7. S. Khuller, B. Raghavachari, and A. Rosenfeld. Landmarks in graphs. Discrete Applied Mathematics 70 200–207 (1996). ˇ 8. J. Kratica, V. Kovaˇcevic´c-Vujˇci´c and M. Cangalovi´ c. The strong metric dimension of some generalized Petersen graphs. Applicable Analysis and Discrete Mathematics 11 1–10 (2017). 9. D. Kuziak, M. L. Puertas, J. A. Rodríguez-Velázquez and I. G. Yero. Strong resolving graphs: The realization and the characterization problems. Discrete Applied Mathematics 236 270–287 (2018). 10. T. May and O. R. Oellermann. The strong metric dimension of distance-hereditary graphs. Journal of Combinatorial Mathematics and Combinatorial Computing 76 59–73 (2011). 11. L. Mol, M. J. H. Murphy and O. R. Oellermann. The threshold dimension of a graph. Discrete Applied Mathematics 287 118–133 (2020) (also available at arXiv:2001.09168 (2020)).

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12. L. Mol, M. J. H. Murphy O. R. Oellermann. The threshold dimension of a graph and irreducible graphs. To appear in Discussiones Mathematicae Graph Theory (also available at: arXiv:2002.11048 (2020). doi:10.7151/dmgt.2359 13. O. R. Oellermann and J. Peters-Fransen. The strong metric dimension of graphs and digraphs. Discrete Applied Mathematics 155(3) 356–364 (2007). 14. A. Sebö and E. Tannier. On metric generators of graphs. Mathematics of Operations Research 29(2) 383–393 (2004). 15. P. J. Slater. Leaves of trees. Congressus Numerantium. 14 549–559 (1975). 16. G. Sudhakara and A. R. Hemanth Kumar. Graphs with metric dimension two – a characterization. World Academy of Science, Engineering and Technology 36 622–627 (2009).

Chapter 5

Symmetry Parameters for Mycielskian Graphs Debra Boutin, Sally Cockburn, Lauren Keough, Sarah Loeb, K. E. Perry, and Puck Rombach

5.1 Introduction A coloring of the vertices of a graph G with colors from {1, . . . , d} is called a d-distinguishing coloring if no nontrivial automorphism of G preserves the color classes. A graph is called d-distinguishable if it has a d-distinguishing coloring. The distinguishing number of G, denoted Dist(G), is the smallest number of colors necessary for a distinguishing coloring of G. Albertson and Collins introduce graph distinguishing in [4]. Independently, in [6], Babai defines the same idea, but calls it an asymmetric coloring. Here, we continue the terminology of Albertson and Collins. A substantial amount of work in graph distinguishing in the last few decades proves that, for a large number of graph families, all but a finite number of

D. Boutin () · S. Cockburn Department of Mathematics and Statistics, Hamilton College, Clinton, NY, USA e-mail: [email protected]; [email protected] L. Keough Department of Mathematics, Grand Valley State University, Allendale, MI, USA e-mail: [email protected] S. Loeb Department of Mathematics and Computer Science, Hampden-Sydney College, Hampden-Sydney, VA, USA e-mail: [email protected] K. E. Perry Department of Mathematics, Soka University of America, Aliso Viejo, CA, USA e-mail: [email protected] P. Rombach Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_5

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members are 2-distinguishable. Examples of such families of finite graphs include: hypercubes Qn with n ≥ 4 [8], Cartesian powers Gn for a connected graph G = K2 , K3 and n ≥ 2 [2, 20, 22], and Kneser graphs Kn:k with n ≥ 6, k ≥ 2 [3]. Examples of such families of infinite graphs include: the denumerable random graph [21], the infinite hypercube [21], and denumerable vertex-transitive graphs of connectivity 1 [26]. In 2007, Imrich asked in personal communication if, within 2-distinguishable colorings, we could find the minimum number of times a second color must be used. In response in [10], Boutin defines the cost of 2-distinguishing G, denoted ρ(G), to be the minimum size of a color class over all 2-distinguishing colorings of G. Some of the graph families with known or bounded 2-distinguishing cost are hypercubes with log2 n+1 ≤ ρ(Qn ) ≤ 2log2 n−1 for n ≥ 5 [10], Kneser graphs with ρ(K2m −1:2m−1 −1 ) = m+1 [13], and the Cartesian product of K2m and an asymmetric graph on m vertices H with ρ(K2m H ) = m · 2m−1 [16]. A determining set is a useful tool in finding the distinguishing number and, when relevant, the cost of 2-distinguishing. A subset S of V (G) is a determining set for G if the only automorphism that fixes the elements of S pointwise is the trivial automorphism. Equivalently, S is a determining set for G if, whenever ϕ and ψ are automorphisms of G with ϕ(s) = ψ(s) for all s ∈ S, then ϕ = ψ [9]. The determining number of a graph G, denoted Det(G), is the size of a smallest determining set. Intuitively, if we think of automorphisms of a graph as allowing vertices to move among their relative positions, the determining number is the fewest pins needed to “pin down” the graph. For some graph families, we only have bounds on the determining number. For instance, for Kneser graphs, log2 (n+1) ≤ Det(Kn:k ) ≤ n−k, with both bounds sharp [9]. However, there are families for which we know the determining numbers of its members exactly. For example, for hypercubes Det(Qn )=log2 n+1 [11], and for generalized Petersen graphs Det(G(n, k)) = 2 if (n, k) = (4, 1), (5, 2), (10, 3) and Det(G(n, k)) = 3 otherwise [18]. Though distinguishing numbers and determining numbers were introduced by different people and for different purposes, they have strong connections. Albertson and Boutin show in [3] that if G has a determining set S of size d, then giving each vertex in S a distinct color from {1, . . . , d} and coloring the remaining vertices with a (d + 1)st color yields a (d + 1)-distinguishing coloring of G. Thus, Dist(G) ≤ Det(G)+1. Furthermore, for a 2-distinguishing coloring of G, Boutin [12] observes that the requirement that only the trivial automorphism preserves the color classes setwise means that only the trivial automorphism preserves them pointwise. Consequently, each of the color classes in a 2-distinguishing coloring is a determining set for the graph, though not necessarily of minimum size. Thus, if G is 2-distinguishable, then Det(G) ≤ ρ(G). In this paper, we refer to the distinguishing number, determining number, and cost of 2-distinguishing, collectively as the symmetry parameters of a graph. In [24], Mycielski introduces a construction that takes a finite simple graph G and produces a larger graph μ(G) called the (traditional) Mycielskian of G, with the same clique number and a strictly larger chromatic number. In particular, the

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Myielskian construction can be used to produce families of triangle-free graphs with increasing chromatic number. A formal definition can be found in Sect. 5.2. The generalized Mycielskian of graph G, denoted μt (G), is defined by Stiebitz in [27] (cited in [28]) to construct families of graphs with arbitrarily large odd girth and increasing chromatic number. Independently, Ngoc defines it in [29] (cited in [30]). The fundamental difference between the Mycielskian and the generalized Mycielskian of a graph is that the former has 1 level of independent vertices, while the latter has t levels of independent vertices, for some t ≥ 1. The definition of μt (G) appears in Sect. 5.2. One strength of the Mycielskian constructions is their ability to build large families of graphs with a given parameter fixed and other parameters strictly growing. In the last few decades, this has motivated significant work on parameters of μt (G) in terms of the same parameters for G. See for example [1, 7, 14, 15, 17, 19, 23, 25]. In this chapter, for finite simple graphs G, we compare symmetry parameters for G and μt (G). This chapter is organized as follows. Section 5.2 gives the definitions of the Mycielskian constructions. Section 5.3 contains a review previous results on symmetry parameters of Mycielskians of graphs, as well as new results on symmetry parameters of the Mycielskian of graphs with isolated vertices. Lastly, Sect. 5.4 lists some open problems.

5.2 Mycielskian Construction The following is Mycielski’s construction for taking a finite simple graph G, and producing a larger graph called the Mycielskian of G. Definition 5.1 Let G be a graph with vertex set V (G) = {v1 , . . . , vn }. The Mycielskian of G, denoted μ(G), has vertex set V (μ(G)) = {v1 , . . . , vn , u1 , . . . un , w}. For each edge vi vj ∈ E(G), μ(G) has edges vi vj , ui vj and vi uj ; in addition, ui w ∈ E(μ(G)) for all 1 ≤ i ≤ n. Note that μ(G) contains G as an induced subgraph. We refer to the vertices v1 , . . . , vn in μ(G) as original vertices and the vertices u1 , . . . , un as shadow vertices. The vertex w is called the root. See Fig. 5.1 for illustrations of μ(K3 ) and μ(K2 + 3K1 ). The Mycielskian construction can of course be iterated. Beginning with μ1 (G) = μ(G), define μk (G) inductively as μk (G) = μ(μk−1 (G)). Mycielski uses this iterated process to define the classical Mycielskian graphs, Mk = μk (K2 ). Note that indices for this family may differ by publication. This graph family proves the existence of triangle-free graphs with arbitrarily large chromatic number.

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Fig. 5.1 Top: K3 and μ(K3 ), drawn with concentric levels with the root in the middle. Bottom: K2 + 3K1 and μ(K2 + 3K1 ), drawn with vertical levels with the root at the top

The following is the definition of a generalized Mycielskian of G, also known as a cone over G. Definition 5.2 Let G be a graph with vertex set V (G) = {v1 , . . . , vn } and let t ≥ 1. The generalized Mycielskian of the graph, denoted μt (G), has vertex set V (μt (G)) = {u01 , . . . , u0n , u11 , . . . , u1n , . . . , ut1 , . . . , utn , w}. For each edge vi vj in G, the graph μt (G) has edge vi vj = u0i u0j , as well as edges t and usj us+1 usi us+1 j i , for 0 ≤ s < t. Finally, μt (G) has edges ui w for 1 ≤ i ≤ n.

