An Invitation to Pursuit-Evasion Games and Graph Theory [1 ed.] 1470467631, 9781470467630

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STUDENT MATHEMATICAL LIBRARY Volume 97

An Invitation to Pursuit-Evasion Games and Graph Theory Anthony Bonato

An Invitation to Pursuit-Evasion Games and Graph Theory

STUDENT MATHEMATICAL LIBRARY Volume 97

An Invitation to Pursuit-Evasion Games and Graph Theory Anthony Bonato

EDITORIAL COMMITTEE John McCleary Rosa C. Orellana (Chair)

Paul Pollack Kavita Ramanan

2020 Mathematics Subject Classification. Primary 05C57, 05C05, 05C90, 68R10. For additional information and updates on this book, visit www.ams.org/bookpages/stml-97 Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/stml/097 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission @ams.org.

c 2022 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

27 26 25 24 23 22

To Douglas

Contents

List of Figures Preface

xi xvii

Chapter 1.

Introduction

1

§1.1.

Introduction to pursuit-evasion games

1

§1.2.

Graph theory

6

§1.3.

Exercises

Chapter 2.

Cops and Robbers

13 17

§2.1.

Introduction to the game of Cops and Robbers

19

§2.2.

The structure of cop-win graphs

23

§2.3.

Bounds on the cop number and guarding

28

§2.4.

The cop number of planar graphs

35

§2.5.

Meyniel’s conjecture on the cop number

41

§2.6.

Capture time

50

§2.7.

Exercises

54

§2.8.

Projects

57

Chapter 3.

Graph Searching

59

§3.1.

Introduction to graph searching

59

§3.2.

Classes and bounds

64 vii

viii

Contents

§3.3.

Monotonicity

68

§3.4.

Node search and pathwidth

73

§3.5.

The search number of trees

80

§3.6.

Helicopter Cops and Robbers

84

§3.7.

Exercises

87

§3.8.

Projects

90

Chapter 4.

Graph Burning

91

§4.1.

Introduction to graph burning

91

§4.2.

Burning trees

95

§4.3.

The burning number conjecture

100

§4.4.

Burning spiders

107

§4.5.

Burning graph classes

114

§4.6.

Exercises

118

§4.7.

Projects

120

Chapter 5.

The Localization Game

121

§5.1.

Introduction to the Localization game

124

§5.2.

Localization on bipartite graphs and trees

126

§5.3.

Connections with other graph parameters

129

§5.4.

Localization on hypercubes

136

§5.5.

Localization on graph classes

139

§5.6.

Localization capture time

153

§5.7.

Exercises

158

§5.8.

Projects

161

Chapter 6.

Firefighter

163

§6.1.

Introduction to Firefighter

163

§6.2.

Core results on Firefighter

165

§6.3.

Firefighting on grids

171

§6.4.

Surviving rate of a graph

176

§6.5.

Exercises

181

§6.6.

Projects

183

Contents Chapter 7.

ix Invisible Robber Games

185

§7.1.

Introduction to invisible robber games

185

§7.2.

0-visibility Cops and Robbers

188

§7.3.

Limited-visibility Cops and Robbers

194

§7.4.

Hyperopic Cops and Robbers

200

§7.5.

Exercises

206

§7.6.

Projects

209

Chapter 8.

Variants of Pursuit-Evasion Games

211

§8.1.

Lazy Cops and Robbers

213

§8.2.

Cops and Eternal Robbers

216

§8.3.

Burning the plane

221

§8.4.

Constrained Firefighter

224

§8.5.

Angel and Devil

227

§8.6.

Pursuing the pursuers

230

§8.7.

Exercises

234

§8.8.

Projects

235

Bibliography

237

Index

251

List of Figures

1.1

Two cops cannot win playing Cops and Robbers on the dodecahedron, but three can.

2

The firefighters, represented by squares, block the spread of a fire, represented by gray vertices, in a grid.

4

If the robber is invisible and knows all the future moves of the cop, then a single cop cannot capture the robber on the depicted graph.