We say that vertex usi is at level s. In addition, we make the identification u0i = vi and refer to the vertices at level zero as original vertices and the vertices at level s ≥ 1 as shadow vertices and to w as the root. Notice that the root is only adjacent to the shadow vertices at level t. We call level t the top level. Observe that μ1 (G) = μ(G). Thus, for ease of notation, we omit the subscript when t = 1. See Fig. 5.2 for illustrations of μ2 (K3 ) and μ2 (K2 + 3K1 ). To help understand the structure of the Mycielskian and generalized Mycielskian, it is useful to understand the degrees of the vertices. Denote the degree of vertex v ∈ V (G) by degG (v) and the degree of x ∈ V (μ(G)) by degμ(G) (x). By the generalized Mycielskian construction, for each original vertex u ∈ V (G), degμt (G) (u) = 2 degG (u). Further, for 1 ≤ i ≤ t − 1, for the shadow of u at level i, degμt (G) (ui ) = 2 degG (u). However, for the shadow of u at the top level t, degμt (G) (ut ) = degG (u) + 1. Finally, degμt (G) (w) = |V (G)|. To address symmetry parameters of μt (G), it is useful to be able to discuss the automorphisms of μt (G) in terms of the automorphisms of G. Boutin, Cockburn, Keough, Loeb, Perry and Rombach prove in [14, 15] that if G is not a star graph, that is, not K1,m for any m ≥ 0, then each automorphism of μt (G) fixes the root,

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Fig. 5.2 Top: K3 , μ2 (K3 ), drawn with concentric levels with the root in the middle. Bottom: K2 + 3K1 and μ2 (K2 + 3K1 ), drawn with vertical levels with the root at the top

preserves the levels of the vertices, and restricts to an automorphism of G. This is stated formally in the following lemma. Lemma 5.3 ([14, 15]) Let G = K1,m for any m ≥ 0, and suppose that G has n vertices and no isolated vertices. Let t ≥ 1, and let # α be an automorphism of μt (G). Then, (i) # α (w) = w, (ii) # α preserves the level of vertices, that is, # α ({us1 . . . , usn }) ⊆ {us1 . . . , usn } for all 0 ≤ s ≤ t, and (iii) # α restricted to {u01 , . . . , u0n } = {v1 , . . . , vn } is an automorphism of G.

5.3 Distinguishing and Determining Mycielskians In [5], Alikhani and Soltani show that the classical Mycielskian graphs defined in Sect. 5.2 satisfy Dist(Mk ) = 2 for all k ≥ 2. To find symmetry parameters of Mycielskians of arbitrary graphs, they consider the role of twin vertices. Two vertices in a graph are said to be twins if they have the same open neighborhood, and a graph is said to be twin-free if it does not contain any twins. In particular, Alikhani and Soltani prove that if G is twin-free with at least two vertices, then Dist(μ(G)) ≤ Dist(G)+1. Further, in [5] the authors conjecture that for all but a finite number of connected graphs G with at least 3 vertices, Dist(μ(G)) ≤ Dist(G). In [14], Boutin, Cockburn, Keough, Loeb, Perry, and Rombach prove the Alikhani and Soltani conjecture, and follow up in [15] by studying the determining number and cost of 2-distinguishing for μ(G) and μt (G). The main results are summarized in Theorems 5.4–5.6.

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Theorem 5.4 ([14]) Let G = K1 , K2 be a graph with  ≥ 0 isolated vertices. If t > Dist(G), then Dist(μt (G)) = t; otherwise, Dist(μt (G)) ≤ Dist(G). Theorem 5.5 ([15]) Let G be a twin-free graph with no isolated vertices such that Det(G) ≥ 2. Then for t ≥ Det(G) − 1, (i) Dist(μt (G)) = 2, (ii) Det(μt (G)) = Det(G), and (iii) ρ(μt (G)) = Det(G). When G has twins, but no isolated vertices, the same authors characterize the determining number of μt (G). To understand this result, we define a minimum twin cover T of G as a set consisting of all but one vertex from each set of mutually twin vertices. See Sect. 5.3.2 for more details. Theorem 5.6 ([15]) Let G be a graph with no isolated vertices. Let T be a (possibly empty) minimum twin cover of G. Let t ≥ 1. (i) If G = K2 , then Det(G) = 1 and Det(μt (G)) = 2. (ii) If G = K2 , then Det(μt (G)) = t|T | + Det(G). As seen in the results of [14, 15, 19], isolated vertices can play an important role when investigating parameters of Mycielskians and generalized Mycielskians. If a graph G has isolated vertices, then μt (G) has one component that consists of the generalized Mycielskian of the graph induced by the non-isolated vertices of G, and the top-level shadows of isolated vertices as pendant vertices adjacent to the root. However, μt (G) also has t isolated vertices for every isolated vertex in G. More formally, if G = H + K1 , where H is a graph with at least one edge and no isolated vertices, then μt (G) = C + tK1 , where C is a connected component consisting of μt (H ) with  additional pendant vertices adjacent to w. See Fig. 5.3 which shows the generalized Mycielskians of G = K3 + 3K1 when t = 1 and 2. We next provide a lemma that helps us understand the automorphisms and determining numbers of graphs with isolated vertices. We then investigate the role that isolated vertices play when considering the distinguishing and determining number of graphs with and without twin vertices. Lemma 5.7 Let H be a graph with at least one edge and no isolated vertices. For t ≥ 1, let C be a copy of μt (H ) with an additional  ≥ 1 pendant vertices adjacent

Fig. 5.3 The graphs G = K3 + 3K1 , μ(G) and μ2 (G)

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to w: x1 , x2 , . . . , x . If S is a minimum size subset of V (μt (H )) \ {w} such that S ∪ {w} is a determining set for μt (H ), then S ∪ {x2 , . . . , x } is a minimum size determining set for C. Proof Let S be a subset of V (μt (H )) \ {w} of minimum size such that S ∪ {w} is a determining set for μt (H ). Observe that C has exactly  vertices of degree 1 and w is the only vertex adjacent to all of these vertices. Thus, w is fixed by every automorphism of C. If  = 1, then since C has only one pendant vertex, it must be fixed under every automorphism. If  ≥ 2, then any automorphism of C that fixes {x2 , . . . , x } must also fix x1 . In both cases, any α ∈ Aut(C) that fixes {x2 , . . . , x } can act nontrivially on only vertices in V (μt (H )) \ {w}. If α also fixes the vertices in S, then by the assumption that S ∪ {w} is a determining set for μt (H ), α must be the identity. Hence, S ∪ {x2 , . . . , x } is a determining set for C. To show minimality, suppose that B is a minimum size determining set for C such that |B| < |S ∪ {x2 , . . . , x }|. Since x1 , . . . , x are mutually twin, we can assume without loss of generality that {x2 , . . . , x } ⊆ B. As already noted, any automorphism of C fixing these vertices also fixes w and x1 , and so by the assumption of minimality, w, x1 ∈ / B. Then B  = B \ {x2 , . . . , x } is a subset of  V (μt (H )) \ {w} and |B | < |S|. Now, let α be any automorphism of μt (H ) fixing B  ∪ {w}. Since α can be extended to an automorphism of C fixing B, a determining set for C, it follows that α restricted to μt (H ) must be the trivial automorphism and so B  ∪ {w} is a determining set for μt (H ), a contradiction.  Notice that Lemma 5.7 shows that if G = H + K1 and w is not in any minimum determining set for μt (H ), then Det(μt (H )) = Det(C). By Lemma 5.3, if H = K1,m for any m ≥ 0, then all automorphisms of μt (H ) fix w. Thus, if H is not a star graph, then Det(μt (H )) = Det(C).

5.3.1 An Isolated Vertex in Twin-Free G Recall that any collection of vertices are said to be twins if they all share the same open neighborhood. Since there are automorphisms that permute twin vertices while leaving other vertices fixed, the presence of twin vertices affects the determining and distinguishing numbers. Thus, we address graphs with twin vertices separately in Sect. 5.3.2. In this section, we extend results from both [14] and [15] on the determining and distinguishing numbers of twin-free graphs to twin-free graphs with an isolated vertex. Note that a twin-free graph can have at most one isolated vertex. In [14], Boutin, Cockburn, Keough, Loeb, Perry and Rombach show that Dist(μt (G)) ≤ Dist(G) for most graphs G. However, in [15] the same authors show that for most graphs equality holds for the determining number of twin-free graphs with no isolated vertices. In particular, they show the following.

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Theorem 5.8 ([15]) Let G be a twin-free graph with no isolated vertices and let t ≥ 1. (i) If G = K2 then, Det(G) = 1 and Det(μt (G)) = 2. (ii) If G = K2 , then any minimum size determining set for G is a minimum size determining set for μt (G) and Det(μt (G)) = Det(G). The theorem below gives analogous results for twin-free graphs having exactly one isolated vertex. Theorem 5.9 Let G be a twin-free graph and let t ≥ 1. (i) If G = K1 , then Det(G) = 0 and Det(μt (G)) = t. (ii) If G is of the form H + K1 for some graph H with at least one edge, then Det(μt (G)) = Det(G) + t − 1. Proof If G = K1 , then Det(G) = 0 since the only automorphism of G is the trivial one. Note that μt (K1 ) = K2 + tK1 and a minimum size determining set consists of one vertex in K2 and t − 1 of the isolated vertices. Next, suppose G = K1 is of the form H +K1 , where H does not have an isolated vertex because G is twin-free. Then Det(G) = Det(H ). Moreover, μt (G) = C + tK1 , where C is a connected component consisting of μt (H ) with an additional pendant vertex adjacent to w. Thus, Det(μt (G)) = Det(C) + t − 1. It therefore suffices to show that Det(C) = Det(H ). We divide into two cases. If H = K2 , then μt (K2 ) = C2t+3 . As noted earlier, any two vertices on an odd cycle constitute a minimum size determining set. Hence if y = w is any vertex on the cycle, we can apply Lemma 5.7 to conclude that S = {y} is a minimize size determining set for C, which implies Det(C) = 1 = Det(K2 ). Now, suppose H = K2 . Since in addition G is twin-free, H = K1,m for any m ≥ 0. Since H is not a star graph, by Lemma 5.3, every automorphism of μt (H ) fixes w, which means that for every S ⊆ V (μt (H )) \ {w}, S ∪ {w} is a determining set for μt (H ) only if S is. By Lemma 5.7, then, if S is a minimum size determining set for μt (H ), it is also one for C and so Det(C) = Det(μt (H )). Applying Theorem 5.8(ii) to H gives Det(μt (H )) = Det(H ) and we are done.  Theorems 5.8 and 5.9 effectively tell us that if G = K1 or K2 and G is twin-free, then only two behaviors are possible. Corollary 5.10 If G = K1 , K2 is a twin-free graph, then for all t ≥ 1,

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Det(μt (G)) =

 Det(G)

if G does not have an isolated vertex;

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.

The theorem below, from [15], addresses the distinguishing number of Mycielskians of twin-free graphs with no isolated vertices. It asserts that for t sufficiently large, these graphs are 2-distinguishable and provides information on the cost of 2distinguishing. Note that, if Det(G) = 1, then Dist(μt (G)) = 2 and ρ(μt (G)) = 1. Theorem 5.11 ([15]) Let G be a twin-free graph with no isolated vertices. If Det(G) = k ≥ 2, then the following hold. (i) If log2 (k + 1) − 1 ≤ t, then Dist(μt (G)) = 2 and ρ(μt (G)) ≤

(k + 1)log2 (k + 1) . 2

(ii) If t ≥ k − 1, then Dist(μt (G)) = 2 and ρ(μt (G)) = k. In (i), the bound on t and on ρ(μt (G)) are both sharp. If a twin-free graph G has one isolated vertex, then μt (G) has t isolated vertices, which means that for t ≥ 2, μt (G) is no longer twin-free. As shown in Theorem 5.9, this causes the determining number to grow linearly with t. In any distinguishing coloring, mutually twin vertices must receive different colors and so it is not surprising that the distinguishing number also grows linearly with t. Theorem 5.12 Let G be a twin-free graph of the form H + K1 with Det(G) = k ≥ 1. If t ≥ log2 (k + 1) − 1, then Dist(μt (G)) = max(2, t). If in addition t = 1 or 2, then ρ(μt (G)) = k + t − 1. Proof As noted in the proof of Theorem 5.9, μt (G) = C +tK1 where C consists of μt (H ) with one pendant vertex adjacent to w. Since H is twin-free with no isolated vertices, we can apply Theorem 5.5 to conclude that Dist(μt (H )) = 2. We can extend any 2-distinguishing coloring of μt (H ) to a 2-coloring of C by coloring the pendant vertex either of the two colors. Every automorphism of C must fix the pendant vertex; every automorphism α of C that fixes these 2 colors classes of C therefore restricts to an automorphism of μt (H ) that fixes the 2 colors classes in the 2-distinguishing coloring. By definition, the restriction of α acts as the identity on μt (H ) and therefore α is the identity on C. Hence this 2-coloring is distinguishing on C and so Dist(C) ≤ 2. From the proof of Theorem 5.9, Det(C) = Det(H ) = Det(G) ≥ 1. This implies that C has nontrivial automorphisms and so Dist(C) ≥ 2. Thus, Dist(C) = 2. A distinguishing coloring for μt (G) = C + tK1 must assign a different color to each isolated vertex. By reusing two colors used in a 2-distinguishing coloring of C, we need at most max(0, t − 2) additional colors. If t = 1 or 2, then μt (G) is 2-distinguishable and we can compute the cost. By Theorem 5.5 and the fact that Det(H ) = Det(G) = k, we know ρ(μt (H )) = k. Let

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S ⊂ V (μt (H )) be a set of k vertices in a smallest color class of a 2-distinguishing coloring, and let c1 be the color used for S, with c2 being the other color. Since the pendant vertex in C is fixed by any automorphism of C, we can color it with c2 . Thus, ρ(C) = ρ(μt (H )) = k. If t = 1, then we can also color the one isolated vertex in μ1 (G) = μ(G) = C + K1 with c2 and so ρ(μt (G)) = k. If t = 2, then we must use both colors on the isolated vertices and so ρ(μt (G)) = k + 1. 

5.3.2 Isolated Vertices in G with Twins For an analysis of determining numbers of Mycielskians of graphs with twin vertices and without isolated vertices, see [15]. Here we consider graphs with twin vertices that may have isolated vertices. The proofs in this section are similar to [15], and we begin with a more condensed version of the background here. The following useful observation about the relationship between twin vertices in G and those is μt (G) appears in [15]. Observation 5.13 ([15]) If original vertices vi and vj are twins in G, then usi and usj are twins in μt (G) for all 0 ≤ s ≤ t. Similarly, if {usi , usj } are twins in μt (G) for some 0 ≤ s ≤ t, then they are twins for all 0 ≤ s ≤ t; in particular, {vi , vj } are twins in G. Let x ∼ y be the equivalence relation on V (G) that indicates that x and y are $ has as its vertices the twin vertices. The quotient graph for relation ∼, denoted G, set of equivalence classes of the form [x] = {y ∈ V (G) | x ∼ y} with [x] adjacent $ if and only if there exist p ∈ [x] and q ∈ [z] such that pq ∈ E(G). We to [z] in G see that NG $ ([x]) = {[z] | z ∈ NG (x)}. $ is twin-free, and, if G is twin-free, then G $ = G. Furthermore, note that G Since automorphisms preserve neighborhoods, every automorphism α of G $ given by $ induces an automorphism $ α of G α ([x]) = [α(x)]. However, there may be $ automorphisms of G that are not of this form, as is shown in Fig. 5.4 [15] where the only nontrivial automorphism of G is the one interchanging the twin vertices x and $ = P4 has a reflectional nontrivial automorphism. y, but G $ and H $ to denote the quotient graphs Throughout the rest of this section, we use G of graphs G and H , respectively, and $ α to denote an automorphism of a quotient graph. Vertex sets with tilde notation represent subsets of vertices in a quotient $ = {[x] ∈ V (G) $ | x ∈ B}. graph. In particular, if B ⊆ V (G), then B We call a minimum size subset of V (G) containing at least one vertex from every pair of twin vertices a minimum twin cover. If G has no twins, then a minimum twin cover is the empty set. Every determining set of a graph G must contain all but one vertex in every collection of mutual twins, i.e. a minimum twin cover. Thus, if a minimum twin cover is a determining set, it must be a minimum size determining

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$ [15] Fig. 5.4 A graph G and its quotient graph G Fig. 5.5 A graph G for which no minimum twin cover is determining [15]

set. In this case, every minimum twin cover is a minimum size determining set, since all minimum twin covers have the same cardinality. To see that this is true, note that a minimum twin cover contains all but one vertex from every equivalence class [x], x ∈ V (G). In other words, every minimum twin cover has cardinality $ Figure 5.5 shows an example of a graph in which minimum twin |V (G)| − |V (G)|. covers are not determining sets. The following three results, from [15], establish relationships between minimum twin covers of G, minimum determining sets for G, and minimum determining sets $ for G. Corollary 5.14 Let G be a graph. If S is a determining set for G, then $ S is a $ determining set for G. Observe that a superset of a determining set is still a determining set. Therefore, $ is a determining set for G, $ then T$ ∪ D $ is a determining set for G $ containing T$. if D $ Similarly, if S is a determining set for G containing T , then S is a determining set $ for G. Theorem 5.15 ([15]) Let T be a minimum twin cover of G and let $ S be a $ containing T$. Let R = {x ∈ V (G) | [x] ∈ $ determining set for G S \ T$}. Then S = T ∪ R is a determining set for G. Furthermore, if $ S is of minimum size among

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$ that contain T$, then S is a minimum size determining set for determining sets for G G. In particular, if $ S = T$, then T is a minimum size determining set for G. This observation allows us to see that Theorem 5.15 yields natural bounds on Det(G) in $ terms of |T | and Det(G). Corollary 5.16 Let T be a minimum twin cover of G. Then $ |T | ≤ Det(G) ≤ |T |+ Det(G), with both bounds sharp. To use these results in the context of generalized Mycielskian constructions, we now investigate how applying the generalized Mycielskian construction affects $ and the size of a minimum twin cover as well as the relationship between Det(G)  Det(μ t (G)). This is done in the next two lemmas. In [15], it is shown that if T is a minimum twin cover in a graph G with no isolated vertices, the set consisting of vertices in T and all of their shadows, Tt = {usi | vi ∈ T , 0 ≤ s ≤ t}, is a minimum twin cover of μt (G) of size (t+1)|T |. If G has isolated vertices, then an extra t − 1 vertices need to be added to T$ to obtain a minimum twin cover of μt (G). This is because if G has  ≥ 1 isolated vertices, only  − 1 of them are in any minimum twin cover of G. Thus, there is always one isolated vertex in G not in T . If u is the isolated vertex in G not in the minimum twin cover, μt (G) has t − 1 shadows of u that are mutually twin with the other isolated vertices. This behavior is different from non-isolated vertices, which cannot have twins that are on different levels. Lemma 5.17 Let T be a minimum twin cover of G. If G has at least one isolated vertex and u is the isolated vertex that is not in T , then for t ≥ 1, Tt ∪ {us | 1 ≤ s ≤ t−1} = {usi | vi ∈ T , 0 ≤ s ≤ t} ∪ {us | 1 ≤ s ≤ t−1} is a minimum twin cover of μt (G) of size (t+1)|T |+t−1. Proof First suppose G = K1 for  ≥ 1. Then μt (G) = K1, + tK1 . Since T must contain −1 of the isolated vertices, the set Tt = {usi | vi ∈ T , 0 ≤ s ≤ t} contains t (−1) of the isolated vertices and −1 of the leaves of the K1, . The set {us | 1 ≤ s ≤ t−1} contains an additional t−1 of the isolated vertices in μt (G). Thus, their union gives a minimum twin cover of μt (G). Next, suppose G = H + K1 , where H has no isolated vertices and  ≥ 1. Then μt (G) = C+tK1 where C is μt (H ) with an additional  pendant vertices adjacent to w. By Observation 5.13 and the fact that the isolated vertices are mutually twin in G, every pair of twins in C has at least one member in Tt . Furthermore, since

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T excludes exactly one vertex from each set of mutual twins in G, Tt ∩ V (C) is a minimum twin cover of C. Finally, a minimum twin cover of μt (G) must contain t − 1 of the isolated  vertices. The sets Tt \ V (C) and {us |1 ≤ s ≤ t−1} give these. If a graph G has twins but no isolated vertices, then it is shown in [15] that the process of applying the canonical quotient map commutes with the process of applying the generalized Mycielskian construction; that is,  $ =μ μt (G) t (G). However, this is not the case for graphs with isolated vertices. As before, an extra t − 1 vertices are needed. $ Then for Lemma 5.18 Let G be a graph with isolated vertices such that G = G. t ≥ 1,  $ =μ μt (G) t (G)+(t−1)K1 . Proof Let G = H + K1 , where H has no isolated vertices and  ≥ 1. Then μt (G) = C + tK1 , where C is a connected graph consisting of a copy of μt (H )  $ + K1 . It is straightforward with  pendant vertices adjacent to w. Thus μ t (G) = C  $ $ to see that C consists of a copy of μt (H ) = μt (H ) with a single extra pendant vertex adjacent to [w]. $=H $ +K1 , where H $ has no isolated vertices. So μt (G) $ is a On the other hand, G $ connected graph consisting of a copy of μt (H ) with a single pendant vertex adjacent  $ =μ $ plus t isolated vertices. Thus, μt (G)  to [w], which is C, t (G)+(t−1)K1 . In [15], it is shown that if a graph G has twins, then we can find the determining $ that contains T$. The result is the number of μt (G) by using a determining set in G following. Theorem 5.19 ([15]) Let G be a graph with twins and no isolated vertices. Let T be a minimum twin cover of G. Then for t ≥ 1, Det(μt (G)) = t|T | + Det(G). If G has isolated vertices, then by Lemma 5.17, a minimum twin cover of μt (G) requires an extra t − 1 vertices beyond the vertices in a minimum twin cover of G and their shadows, and so it is not surprising that its determining number must also increase by t − 1. Theorem 5.20 Let G be a graph with twins and isolated vertices and let T be a minimum twin cover of G. Then for t ≥ 1, Det(μt (G)) = t|T | + Det(G) + t − 1.