5

1.4

A drawing of the Heawood graph.

7

1.5

A graph of diameter 4.

8

1.6

Two isomorphic graphs. The depicted graph is named the Petersen graph. 10

1.7

From left to right, the hypercube graphs Q0 , Q1 , Q2 , and Q3 .

10

1.8

An example of a digraph.

11

2.1

The pursuer chases the evader on P5 , capturing them after two moves. 17

2.2

An infinite one-way path.

21

2.3

The subgraph induced by {1, 2} is a retract, with the retraction mapping 3 to 1, 4 to 2, and fixing 1 and 2.

25

1.2 1.3

xi

xii

List of Figures

2.4

A cop-win labeling of a cop-win graph.

28

2.5

A cop-win graph G.

29

2.6

A graph along with an optimal tree decomposition.

33

2.7

A maze and its corresponding graph.

35

2.8

An outerplanar graph.

37

2.9

The path between x1 and x2 .

39

2.10 Expanding the cop territory in the final case. The shaded vertices are cop territory.

40

2.11 An embedding of K5 on the torus.

41

2.12 The Fano plane and its incidence graph. Vertices corresponding to lines are represented by triples.

42

2.13 A cop guards a star X, while the robber moves into a component G0 in G − X.

45

2.14 A perfect matching in the hypercube Q4 .

47

2.15 The cop-win graph H7 . Notice that vertex 1 is the unique corner. 52 2.16 The graph G.

55

2.17 The latin square graph of L3 . Cells are labeled Ei , where 1 ≤ i ≤ 9.

56

3.1

The layout of a top-secret building represented as a graph. Corridors, both hallways and tunnels, correspond to edges, and vertices correspond to their intersection. 60

3.2

The searcher S at v must slide to x1 and can never leave there.

66

3.3

Contracting the edge uv to a vertex x.

67

3.4

In mixed search, an edge is cleared by either sliding a searcher along it or by placing searchers on its endpoints.

69

3.5

The set X consists of the bold edges, and b(X) are the gray vertices. 70

3.6

We have that ns(K3,3 ) = 4, while s(K3,3 ) = 5.

74

3.7

A graph with pathwidth 2 and its corresponding path decomposition.

76

List of Figures

xiii

3.8

Two layouts of K3,3 . The top one has vertex separation 4, while the bottom one has vertex separation 3. 78

3.9

An interval graph representation of a graph. Overlapping intervals correspond to adjacent vertices.

79

3.10 The subtree S + v is the subgraph induced by V (S) and v. 80 3.11 The trees in T1 , T2 , and T3 .

83

3.12 In the tree T, one cop C1 stays at a fixed vertex v that acts like a root, while C2 moves to the subtree of T − v containing R. In this way, the robber is captured.

85

3.13 The tree T2 with exclusive search number 2.

89

4.1

Burning the path P4 . The circled vertices are sources.

92

4.2

A rooted tree partition, with roots r1 , r2 , and r3 .

97

4.3

A spider graph with an optimal burning sequence.

100

4.4

A caterpillar graph.

101

4.5

The subtree rooted at p, denoted by Tp .

105

4.6

The Cartesian grid G4,6 .

115

4.7

The strong grid P4  P4 .

115

4.8

The generalized Petersen graph G(10, 3), which is called the Desargues graph.

117

Detecting an invisible spy with receivers. The spy, represented by the gray square, is distance 2 and 5 from the receivers in the right hallway.

122

5.2

Identifying an invisible robber on a star.

123

5.3

The subtrees Ti in T.

129

5.4

The tree T3 with localization number 2.

130

5.5

The pretzel graph.

131

5.6

A graph with degeneracy 3.

131

5.7

The robber may move to w or x in H to avoid capture.

133

5.8

A cop occupying 000 in Q3 . The robber moves from 101 to 100 and C0 detects that the robber has come closer. 138

5.1

xiv 5.9

List of Figures Scanning on a bipartite graph. The cop C4 moves from vertex-to-vertex until R is identified.