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$ Then T contains  − 1 Proof First suppose that G = K1 with  ≥ 2 since G = G. of the isolated vertices of G; such a set is also a minimum determining set for G and so Det(G) = |T |. Observe that μt (G) = K1, + tK1 . It is clear that a minimum determining set for μt (G) consists of  − 1 of the pendant vertices of K1, , together with t − 1 of the isolated vertices. It follows that Det(μt (G)) = (t + 1) − 2 = (t + 1)|T | + t − 1 = t|T | + Det(G) + t − 1. Now suppose G = H + K1 with  ≥ 1 and where H has at least one edge and no isolated vertices. By construction, μt (G) = C + tK1 , where C consists of μt (H ) with  pendant vertices adjacent to w. Then, Det(μt (G)) = Det(C) + Det(tK1 ) = Det(C) + t − 1. Since H has no isolated vertices, a minimum twin cover T of G must consist of a minimum twin cover TH of H plus  − 1 of the isolated vertices. Thus, |T | = |TH | +  − 1. In the following we show that Det(C) = t|TH | + Det(H ) +  − 1. We divide into two cases, depending on whether H is a star graph of the form K1,m for m ≥ 1. $ = H $ + K1 . Let A $ be a First suppose H = K1,m . Since G = H + K1 , G $ containing T$ and let R = {x ∈ V (G) | [x] ∈ A\ $ T$}. minimum determining set for G Notice that, if  = 1, one isolated vertex is not in T and its equivalence class is not $ and if  ≥ 2, the isolated vertex in G $ is in T$. Thus, R ⊆ V (H ). in A $ is a minimum size determining set for G $ containing T$, A $ ∩ V (H $) = Since A $ % R ∪ T% is a minimum size determining set for H containing T . By the proof of H H Theorem 5.19, we then get that R ∪ TH is a determining set for μt (H ) and thus, Det(μt (H )) = (t + 1)|TH | + |R| = t|TH | + Det(H ). Now, since C is a copy of μt (H ) with  pendant vertices adjacent to w, any minimum determining set for C must include −1 of the pendant vertices. It follows that Det(C) ≤ Det(μt (H )) +  − 1. Since H = K1,m , any automorphism of μt (H ) fixes w by Lemma 5.3. Thus, any minimum determining set for μt (H ) does not contain w. Thus, by Lemma 5.7, Det(C) = Det(μt (H )) +  − 1 = t|TH | + Det(H ) +  − 1. Now suppose that H is a star graph. We begin with the case that H = K1,m $ = K2 . A minimum twin cover T of G consists of m − 1 of for m ≥ 2. Then H the pendant vertices in H and  − 1 of the isolated vertices. In particular, TH = T ∩ V (H ) is a minimum twin cover of H that is also a determining set for H . Thus, Det(H ) = |TH |. Since H has no isolated vertices, (TH )t is a minimum twin cover of μt (H ). We next show that (TH )t is also a determining set for μt (H ).

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In μt (H ) the minimum twin cover contains m − 1 vertices on each level. Let v be the vertex in H of degree m, and v1 , . . . , vm−1 be the vertices in TH . In the following, we show that any automorphism of μt (H ) that fixes (TH )t fixes w. Observe that the neighbors of w in μt (H ) that are in (TH )t are ut1 , . . . , utm−1 . The only other vertex with this property is ut−1 . However, the vertex ut−1 has degree 2m and w has degree m + 1. Therefore, any automorphism of μt (H ) that fixes (TH )t must also fix w. Since automorphisms preserve distance, this implies that such an automorphism preserves levels. Within a level, fixing the vertices in (TH )t fixes everything. Therefore, (TH )t is a determining set for μt (H ). Since a minimum twin cover is a determining set, all minimum size determining sets are minimum twin covers, each of which excludes w. In particular, (TH )t is a minimum size subset of V (μt (H )) \ {w} such that (TH )(t) ∪ {w} is a determining set. By Lemma 5.7, (TH )t plus  − 1 of the pendant vertices adjacent to w constitute a minimum size determining set for C. Thus, Det(C) = (TH )t +  − 1 = (t + 1)(TH ) +  − 1, as desired. Finally, we consider the case where H is the star graph K1,1 = K2 . Since we are assuming that G has twins, G = K2 + K1 for some  ≥ 2. A minimum twin cover T of G consists of  − 1 of the isolated vertices, but no vertices of H . Thus, TH is empty and so |TH | = 0. Also, T is not a determining set for G; instead the determining set for G consists of one of the two vertices in K2 together with T . It follows then that Det(H ) = 1. In this case, μt (G) = C + tK1 , where C is a copy of μt (K2 ) = C2t+3 with an additional  pendant vertices x1 , . . . , x adjacent to w. This is illustrated in Fig. 5.6. If y = w is any vertex in the cycle, then {y} ∪ {w} is a minimum size determining set for μt (K2 ) and so by Lemma 5.7, {y, x2 , . . . , x } is a minimum size determining set for C. Thus, Det(C) =  = t|TH | + Det(H ) + ( − 1). Finally, Det(μt (G)) = Det(C) + t − 1 & ' = t|TH | + Det(H ) + ( − 1) + t − 1 & ' & ' = t|TH | + t − t + Det(H ) + ( − 1) + t − 1 & ' & ' = t|TH | + t ( − 1) + Det(H ) + ( − 1) + t − 1 = t|T | + Det(G) + t − 1, as desired.



If a minimum twin cover T is a determining set for G, then by Corollary 5.14, T$ $ Thus |T | = Det(G) and |R| = 0, and so Theorem 5.20 is a determining set for G. implies the following corollary.

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Fig. 5.6 The graphs G = K2 + 3K1 , μ(G) and μ2 (G)

$ If G has Corollary 5.21 Let G be a graph with isolated vertices such that G = G. a minimum twin cover that is also a determining set, then for t ≥ 1, Det(μt (G)) = (t+1) Det(G) + t − 1. $ If no minimum Note that for R in the hypothesis of Theorem 5.20, |R| ≤ Det(G). $ contains any vertex of T$, then |R| = Det(G). $ This is the determining set for G case, for example, for the graph G in Fig. 5.5 or for G = K2 + K1 with  ≥ 2. Combining this observation with Corollary 5.21 gives the following corollary, $ with isolated vertices. bounding Det(μt (G)) for all graphs G = G $ Let T be Corollary 5.22 Let G be a graph with isolated vertices such that G = G. a minimum twin cover of G. Then for t ≥ 1, $ (t+1)|T |+t−1 ≤ Det(μt (G)) ≤ Det(G)+(t+1)|T |+t−1. Both bounds are sharp. We end our discussion on the determining number of the generalized Mycielskians of graphs with isolated vertices by combining Theorems 5.20 and 5.9. Observe that if G is twin-free, then the minimum twin cover is empty and |R| = Det(G). Theorem 5.23 Let G be a graph with isolated vertices and let T be a (possibly $ containing T$, empty) minimum twin cover of G. Among all determining sets for G $ $ $ let A be one of minimum size. Let R = {x ∈ V (G) | [x] ∈ A \ T }. Let t ≥ 1. (i) If G = K1 then, Det(G) = 0 and Det(μt (G)) = t. (ii) If G = K1 , then Det(μt (G)) = t|T | + Det(G) + t − 1.

5.4 Open Problems There are a significant number of results on distinguishing graph families in the literature. However, relatively little work has been done on the cost of 2distinguishing or on finding determining numbers.

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Some of the existing work on determining numbers is related to Cartesian products. The Cartesian product of graphs G and H has vertex set V (G) × V (H ) with an edge between vertices (x, u) and (y, u) if x is adjacent to y in G and between vertices (x, u) and (x, v) if u is adjacent to v in H . The Cartesian power Gk of G is the Cartesian product of G with itself k times. A graph G is prime with respect to the Cartesian product if it cannot be written as the Cartesian product of two smaller graphs. Theorem 5.24 ([12]) If Gk is a 2-distinguishable Cartesian power of a prime connected graph G on at least three vertices with Det(G) ≤ k and max{2, Det(G)} < Det(Gk ), then ρ(Gk ) ∈ {Det(Gk ), Det(Gk ) + 1}. Problem 5.1 ([12]) Classify the 2-distinguishable Cartesian powers Gk so that ρ(Gk ) = Det(Gk ), or so that ρ(Gk ) = Det(Gk ) + 1. Problem 5.2 ([12]) More generally, classify the 2-distinguishable graphs with ρ(G) ∈ {Det(G), Det(G) + 1}. One could choose to focus on either ρ(G) = Det(G) or ρ(G) = Det(G) + 1 for the classification instead. For Mycielskian graphs, some work has been done on the edge version of distinguishing. A coloring of the edges of a graph G with colors from {1, . . . , d} is called a d-distinguishing edge coloring if no nontrivial automorphism of G preserves the color classes. No such coloring exists when G has K2 as a component nor when G has two or more K1 components. For all other graphs G, the distinguishing index of G, denoted Dist (G), is the smallest number of colors in a d-distinguishing edge coloring of G. Theorem 5.25 ([5]) Let G be a twin free graph with at least three vertices, no K2 component and at most one K1 component. Then Dist (μ(G)) ≤ Dist (G) + 1. Conjecture 5.26 ([5]) Let G be a connected graph with at least three vertices. Then Dist (μ(G)) ≤ Dist (G) except for a finite number of graphs. Problem 5.3 Express Dist (μt (G)) in terms of Dist (G) and possibly other parameters of G. Problem 5.4 For each graph G, determine t for which Dist (μt (G)) = 2. Acknowledgments This work is a result of a collaboration made possible by the Institute for Mathematics and its Applications’ Workshop for Women in Graph Theory and Applications, August 2019.