142

5.10 A polarity of the Fano plane. The 3-tuples enclosed in rectangles correspond to images of the points. Note that 7 maps to 246. 145 5.11 The polarity graph derived from the polarity given in Figure 5.10.

146

5.12 The Petersen graph represented as the Kneser graph K(2, 5).

148

5.13 The Hoffman-Singleton graph.

150

5.14 We have that ζ(K1,5 ) = 1 and lcapt(K1,5 ) = 4.

154

5.15 The graph G where T3 is not a hideout.

159

6.1

Six vertices (representing trees) are saved by first protecting w, versus nine vertices saved by first protecting u. Protected vertices are squares and those on fire are gray. 164

6.2

Protecting u versus protecting v. Protected vertices are squares.

166

6.3

In Q3 , the fire starts at the gray vertex labeled v. Labels correspond to the round where vertices catch fire or are protected. Protected vertices are squares. 168

6.4

The fire breaks out at v, and protected vertices are squares. The greedy strategy is depicted on the left, which saves seven vertices, while the optimal strategy is on the right saves nine vertices. Labels correspond to the round where vertices catch fire or are protected. 170

6.5

Moving clockwise from the top left, the Cartesian, strong, hexagonal, and triangular grids. 172

6.6

Building a fire wall in G with two firefighters that results in 18 vertices on fire. The fire breaks out at v. Labels correspond to the round where vertices either catch fire or are protected. 174

6.7

The vertex v is big, while vertices u and w are small.

179

6.8

The ladder graph K2 P6 .

183

List of Figures

xv

7.1

An invisible robber evades a single cop.

7.2

A robber evades a single cop in Hyperopic Cops and Robbers. The robber is visible to the cop in their present position, but if the cop enters the triangle, the robber is invisible. 187

7.3

The robber evades the cop by moving between a, b, and c. 193

7.4

A tree T with height 4 and c1 (T ) = 2.

198

7.5

The cop C1 vibrates on an edge and the remaining cops clean Tx .

199

7.6

The Heawood graph has hyperopic cop number 3.

204

7.7

The tree T2 .

208

8.1

The robber captures a cop on the dodecahedron, eliminating them from the game. Two cops are not able to capture the robber, and so the robber wins. 212 ∼ For the graph G = K3 K3 , c(G) = 2 and cL (G) = 3. 214

8.2

186

8.3

The cop C captures the first robber R1 and then a second robber R2 appears. 217

8.4

The growing grids G1 , G2 , and G3 as bold squares. Gray vertices are sources, and circled vertices are those burned in the initial or second round. 223

8.5

The graph G, where the fire breaks out at x, and a is protected in the initial round. The outcome of 1-Firefighter depends on whether the fire burns b or y. 225

8.6

The squares are the blocked vertices, and the Angel occupies the gray vertex. If the Angel has power 1, then with these blocked vertices, they will eventually lose. 228

8.7

The graph G with Wct (G) = 3.

230

8.8

A graph G with c(G) = cc(G) = 2 and γ(G) = 3.

232

8.9

The graph H with c(H) = 2 and cc(H) = 4.

233

Preface

Graphs measure interactions between objects, from follows on Twitter, to transactions between Bitcoin users, and to the flow of energy in a food chain. Whether we think of graphs as abstract collections of dots and lines, or view them as modeling complex interactions in the real world, there is no question that they play an essential role in understanding nature. Graph theory is a robust topic within mathematics, and sets out to formalize the science of interactions. While graphs statically represent interacting systems, we may also use them to model the dynamic interactions within those systems. For example, imagine an invisible evader loose on a graph, leaving behind only breadcrumb clues to their whereabouts. You set out with pursuers of your own, seeking out the evader’s location. Would you be able to detect their location? If so, then how many resources are needed for detection, and how fast can that happen? These basic-seeming questions point towards the broad conceptual framework of the field of pursuit-evasion games. In pursuit-evasion games, a set of pursuers attempts to locate, eliminate, or contain the threat posed by an evader. Pursuit-evasion has a long history, including the early work by Pierre Bouguer in 1732 on a pirate ship escaping a merchant vessel; see the book Chases and Escapes: The Mathematics of Pursuit and Evasion [157] for more discussion.