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References 1. A. Mohammed Abid and T. R. Ramesh Rao. Dominator coloring of Mycielskian graphs. The Australasian Journal of Combinatorics 73 274–279 (2019). 2. M. O. Albertson. Distinguishing Cartesian powers of graphs. Electronic Journal of Combinatorics 12 N17 (2005). 3. M. O. Albertson and D. L. Boutin. Using determining sets to distinguish Kneser graphs. Electronic Journal of Combinatorics 1 4 R20 (2007). 4. M. O. Albertson and K. L. Collins. Symmetry breaking in graphs. Electronic Journal of Combinatorics 3 (1) R18 (1996). 5. S. Alikhani and S. Soltani. Symmetry breaking in planar and maximal outerplanar graphs. Discrete Mathematics, Algorithms and Applications 11 (1) 1950008 (2019) doi:10.1142/S1793830919500083. 6. L. Babai. Asymmetric trees with two prescribed degrees. Acta Mathematica. Academiae Scientiarum Hungaricae 29 (1-2) 193–200 (1977). 7. R. Balakrishnan and S. Francis Raj. Bounds for the b-chromatic number of the Mycielskian of some families of graphs. Ars Combinatoria 122 89–96 (2015). 8. B. Bogstad and L. J. Cowen. The distinguishing number of the hypercube. Discrete Mathematics 283 (1-3) 29–35 (2004). 9. D. L. Boutin. Identifying graph automorphisms using determining sets. Electronic Journal of Combinatorics 13 (1) Research Paper 78 (2006). 10. D. L. Boutin. Small label classes in 2-distinguishing labelings. Ars Mathematica Contemporanea 1 (2) 154–164 (2008). 11. D. L. Boutin. The determining number of a Cartesian product. Journal of Graph Theory 61 (2) 77–87 (2009). 12. D. L. Boutin. The cost of 2-distinguishing Cartesian powers. Electronic Journal of Combinatorics 20 (1) Research Paper 74 (2013). 13. D. L. Boutin. The cost of 2-distinguishing selected Kneser graphs and hypercubes. Journal of Combinatorial Mathematics and Combinatorial Computing 85 161–171 (2013). 14. D. Boutin, S. Cockburn, L. Keough, S. Loeb, K. E. Perry and P. Rombach. Distinguishing Generalized Mycielskian Graphs. arXiv:2006.03739 (2020). 15. D. Boutin, S. Cockburn, L. Keough, S. Loeb, K.E. Perry and P. Romback. Determining Number and Cost of Generalized Mycielskian Graphs. arXiv:2007.15284 (2020). 16. D. Boutin and W. Imrich. The cost of distinguishing graphs. In: Groups, graphs and random walks. London Mathematical Society Lecture Note series 436 104–119, Cambridge Univ. Press, Cambridge (2007). 17. X.-G. Chen and H.-M. Xing. Domination parameters in Mycielski graphs. Utilitas Mathematica 71 235–244 (2006). 18. Angsuman Das. Determining number of generalized and double generalized Petersen Graph. In: Algorithms and Discrete Applied Mathematics, 6th International Conference, CALDAM 2020, Hyderabad, India, February 13–15, 2020, Proceedings, Lecture Notes in Computer Science, Springer, 131–140 (2020). 19. D. C. Fisher, P. A. McKenna and E. D. Boyer. Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski’s graphs. Discrete Applied Mathematics 84 (1-3) 93–105 (1998). 20. W. Imrich and S. Klavžar. Distinguishing Cartesian powers of graphs. Journal of Graph Theory 53 (3) 250–260 (2006). 21. W. Imrich, S. Klavžar and V. Trofimov. Distinguishing infinite graphs. Electronic Journal of Combinatorics 14 Research Paper 36 (2007). 22. Sandi Klavžar and Xuding Zhu. Cartesian powers of graphs can be distinguished by two labels. European Journal of Combinatorics 28 (1) 303–310 (2007). 23. W. Lin, J. Wu, P. C. B. Lan and G. Gu. Several parameters of generalized Mycielskians. Discrete Applied Mathematics 154 (8) 1173–1182 (2006).

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24. J. Mycielski. Sur le coloriage des graphs. Colloquium Mathematicum 3 161–162 (1955). 25. Z. Pan and X. Zhu. Multiple coloring of cone graphs. SIAM Journal on Discrete Mathematics 24 (4) 1515–1526 (2010). 26. S. M. Smith, T. W. Tucker and M. E. Watkins. Distinguishability of infinite groups and graphs. Electronic Journal of Combinatorics 19 (2) Paper 27 (2012). 27. M. Stiebitz. Beiträge zur Theorie der färbungskritischen Graphen. Ph. D. Thesis. Technical University Ilmenau (1985). 28. C. Tardif. Fractional chromatic numbers of cones over graphs. Journal of Graph Theory 38 (2) 87–94 (2001). 29. N. Van Ngoc. On graph colourings. Ph. D. Thesis. Hungarian Academy of Sciences (1987). 30. N. Van Ngoc and Z. Tuza. 4-Chromatic graphs with large odd girth. Discrete Mathematics 138 (1–3) 387–392 (1995).

Chapter 6

Reconfiguration Graphs for Dominating Sets Kira Adaricheva, Chassidy Bozeman, Nancy E. Clarke, Ruth Haas, Margaret-Ellen Messinger, Karen Seyffarth, and Heather C. Smith

6.1 Introduction Given a problem and a set of feasible solutions, the associated reconfiguration problem involves determining whether or not one feasible solution can be transformed to another by following a predetermined rule. Many problems can be framed as reconfiguration problems. Examples from graph theory include independent sets, cliques, and vertex covers of graphs, as well as graph coloring. It is easy to see that any reconfiguration problem can be modeled with a reconfiguration graph, where vertices represent feasible solutions, and two vertices are adjacent if and only if the corresponding feasible solutions can be transformed from one to the other via one application of the predetermined rule. We consider only finite simple graphs. For basic graph theory notation and terminology, see [5]. The fundamental questions in reconfiguration in most contexts are about connectivity. Is it possible to get from one feasible solution to another? Is it possible to get

K. Adaricheva Department of Mathematics, Roosevelt Hall, Hofstra University, Hempstead, NY, USA e-mail: [email protected] C. Bozeman Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA, USA e-mail: [email protected] N. E. Clarke Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada e-mail: [email protected] R. Haas () Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI, USA e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Ferrero et al. (eds.), Research Trends in Graph Theory and Applications, Association for Women in Mathematics Series 25, https://doi.org/10.1007/978-3-030-77983-2_6

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between any pair of feasible solutions? What is the complexity of determining these questions for a particular instance? Branching out from these, many researchers have begun examining the structure of reconfiguration graphs. One of the most well-studied reconfiguration problems is vertex coloring. Given a graph G and a positive integer k, where k ≥ χ (G), the feasible solutions are all the proper colorings of G with k or fewer colors, and the reconfiguration rule is to change the color on exactly one vertex. The reconfiguration graph, in this case, is called the k-coloring graph (or simply the coloring graph when k is at least the number of vertices in G), and it arises naturally in theoretical physics when studying the Glauber dynamics of an anti-ferromagnetic Potts model at zero temperature [21]. A key result on the connectivity of the k-coloring graph is shown by Cereceda et al. [10]. They show that the k-coloring graph is connected whenever k > 1+col(G). Here col(G) is the coloring number of G, the smallest integer t for which there exists an ordering of the vertices v1 , . . . , vn (where n is the number of vertices of G) such that for all i, the degree of vi in the subgraph induced by {v1 , . . . , vi } is less than t. That is, there exists an ordering of the vertices so that G can be greedily colored using t colors. In Beier et al. [4], the question of which graphs can be coloring graphs is addressed. One important question about the structure of reconfiguration graphs concerns hamiltonicity, that is, can one find a Hamilton path or Hamilton cycle in a reconfiguration graph? A Hamilton path or Hamilton cycle in the reconfiguration graph is a “combinatorial Gray code,” that is, it is a listing of all the objects in a set so that successive objects differ in some prescribed minimal way. The term Gray code was first used to describe an ordered list of fixed length binary strings such that consecutive strings differ in exactly one bit. In [12], MacGillivray and Choo show that for any graph G, the k-coloring graph of G has a Hamilton cycle if k > col(G) + 2. Additionally, they give the smallest value of k for which there is a Gray code through the k-colorings of a tree. Celaya et al. [9] give conditions on k that ensure that the k-coloring graph of a complete bipartite graph has a Hamilton cycle. Seyffarth and Cavers [8] determine the smallest k for which the k-coloring graph of a 2-tree has a Hamilton cycle.

M.-E. Messinger Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB, Canada e-mail: [email protected] K. Seyffarth Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada e-mail: [email protected] H. C. Smith Department of Mathematics and Computer Science, Chambers 3039, Davidson College, Davidson, NC, USA e-mail: [email protected]

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A classic graph theoretic concept that can be studied as a reconfiguration problem is domination. Let G be a graph with vertex set V (G). A dominating set of G is a set D ⊆ V (G) such that every vertex of V (G)\D is adjacent to a vertex of D. The domination number of G, denoted γ (G), is the minimum cardinality of a dominating set of G. As the associated decision problem is NP-complete, much focus has been on bounding the domination number with respect to other graph parameters, and on determining the domination numbers for various classes of graphs. In this paper we consider reconfiguration of dominating sets under the token addition/removal (TAR) model, first considered in 2014 by Haas and Seyffarth [17]. Following the notation of [23], we define the k-dominating graph of a graph G, denoted Dk (G), as follows. The vertices of Dk (G) are the (not necessarily minimal) dominating sets of G that have cardinality k or less; thus, each dominating set of G of cardinality at most k is represented by a vertex in Dk (G). Let X and Y be dominating sets of cardinality at most k in graph G, represented by x, y ∈ V (Dk (G)), respectively. Then xy is an edge of Dk (G) if and only if dominating set Y can be obtained by adding a vertex of G to, or deleting a vertex of G from, dominating set X. When k = |V (G)|, we simplify notation by writing D(G) instead of D|V (G)| (G) and call D(G) the dominating graph of G. There are several recent survey papers that examine different aspects of reconfiguration. A paper by Nishimura [26] summarizes the state of understanding of algorithmic and complexity questions in a wide range of reconfiguration settings. Reconfiguration of graph coloring problems and dominating set problems are surveyed in a recent paper of Mynhardt and Nasserasr [23].

6.2 Previous Work on Reconfiguration of Dominating Sets There has been some work on several other kinds of reconfiguration for domination. In [15, 27], attention is restricted to reconfiguring between minimal dominating sets. In both these papers, the γ -graph of G, denoted γ [G], has vertices corresponding to the dominating sets of cardinality γ (G), called γ -sets of G, but the edges are defined in different ways. In [27] there is an edge between two γ -sets S and T if and only if S is obtained from T by exchanging any one vertex for another (token jumping), while Fricke et al. [15] define the sliding γ -graph to have an edge between two γ -sets S and T only if the swapped vertices are adjacent in the original graph (token sliding). Connelly et al. [13] show that every graph is the sliding γ -graph of some graph. In [14], Edwards et al. investigate the order, diameter, and maximum degree of jump and sliding γ -graphs of trees. Other papers consider different types of domination. Reconfiguration of total dominating sets is considered by Alikhani et al. in [1]. In [24] Mynhardt and Teshima study slide reconfiguration graphs for several concepts related to domination, including irredundant sets.

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Domination reconfiguration problems under the TAR model were first considered in 2014 by Haas and Seyffarth [17]. Since then there have been multiple papers on the topic by a variety of authors including Alikhani et al. [2], Mynhardt et al. [25], Suzuki et al. [28], Bousquet et al. [6], as well as a second paper by Haas and Seyffarth [18]. We now turn our attention to the k-dominating graph. As is typical for reconfiguration problems, one of the first questions to be asked is that of determining the values of k for which the k-dominating graph is connected. In their first paper on the subject, Haas and Seyffarth [17] show that this is not a monotone property; that is, it is possible for Dk (G) to be connected while Dk+1 (G) is not. However, they show that if k is sufficiently large, then there is such a property. Let (G) denote the size of a largest minimal dominating set of G. Lemma 6.1 ([17]) For any graph G, if k > (G) and Dk (G) is connected, then Dk+1 (G) is connected. Define d0 (G) to be the smallest integer such that Dk (G) is connected for all k ≥ d0 (G). Observe that d0 (G) exists because the dominating graph, D(G), is connected. Thus, trivially, d0 (G) ≤ |V (G)|. In [17] Haas and Seyffarth give a better upper bound for d0 (G). Theorem 6.2 ([17]) For any graph G with at least two edges having no common endpoint, if k ≥ min{|V (G)| − 1, (G) + γ (G)}, then Dk (G) is connected. They also give a lower bound for d0 (G). Lemma 6.3 ([17]) For any graph G with at least one edge, D(G) (G) is not connected. Together, Theorem 6.2 and Lemma 6.3 give the following bounds on d0 (G). Corollary 6.4 ([17]) For any graph G with at least two edges having no common endpoint, (G) + 1 ≤ d0 (G) ≤ min{|V (G)| − 1, (G) + γ (G)}. In fact, the lower bound in Corollary 6.4 is equal to d0 (G) when G satisfies one of the following conditions [17, 18]: is a bipartite graph or a chordal graph; has α(G) ≤ 2; is perfect and irredundant perfect; is well-covered with neither a 4-cycle nor a 5-cycle as a subgraph; is well-covered and has girth at least five; is well-covered, claw-free, and has no 4-cycle as a subgraph; is a well-covered plane triangulation. Recall that a set S ⊆ V (G) is independent if no pair of vertices in S is adjacent. The independence number of G, denoted α(G), is the size of a largest independent set of G. Note that any maximal independent set is a minimal dominating set, and thus for any graph G, α(G) ≤ (G). Thus one can frequently exploit information about α(G) when studying domination. In their second paper on reconfiguration of dominating sets, Haas and Seyffarth show the following. Theorem 6.5 ([18]) For any graph G, d0 (G) ≤ (G) + α(G) − 1. Furthermore, if G is triangle-free, then d0 (G) ≤ (G) + α(G) − 2.

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Suzuki et al. [28] were the first to find a graph G for which d0 (G) > (G) + 1. They construct an infinite family of graphs of varying tree-width, including a planar graph, all with d0 (G) = (G) + 2. Another example with d0 (G) = (G) + 2, is given in [18]. Mynhardt, Teshima, and Roux [25] construct two important classes of graphs. The first class contains graphs with arbitrary upper and lower domination numbers satisfying (G) ≥ γ (G) ≥ 2, having the property that d0 (G) = (G) + γ (G) − 1. A second class consists of graphs with arbitrary upper and lower domination numbers satisfying (G) ≥ γ (G) + 1 ≥ 2, having the property that d0 (G) = (G) + γ (G). A relationship between the connectedness of the k-dominating graph and matchings is described in Suzuki et al. [28]. They show that if G is an n-vertex graph that has a matching of size (at least) p + 1, then Dn−p (G) is connected. From this, they conclude that the diameter of Dn−p (G) is O(n) for an n-vertex graph with a matching of size p+1. In contrast, they also construct an infinite family of graphs of order n = 63r −6 for which each graph G has Dγ (G)+1 (G) of exponential diameter, (2n ). In a recent paper, Bousquet et al. [6] improve known results by showing that if k = γ (G) + α(G) − 1 then not only is Dk (G) connected, but the diameter of the graph is linear in the number of vertices of G. They prove linear diameter for smaller k in some special classes of graphs, and show that for any planar graph G, d0 (G) ≤ (G) + 3. The complexity of determining whether or not Dk (G) is connected is PSPACEcomplete [16], as shown by Haddadan et al. [19]. This is true even for graphs of bounded bandwidth, split graphs, planar graphs, and bipartite graphs. The same paper provides linear-time algorithms for determining the connectedness of Dk (G) if G is a cograph, a tree, or an interval graph. Some of the same authors plus others have considered parameterized complexity in [20] and [22]. Haas and Seyffarth [17] briefly consider the question of which graphs are realisable as k-dominating graphs, and observe that for n ≥ 4, D2 (K1,n−1 ) ∼ = K1,n−1 . Alikhani et al. [2] subsequently prove that these stars are the only graphs for which Dk (G) ∼ = G for some k. They also show that the only paths that occur as k-dominating graphs are P1 and P3 , and the only cycles that occur as k-dominating graphs are C6 , C8 .

6.3 Properties of D(G) The underlying motivation for our work in this paper is an interest in Hamilton paths and Hamilton cycles in k-dominating graphs. Results about existence of Hamilton paths and Hamilton cycles in graphs usually require specific knowledge about the structure of the graphs, and as such, we want as much structure as possible. This is the reason that, for the remainder of this paper, we consider only the dominating graph, D(G), i.e., the k-dominating graph when k = |V (G)|. We typically use

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capital letters to denote dominating sets of G and lower-case letters to denote the corresponding vertices in D(G). In this section, we focus our attention on properties of D(G). It is easy to see that, for any graph G, D(G) is connected because V (G) is a dominating set for G. We also note that D(G), and more generally Dk (G), is bipartite with partite sets based on the parity of the dominating sets. We begin by considering dominating graphs for two specific classes of graphs. In the complete graph on n ≥ 1 vertices, Kn , every non-empty subset of the vertices is a dominating set. Therefore D(Kn ) is isomorphic to the graph obtained from hypercube, Qn , by deleting one vertex. Figure 6.1a illustrates D(K4 ) where the binary string labels indicate the set of vertices in the corresponding dominating set. We next consider the dominating graph of K1,n , n ≥ 1, with bipartition X = {x} and Y = {y1 , . . . , yn }. The graph K1,n has 1 + 2n dominating sets: the set Y , as well as every set of the form X ∪ Y1 where Y1 ⊆ Y . Hence, D(K1,n ) is isomorphic to the graph obtained from Qn by adding one vertex (corresponding to dominating set Y ) and an edge from that vertex to the vertex of Qn corresponding to the dominating set X ∪ Y . As an example, the dominating graph D(K1,4 ) is illustrated in Fig. 6.1b, where binary string labels correspond to dominating sets that contain x and some subset of Y , and vertex  corresponds to the dominating set Y . We now consider diameter and connectivity of dominating graphs. Theorem 6.6 If G is a connected graph on n ≥ 2 vertices, then D(G) has diameter n. Proof Let D1 and D2 be dominating sets of G with corresponding vertices d1 and d2 , respectively, in D(G). Let d be the vertex in D(G) corresponding to D1 ∪ D2 . Let X1 = D1 \ D2 and X2 = D2 \ D1 . Observe that any superset of a dominating set of G is a dominating set of G. Starting with D1 , adding the vertices of X2 , one at a time, produces a shortest path in D(G) between d1 and d. Likewise, from D2 , by adding the vertices of X1 one at a time, we obtain a shortest path in D(G) between 1111

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(a) Fig. 6.1 Dominating graphs for K4 and K1,4 . (a) D(K4 ). (b) D(K1,4 )

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d2 and d. By the definitions of X1 and X2 , the concatenation of these two paths is a shortest path in D(G) from d1 to d2 , and has length |X1 | + |X2 | ≤ n. Therefore the distance between any two vertices in D(G) is at most n. To see that this bound is tight, let D1 be a minimal dominating set of G and D2 = V (G) \ D1 . Because G is connected, D2 is a dominating set. Hence a shortest path between d1 and d2 in D(G) has length n.  Lemma 6.7 Let G be a connected graph. Then D(G) has no cut vertex, other than possibly the vertex corresponding to the dominating set V (G). Proof Let G be a connected graph. We first note that the vertex of D(G) corresponding to the dominating set V (G) is sometimes a cut vertex in D(G), as is the case when G ∼ = K1,n for n ≥ 1. Suppose x is a cut vertex in D(G) that corresponds to dominating set X in G and suppose X = V (G). Since x is a cut-vertex, the degree of x in D(G) is at least two. Let a, b ∈ N(x) be such that a and b are in different components of D(G) − x. Let A, B be the dominating sets of G that correspond to vertices a, b in D(G), respectively. Since X = V (G), there exists a c ∈ V (G) \ X. Observe that A ∪ {c}, X ∪ {c}, and B ∪ {c} are dominating sets in G; let a  , x  , b , respectively, represent the corresponding vertices in D(G). Note that x  = x, though it is possible that x  ∈ {a, b}. If x  ∈ {a, b}, then aa  x  b b is a path in D(G) from a to b that does not include x. Otherwise, assume without loss of generality that x  = a. Then x  b b = ab b is a path in D(G) from a to b that does not include x. In either case, this contradicts the assumption that x is a cut vertex in D(G).  Theorem 6.8 Let G be a connected graph. The graph D(G) has a cut-vertex if and only if G ∼ = K1,n for some n ≥ 1. Proof From the earlier description of D(K1,n ), n ≥ 1, it is clear that this graph has a cut vertex. For the converse, let G be a connected graph, and suppose that D(G) has a cut vertex, x. By Lemma 6.7, x corresponds to the dominating set V (G) of G. Suppose a, b are neighbours of x in D(G) such that a and b lie in different components of D(G) − x. Let A = V (G)\{α} and B = V (G)\{β1 }, α, β1 ∈ V (G), be the dominating sets of G that correspond to vertices a and b in D(G). Since x is a cut vertex in D(G), it follows that V (G)\{α, β1 } is not a dominating set of G. Since G is connected, this implies one of α, β1 has degree one in G, and that αβ1 ∈ E(G). Without loss of generality, suppose d(β1 ) = 1. If |V (G)| > 2, then α is adjacent to some vertex β2 in G. Observe that V (G)\{β2 } is a dominating set of G, and that since d(β1 ) = 1, V (G)\{β1 , β2 } is also a dominating set of G. However, V (G)\{α, β2 } is not a dominating set of G. If it were, then there would be a path in D(G) − x corresponding to the dominating sets A, V (G)\{α, β2 }, V (G)\{β2 }, V (G)\{β1 , β2 }, B, contradicting the fact that x is a cut vertex. Consequently, one of α, β2 has degree one in G, but since d(α) > 1, d(β2 ) = 1. Repeating this argument for all remaining  neighbours of α in G, it follows that G ∼ = K1,n for some n ≥ 1. We now state an unpublished result of Brouwer et al. [7].