xvii

xviii

Preface

Our focus is on discrete versions of pursuit-evasion games played on graphs. The rules, specified from the outset, greatly determine the difficulty of the questions posed on page xv. For example, the evader may be visible, but the pursuers may have limited movement speed, only moving to nearby vertices adjacent to them. Such a paradigm leads to the game of Cops and Robbers, and deep topics like Meyniel’s conjecture on the cop number of a graph. Central to pursuit-evasion games is the optimization of certain parameters, whether they are the cop number, burning number, or localization number, for example. Finding the values, bounds, and algorithms to compute these graph parameters leads to fascinating topics intersecting classical graph theory, geometry, and combinatorial designs. The book aims to provide a friendly invitation to both pursuitevasion games and graph theory. An effort is made to make the definitions, examples, and proofs clear and readable. Readers may be undergraduate or graduate students who have taken a previous course in discrete mathematics or graph theory. While our focus is on pursuit-evasion games, we will reveal many fascinating topics in graph theory. Along the way, readers will learn about topics such as treewidth, product graphs, planar graphs, and retracts, to name a few. We summarize the graph theory concepts that will be discussed at the beginning of each chapter. Professional mathematicians and theoretical computer scientists who want to learn about the central pursuit-evasion games topics in one place will find the book a valuable resource, and a trove of conjectures and open problems. Applications of pursuit-evasion games range from robotics [69], to mobile computing [98], and even to programmable matter [78]. Those working in the computing sciences and engineering interested in the mathematical foundations of pursuit-evasion games will find the book a helpful resource. The book begins with an introduction to pursuit-evasion games and graph theory in Chapter 1. More advanced readers may skip this chapter, or use it as a reference for notation. We then focus on one of the most famous pursuit-evasion models in Chapter 2, the game of Cops and Robbers played on graphs. We consider the search number in Chapter 3, where the robber moves infinitely fast and

Preface

xix

searchers must be employed to capture them. The search number and its variants tie in closely with parameters such as pathwidth and treewidth. Graph burning is discussed in Chapter 4, where the pursuer is missing and an evader attempts to burn the graph as fast as possible. Graph burning models the spread of contagion in a network, ranging from memes to viruses. The Localization game is studied in Chapter 5, where the robber is invisible but detectable by the cops via distance probes giving partial information about their whereabouts. Firefighter is discussed in Chapter 6, and is a process analogous to graph burning, but the pursuers attempt to contain a fire spreading in a graph. Chapter 7 considers the situation where the robber is invisible when they are close or far from the cop. Our final chapter collects several recent variants of pursuit-evasion games, such as Cops and Eternal Robbers, Angel and Devil, and Lazy Cops and Robbers. For a typical twelve-week course, the core topics are contained in the first four chapters. With full proofs covered and moving at a leisurely pace, this could take about eight to ten weeks. Topics for the remaining weeks could be taken selectively from the remaining chapters. If certain sections and proofs are skipped at the discretion of the instructor, then the entire book could be covered in either a one- or two-term course. The book may also be used as an adjunct in a graph theory course if the instructor would like to introduce topics in pursuit-evasion games. While we discuss cutting-edge mathematics, we aim to make the book self-contained, understandable, and accessible to a broad mathematical audience. To aid the reader, we have included dozens of figures explaining concepts and proofs. Each chapter contains several exercises that will be helpful to those wishing to expand or polish their skills, and for devising assignments. There are over 170 exercises of varying degrees of difficulty. The book contains a comprehensive bibliography that includes the relevant references for the topics in each chapter. Theorems without proofs are cited, so that an ambitious reader can read those outside the book. An innovation we include at the end of each chapter after the first are research projects that are of a larger scope than those found

xx

Preface

in the exercises. Projects include citations, where the reader is directed to further reading. The reader will likely need to consult a small number of other sources for the additional background to complete these projects. These projects may be done for credit at the end of a course, or used as topics towards an undergraduate thesis. Projects are entirely optional and may be skipped without upsetting the flow of the chapters. The projects may also serve as the basis for an NSF Research Experience for Undergraduates (REU) or NSERC Undergraduate Student Research Award (USRA). Although pursuit-evasion games might be viewed as an emerging topic, its literature is vast, so we have omitted certain advanced directions. We do not consider the robust literature on graph algorithms in pursuit-evasion games. However, along the way, we will discuss a few algorithmic approaches. Further, we focus on deterministic results. A treatment of pursuit-evasion games using the probabilistic method and stochastic models may be found in [49]. Lastly, we restrict the majority of our attention to finite, undirected graphs. I want to thank the reviewers for their excellent feedback on the book, which truly helped the present book into the form it is now. I want to thank my family, friends, students, and co-authors for their generous support. A warm thank you Ina Mette, Marcia Almeida, Erin Donahue, John F. Brady Jr, and the team at the AMS for making this book a reality. A thank you to Melissa Huggan and Trent Marbach for their edits. I would especially like to thank my husband Douglas, who makes my mathematical practice possible. One last thank you goes to you, the reader. I hope your experience reading this book will be as fun and inspiring as mine was writing it.

Chapter 1

Introduction

1.1. Introduction to pursuit-evasion games Imagine searching for a lost companion in a complex cave system, directing a driverless car to avoid obstacles in an urban terrain, or modeling the spread of social contagion on Facebook. These disparate problems have a common mathematical foundation in graphs, consisting of vertices and edges linking them. In pursuit-evasion, we consider games and processes where there is an evader loose on a graph. Time is discrete, measured in rounds, and indexed by nonnegative integers. Round 0 is also called the initial round. Depending on the rules, the evader can move in each round from vertex-to-vertex along edges, jump far ahead along paths, or create contagion that spreads throughout the graph. The evader may also be either visible or not. To complement the evader, we have a set of pursuers. The way pursuers move also varies depending on the type of pursuit-evasion; they may move from vertex-to-vertex, move along paths, or be off the graph altogether, appearing at a vertex. Note that we have a set of pursuers, and so in some scenarios, there could be no pursuers, leaving the evader up to their own devices. The pursuer’s goals may be landing on the vertex of the evader and thereby capturing or eliminating them, detecting their exact location, or surrounding them. We may ask to optimize the number of pursuers

1

2

1. Introduction

needed to win, measure the number of rounds it takes the pursuers to win, or minimize the time required for the evaders to reach a specified goal (such as spreading their influence to each vertex of the graph). One of the best-known and earliest studied topics in pursuitevasion on graphs is the game of Cops and Robbers. In this game, the cops and robber play on vertices, and move at alternate ticks of the clock, with the cops going first. The robber is visible, and players move to adjacent vertices or stay put. The cops win if they can land on the vertex of the robber; otherwise, the robber wins. For example, consider the dodecahedron graph in Figure 1.1. Two cops cannot win there, since at a vertex where the robber is situated, there is always an edge where the robber may escape. However, three cops can win on this graph; see Exercise 1.4.

C C

R

Figure 1.1. Two cops cannot win playing Cops and Robbers on the dodecahedron, but three can.