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Theorem 6.9 ([7]) The number of dominating sets of a finite graph is odd. For completeness, we include an expanded version of their proof. Proof Let G be a graph and let A := {(S, T ) : S, T ⊆ V (G), S ∩ T = ∅, st ∈ E(G) for each s ∈ S and t ∈ T }. Let S ⊆ V (G) be arbitrary, and let CS = V (G)\N[S]. Then CS = ∅ if and only if N [S] = V (G), that is, S is a dominating set. Thus (S, T ) ∈ A if and only if T ⊆ CS . Therefore |{T : (S, T ) ∈ A}| = 2|CS | . Notice that this quantity is even unless |CS | = 0. Therefore |A| and the number of sets S with |CS | = 0 have the same parity. It follows that |A| and the number of dominating sets of G have the same parity. Note that A is a symmetric relation. Further, the only time (S, T ) = (T , S) is when S = T = ∅. Therefore |A| is odd, and hence the number of dominating sets of G is odd.  Recall that any dominating graph is bipartite, so all cycles have even length. In particular, if G is a graph for which D(G) has a Hamilton cycle, then the number of dominating sets of G is even. Combining this with Brouwer’s result gives us the following. Corollary 6.10 For any graph G, the dominating graph D(G) has no Hamilton cycle. We conclude this section with a discussion of the minimum degree of D(G) and the implications for G. Lemma 6.11 Let G be a connected graph on n vertices. Then δ(D(G)) < n/2. Proof By Dirac’s Theorem, if δ(D(G)) ≥ n/2, then D(G) has a Hamilton cycle. However, Corollary 6.10 states that for any graph G, D(G) has no Hamilton cycle. Thus for any graph G, δ(D(G)) < n/2.  Theorem 6.12 Let G be a connected graph on n vertices. If δ(D(G)) = k for some integer k ≥ 1, then α(G) ≥ n − 2k + 1. Furthermore this bound is tight. Proof Let G be a connected graph on n vertices such that δ(D(G)) = k for some integer k ≥ 1. Let u be a vertex in D(G) that has degree k. By Lemma 6.11, k < n/2 Let U be the dominating set of G that corresponds to vertex u in D(G). Observe that if any vertex of S = V (G)\U is added to set U , the resulting set will be a dominating set of G. Thus k ≥ |S| = n − |U |, so |U | ≥ n − k. Let R = {r ∈ U : U \{r} is a dominating set of G}. Then k = |S| + |R|. It follows that |U \ R| = |U | − |R| ≥ (n − k) − (k − |S|) > n − 2(n/2) + |S| = |S|.

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For each x ∈ U \ R, U \{x} is not a dominating set of G; thus some vertex of N (x) ∪ {x} is not in the neighborhood of any other vertex in U . Let P = {p ∈ U \ R : N(p) ∩ U = ∅}. Note that P is an independent set. Let Q = (U \ R) \ P . These are the vertices q ∈ U \ R such that NG (q) ∩ U = ∅, but there is an s(q) ∈ S, such that NG (s(q)) ∩ U = {q}. Therefore |Q| ≤ |S| because s(q) = s(q  ) for distinct q, q  ∈ Q. Observe |(U \ R) \ P | = |U | − |R| − |P | = |Q| ≤ |S| < |U \ R|, i.e., |(U \ R) \ P | < |U \ R|, and hence P = ∅. Let p ∈ P . Since G is connected and NG (p) ∩ U = ∅, there exists s(p) ∈ S adjacent to p. Note that s(p) = s(q) for any q ∈ Q, implying |Q| ≤ |S| − 1, and hence |U | − |R| − |P | = |Q| ≤ |S| − 1. Rearranging, we obtain |P | ≥ |U | − (|R| + |S|) + 1 = |U | − k + 1 ≥ n − 2k + 1. Since P is an independent set, we have the desired result. When G ∼ = K1,n−1 , then D(G) is isomorphic to the graph obtained from Qn−1 by appending one pendant vertex. In this case, k = 1 so |P | = n − 1 = α(K1,n−1 ).  We can now establish some additional information about the structure of D(G). Recall that the complement of a graph G is denoted G. Also, for any two graphs G and H , the join of G and H , denoted G ∨ H , is obtained from disjoint copies of G and H by adding the edge ab for each a ∈ V (G) and b ∈ V (H ). Corollary 6.13 Let G be a connected graph on n ≥ 2 vertices such that δ(D(G)) = k. Then G is a connected subgraph of K2k−1 ∨ Kn−2k+1 . Furthermore, if δ(D(G)) = 1, then G ∼ = K1,n−1 . Proof By Theorem 6.12, G has an independent set of size at least n − 2k + 1, so G is a connected subgraph of K2k−1 ∨ Kn−2k+1 . If δ(D(G)) = 1, the only connected subgraph of K1 ∨ Kn−1 ∼  = K1,n−1 is K1,n−1 , giving the desired conclusion. Corollary 6.13 implies that K1,n is the only connected graph G whose dominating graph has a vertex of degree one.

6.4 Hamilton Paths in Dominating Graphs As pointed out in Corollary 6.10, a dominating graph never has a Hamilton cycle. Hence, for the remainder of this paper, we focus our attention on Hamilton paths of dominating graphs. We begin with dominating graphs of complete bipartite graphs, and then turn our attention to dominating graphs of paths.

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6.4.1 Hamilton Paths in D(Km,n ) Recall from Sect. 6.3 that the dominating graph of the complete graph, Kn , is isomorphic to the hypercube Qn with one vertex removed (see Fig. 6.1a for an example). We also saw that D(K1,n ) is isomorphic to the graph obtained from the hypercube, Qn , by adding one new vertex and a single edge from that vertex to a vertex of Qn . It is well-known that for n ≥ 2, Qn , has a Hamilton cycle, and thus the next observation follows directly. Observation 6.14 For any n ≥ 1, D(Kn ) and D(K1,n ) have Hamilton paths. However, not every dominating graph has a Hamilton path. Consider the complete bipartite graph K2,2 . The dominating sets of K2,2 are all S ⊆ V (K2,2 ) with |S| ≥ 2. Recall that the dominating graph is bipartite, with the bipartition based on the parity of the dominating sets. Since K2,2 has seven dominating sets of even size and four dominating sets of odd size, D(K2,2 ) has no Hamilton path. More generally, any complete bipartite graph with both parts even has no Hamilton path. Proposition 6.15 For any even integers m, n ≥ 2, D(Km,n ) has no Hamilton path. Proof To see this, we use the fact that the dominating graph is bipartite, and count the number of dominating sets of each parity. Let A and B be the partite sets of Km,n where |A| = m and |B| = n. The dominating sets of Km,n are A, B, and all S ⊆ V (G) with S ∩ A = ∅ and S ∩ B = ∅. Since both m and n are even, the number of even-sized dominating sets is (2m−1 − 1)(2n−1 − 1) + (2n−1 )(2m−1 ) + 2 = 2m+n−1 − 2m−1 − 2n−1 + 3 while the number of odd-sized dominating sets is (2m−1 − 1)(2n−1 ) + (2m−1 )(2n−1 − 1) = 2m+n−1 − 2m−1 − 2n−1 . Since the number of even-sized dominating sets is three more than the number of  odd sized dominating sets, D(Km,n ) has no Hamilton path. It turns out, however, that if m, n ≥ 2, and at least one of m or n is odd, then D(Km,n ) does have a Hamilton path. The proof relies on the concept of Hamiltonlaceable bipartite graphs. A bipartite graph G with bipartition (A, B) is said to be Hamilton-laceable if for any pair of vertices a ∈ A and b ∈ B, there is a Hamilton path in G with endpoints a and b. Chen and Quimpo [11] prove that for all integers n ≥ 1, the hypercube Qn is Hamilton-laceable. This is key to the proof of the next result. Theorem 6.16 For any integer m ≥ 2 and any odd integer n ≥ 3, D(Km,n ) has a Hamilton path.

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Proof Let Km,n have bipartition (X, Y ) with X = {x1 , . . . , xm } and Y = {y1 , . . . , yn }. The dominating sets of Km,n are easy to describe; they are the set X, the set Y , and every set P ⊆ X ∪ Y such that P ∩ X = ∅ and P ∩ Y = ∅. Note that D(Km,n ) is a subgraph of the (m + n)-cube. We denote subsets of V (Km,n ) as pairs of sets of indices (Xi , Yj ), where Xi ⊆ {1, . . . , m} and Yj ⊆ {1, . . . , n}. For example, ({1, 4}, {2}) denotes the subset {x1 , x4 , y2 }. For any fixed nonempty set Xi ⊆ {1, . . . , m}, let D[Xi ] be the subgraph of D(Km,n ) induced by the set of dominating sets of the form (Xi , Yj ) where Yj ⊆ {1, . . . , n}. When Xi  {1, . . . , m}, then D[Xi ] is isomorphic the graph obtained from Qn by deleting the vertex associated with the empty set, and D[{1, . . . , m}] is isomorphic to Qn . Let R0 , . . . , Rk be a Hamilton path in Qn with R0 = ∅ and Rk = {1, . . . , n}, where k = 2n − 1. This path exists because n is odd and Qn is Hamilton-laceable. Without loss of generality we may assume R1 = {1}. A Hamilton path of D(Km,n ) is constructed by taking the union of the Hamilton paths from all D[Xi ]. The details differ slightly for the two cases m even and m odd. First suppose that m is odd. Let S0 , . . . , Sp be a Hamilton path in Qm with S0 = ∅ and Sp = {1, . . . , m}, where p = 2m − 1. As before, this path exists because Qm is Hamilton-laceable and m is odd. The following sequence of dominating sets lists, in order, the vertices of the desired Hamilton path in D(Km,n ). (Sp , R0 ) (Sp , R1 )(Sp , R2 ) . . . (Sp , Rk ) (Sp−1 , Rk )(Sp−1 , Rk−1 )(Sp−1 , Rk−2 ) . . . (Sp−1 , R1 ) (Sp−2 , R1 )(Sp−2 , R2 ) . . . (Sp−2 , Rk ) .. . (S2 , Rk )(S2 , Rk−1 )(S2 , Rk−2 ) . . . (S2 , R1 ) (S1 , R1 )(S1 , R2 ) . . . (S1 , Rk ) (S0 , Rk ). We now assume that m is even. Let T0 , . . . Tp be a Hamilton path in Qm with T0 = {1}, and Tp = {1, . . . , m}, where p = 2m − 1. As before, this path exist because m is even (so T0 and Tp are in different parts of Qm ) and Qm is Hamiltonlaceable. Note that parity implies that the empty set, ∅, is an odd distance from T0 in Qm , and thus T2α+1 = ∅ for some integer α. To complete the proof, we require a different Hamilton path in D[X], that is, the Hamilton path R0 , . . . , Rk does not have the necessary properties. Let U0 , . . . , Uk be a Hamilton path in Qn with U1 = R1 = {1}, and Uk = ∅. This exists because Qn is Hamilton-laceable. The following sequence of dominating sets lists, in order, the vertices of the desired Hamilton path in D(Km,n ).

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(T0 , R1 )(T0 , R2 ) . . . (T0 , Rk ) (T1 , Rk )(T1 , Rk−1 ) . . . (T1 , R1 ) (T2 , R1 )(T2 , R2 ) . . . (T2 , Rk ) (T3 , Rk )(T3 , Rk−1 ) . . . (T3 , R1 ) .. . (T2α , R1 )(T2α , R2 ) . . . (T2α , Rk ) (T2α+1 = ∅, Rk ) (T2α+2 , Rk )(T2α+2 , Rk−1 )(T2α+2 , Rk−2 ) . . . (T2α+2 , R1 ) (T2α+3 , R1 )(T2α+3 , R2 )(T2α+3 , R3 ) . . . (T2α+3 , Rk ) .. . (Tp−1 , Rk )(Tp−1 , Rk−1 )(Tp−1 , Rk−2 ) . . . (Tp−1 , R1 ) (Tp , U1 = R1 )(Tp , U2 )(Tp , U3 ) . . . (Tp , Uk = ∅). Note that 2α + 2 and p − 1 have the same parity, so the order of the second elements in the ordered pairs beginning with T2α+2 and those beginning with Tp−1 is the same. This completes the proof of the theorem.  Combining Observation 6.14 with Proposition 6.15 and Theorem 6.16 gives a simple characterization: for any integers m, n ≥ 1, D(Km,n ) has a Hamilton path if and only if at least one of {m, n} is odd.

6.4.2 Hamilton Paths in D(Pn ) We next focus on proving that for all n ≥ 1, D(Pn ) has a Hamilton path. To do this, we first introduce some notation and a lemma. Notation 6.17 Assume that Pn , the path on n vertices, has vertex set {1, . . . , n} and edge set {ij : |i − j | = 1}. We encode a subset X of V (Pn ) as an n-digit binary string, x1 x2 . . . xn by setting xi = 1 if and only if i ∈ X, 1 ≤ i ≤ n. We often abuse this notation by not differentiating between a subset of V (Pn ) and its representation as a binary string. Observation 6.18 It is routine to verify that a binary string x1 x2 . . . xn corresponds to a dominating set of Pn if and only if either n = 1 and x1 = 1, or n ≥ 2 and the following three conditions hold: 1. x1 x2 = 00; 2. xn−1 xn = 00; 3. xi−1 xi xi+1 = 000 for any i, 2 ≤ i ≤ n − 2. In addition, dominating sets X and Y of Pn , represented by binary strings x1 x2 . . . xn and y1 y2 . . . yn , respectively, are adjacent in D(Pn ) if and only if there is a unique i, 1 ≤ i ≤ n, for which xi = yi .

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Lemma 6.19 Let σn denote the number of dominating sets of Pn , and thus the number of vertices of D(Pn ). Then σ1 = 1, σ2 = 3, σ3 = 5, and for all n ≥ 4, σn = σn−1 + σn−2 + σn−3 . Proof For each i, 1 ≤ i ≤ n, let Si denote the set of n-digit binary strings that correspond to dominating sets as outlined in Observation 6.18. Then S1 = {1}, S2 = {01, 10, 11}, S3 = {010, 110, 101, 011, 111}, and S4 = {0110, 0101, 1001, 1010, 0111, 1011, 1101, 1110, 1111}. Thus we see that σ1 = 1, σ2 = 3, σ3 = 5, and σ4 = 9 = σ1 + σ2 + σ3 . Let n ≥ 5 and let s = x1 . . . xn ∈ Sn . Define the following subsets of Sn : • • • • • •

Y1 Y2 Y3 Y4 Y5 Y6

= {s = {s = {s = {s = {s = {s

∈ Sn ∈ Sn ∈ Sn ∈ Sn ∈ Sn ∈ Sn

| xn−1 xn = 11}. |xn−2 xn−1 xn = 101}; | xn−3 xn−2 xn−1 xn = 1001}; | xn−2 xn−1 xn = 110}; | xn−3 xn−2 xn−1 xn = 1010}; | xn−4 xn−3 xn−2 xn−1 xn = 10010}.

It is left to the reader to verify that {Y1 , . . . Y6 } is a partition of Sn . Note that any string x1 x2 . . . xn−1 ∈ Sn−1 can be transformed into a string x1 x2 . . . xn−1 1 ∈ Sn . This produces the strings in Sn with xn−1 xn = 11 or xn−2 xn−1 xn = 101, implying that σn−1 = |Y1 |+|Y2 |. Next, any string x1 . . . xn−2 ∈ Sn−2 can be transformed into a string x1 x2 . . . xn−2 10 ∈ Sn . This produces the strings of Sn with xn−2 xn−1 xn = 110 or xn−3 xn−2 xn−1 xn = 1010, so σn−2 = |Y4 | + |Y5 |. The only strings of Sn that have not yet been counted are those with xn−3 xn−2 xn−1 xn = 1001, and those with xn−4 xn−3 xn−2 xn−1 xn = 10010. Let tn−3 denote the number of strings in Sn−3 with xn−3 = 0, and let un−3 denote the number of strings in Sn−3 with xn−3 = 1. Then tn−3 + un−3 = σn−3 . Observe that the strings counted by tn−3 have xn−4 = 1, so appending “010” to the strings of Sn−3 counted by tn−3 produces the strings of Sn with xn−4 xn−3 xn−2 xn−1 xn = 10010, so tn−3 = |Y6 |. Furthermore, appending “001” to the strings of Sn−3 counted by un−3 produces the strings of Sn with xn−3 xn−2 xn−1 xn = 1001, so un−3 = |Y3 |. Thus σn−3 = tn−3 + un−3 = |Y6 | + |Y3 |. Therefore σn = (|Y1 | + |Y2 |) + (|Y4 | + |Y5 |) + (|Y6 | + |Y3 |) = σn−1 + σn−2 + σn−3 , as required.



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Theorem 6.20 For all n ≥ 1, D(Pn ) has a Hamilton path. In particular, there is a Hamilton path in which all strings ending in “01” are listed first, followed by all strings ending in “11”, followed by all strings ending in “010”, followed by all strings ending in “110”. Proof The result is trivial when n = 1. When n = 2, one can verify that H2 = 01, 11, 10 is a Hamilton path in D(P2 ) satisfying the ordering condition on the vertices. Similarly, when n = 3, it is routine to verify that H3 = 101, 111, 011, 010, 110 is a Hamilton path in D(P3 ) that satisfies the condition on the ordering of the strings. We now proceed by induction on n. Fix n ≥ 3, and suppose D(Pn ) has a Hamilton path described by n-digit binary strings Hn = a1n , . . . , apn , b1n , . . . , bqn , c1n , . . . , crn , d1n , . . . , dsn where p + q + r + s = |V (D(Pn ))|, each ain is a string ending in “01”, 1 ≤ i ≤ p; each bin is a string ending in “11”, 1 ≤ i ≤ q; each cin is a string ending in “010”, 1 ≤ i ≤ r; and each din is a string ends in “110”, 1 ≤ i ≤ s. For each din , 1 ≤ i ≤ s, we construct a string dˆin by replacing the “110” at the end of din with “100”. Note that dˆin is not a vertex of D(Pn ). We define ain+1 , 1 ≤ i ≤ 2s + r; bin+1 , 1 ≤ i ≤ p + q; cin+1 , 1 ≤ i ≤ p; and n+1 di , 1 ≤ i ≤ q as follows: ain+1

bin+1

⎧ n ⎨ dˆi 1 n := d2s+1−i 1 ⎩ n c2s+r+1−i 1  n bq+1−i 1 := n ap+q+1−i 1

for 1 ≤ i ≤ s; for s + 1 ≤ i ≤ 2s; for 2s + 1 ≤ i ≤ 2s + r. for 1 ≤ i ≤ q; for q + 1 ≤ i ≤ q + p.

cin+1 := ain 0 for 1 ≤ i ≤ p. djn+1 := bjn 0 for 1 ≤ i ≤ q. We claim that n+1 n+1 n+1 , b1n+1 , . . . , bp+q , c1 , . . . , cpn+1 , d1n+1 , . . . , dqn+1 , a1n+1 , . . . , a2s+r

(6.1)

is a Hamilton path in D(Pn+1 ) satisfying the conditions of the theorem. We begin by observing that the construction ensures that the 2p + 2q + r + 2s vertices listed in (6.1) are distinct. For each j , 1 ≤ j ≤ 2s + r, ajn+1 ends in “01” since it is obtained by appending a “1” to a string ending in “0”. Furthermore, n+1 the construction guarantees that a1n+1 , . . . , a2s+r is a path in D(Pn+1 ). For each j ,

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1 ≤ j ≤ p + q, bjn+1 is obtained by appending a “1” to a string ending in “1”, n+1 is and hence ends in “11”. The construction again guarantees that b1n+1 , . . . , bp+q n+1 n+1 a path in D(Pn+1 ), and that b1 is adjacent to a2s+r . For each j , 1 ≤ j ≤ p, cjn+1 is obtained by appending a “0” to a string ending in “01”, and thus ends in

“010”. The construction guarantees that c1n+1 , . . . , cpn+1 is a path in D(Pn+1 ), and n+1 . Finally, for each j , 1 ≤ j ≤ q, djn+1 is obtained by that c1n+1 is adjacent to bp+q appending a “0” to a string ending in “11”, and thus ends in “110”. The construction guarantees that d1n+1 , . . . , dqn+1 is a path in D(Pn+1 ), and that d1n+1 is adjacent to cpn+1 . Therefore n+1 n+1 n+1 , b1n+1 , . . . , bp+q c1 , . . . , cpn+1 d1n+1 , . . . , dqn+1 Hn+1 := a1n+1 , . . . , a2s+r

is a path in D(Pn+1 ). To prove that Hn+1 is a Hamilton path, it suffices to prove that the number of vertices in D(Pn+1 ) is equal to the number of vertices in the Hn+1 . For each  ≥ 3, let ρ denote the number of vertices in the path H . Then ρ3 = 5 with p = 1, q = 2, r = 1 and s = 1, and we observe that q = r + s. The nature of the construction of Hn+1 from Hn (see (6.1)) ensures that for all  ≥ 3, the number of strings of H ending in “11” is equal to the number of strings of H ending in “010” plus the number of strings of H ending in “110”. We begin with ρn = p + q + r + s; from (6.1), we have ρn+1 = 2p + 2q + r + 2s. Repeating the induction step for ρn+2 and ρn+3 gives us ρn+2 = 3p + 4q + 2r + 4s and ρn+3 = 6p + 8q + 3r + 6s. From this, we see that ρn+3 = ρn+2 + (3p + 4q + r + 2s) = ρn+2 + ρn+1 + (p + 2q) = ρn+2 + ρn+1 + (p + q + r + s) since q = r + s = ρn+2 + ρn+1 + ρn , Comparing this to the result in Lemma 6.19, we see that sequence ρ ,  ≥ 1, satisfies the same recurrence relation with the same initial conditions as the sequence σ ,  ≥ 1. In particular, ρn+1 = σn+1 = |V (D(Pn+1 )|, proving that Hn+1 is a Hamilton path. 

6.5 Additional Results and Future Work Our main results given here are firstly that D(Km,n ) has a Hamilton path if and only if at least one of m or n is odd (Sect. 6.4.1), and secondly that D(Pn ) has a Hamilton path (Theorem 6.20). With respect to the path Pn , not only do we show

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that a Hamilton path exists in D(Pn ), but we give specific structure for the path in which all dominating sets that contain the last vertex,“n”, appear before all the dominating sets that do not contain “n”. These are examples of some of the types of results to be found in our forthcoming paper [3], in which we construct Hamilton paths with additional properties; in addition, we provide results about reconfiguring dominating sets of trees, cycles and some other families of graphs.

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