1.1. Introduction to pursuit-evasion games

3

The earliest publication on Cops and Robbers dates back to the recreational mathematics book Amusements in Mathematics, published in 1917 by Henry Ernest Dudeney [84]. The book contains hundreds of recreational math puzzles. In problem 395, A War Puzzle Game, we find the first historical reference to Cops and Robbers. Since the game is presented as a puzzle, there was no rigorous analysis of it. The first academic work published on the game was by Richard Nowakowski and Peter Winkler in their famous 1983 paper [162]. The game was studied independently by Alain Quilliot back in [170, 171], who considered it in his doctoral thesis. In each of these references, there is only one cop. The first instance of multiple cops was in a paper of Aigner and Fromme from 1984 [3]. From the earlier 1980s to the present day, dozens of papers were published on Cops and Robbers and its variants; see the book The Game of Cops and Robbers on Graphs [47] for a survey up to 2011. Chapter 2 focuses on the game of Cops and Robbers. Another early study of pursuit-evasion games that predates the cited work above on Cops and Robbers was introduced by Tory Parsons in 1976, who, inspired by Richard Breisch [58], modeled explorers lost in a cave system using graph theory. He wrote two papers [165, 166] that provide the framework for the field of graph searching. In Parsons’ models, the invisible evader may be on an edge of the graph, with the graph embedded in three-dimensional space. The goal of the searchers is to sweep the edges continuously, and detect the location of the evader in finite time. This model was independently introduced by Nikolai Petrov in [167]. Since the work of Parsons and Petrov, there have been many similar search models studied. Such searching games connect to important structural properties of graphs and width-type parameters such as treewidth and pathwidth, as we will cover in Chapter 3. See [4, 50, 98] or the more recent [159] for surveys. We may also consider the situation when the pursuer is missing in action and the evader is free to contaminate a graph in a prescribed way. In graph burning, the evader burns vertices over time, and burning vertices spread to their neighbors. Graph burning is motivated by the spread of memes or other social contagion in social networks. The

4

1. Introduction

main challenge in this setting is to minimize the number of rounds where all vertices are burned. We discuss graph burning in Chapter 4. If we add pursuers to the mix, then we arrive at the Firefighting graph process, where the evader’s only job is to burn one vertex, and the pursuers (or firefighters) attempt to minimize the number of vertices that burn. See Figure 1.2. Firefighter will be discussed in Chapter 6.

Figure 1.2. The firefighters, represented by squares, block the spread of a fire, represented by gray vertices, in a grid.

There are many other types of pursuit-evasion games, and we do not attempt to list or survey them all in the book. We consider several popular pursuit-evasion games and processes in-depth in the coming chapters. Apart from those described above, we study the Localization game in Chapter 5, where the robber is invisible but the cops send out distance probes to track their location. The final chapters discuss variants of Cops and Robber such as Lazy Cops and Robbers, where only one cop may move at a time, and games where the cops have limited vision, searching for a robber that only appears

1.1. Introduction to pursuit-evasion games

5

if sufficiently close or far away. See Figure 1.3. For an outline of the chapters, see the Preface.

C R

Figure 1.3. If the robber is invisible and knows all the future moves of the cop, then a single cop cannot capture the robber on the depicted graph.

Our goal is to give the reader a broad introduction to some of the most active topics in pursuit-evasion games. As a first course on pursuit-evasion, our goal is to focus on results whose proofs are accessible to those who completed a first course in graph theory or discrete math. As such, we do not veer into probabilistic graph pursuit-evasion games (see [49] for a detailed treatment), or into the algorithmic aspects of such games. 1.1.1. Graph theory concepts. In the remainder of the chapter, we will cover the relevant introductory properties and notation associated with graphs. Some of the main concepts are the following: (1) Degrees and neighbors. (2) The First Theorem of Graph Theory; see Theorem 1.1. (3) Walks, paths, and cycles. (4) Distance, diameter, radius, and eccentricity. (5) Subgraphs, induced subgraphs, and spanning subgraphs.

6

1. Introduction (6) Graph classes such as complete graphs, null graphs, bipartite graphs, hypercubes, and trees. (7) Graph isomorphisms. (8) Chromatic and domination numbers. (9) Directed graphs.

1.2. Graph theory To keep things self-contained, we begin by covering several basic definitions and concepts from graph theory. We introduce the requisite notation and definitions that will be used throughout the book. A reader familiar with the subject may choose to skip this section, consult it for notation, or at least warm up by completing the exercises at the end. For further background in graph theory, the reader is directed to the texts [83, 185]. If we need additional concepts from graph theory, then these will be introduced in their respective chapters. A graph G is a pair consisting of a nonempty vertex set V (G), an edge set E(G) containing pairs of vertices. Note that E(G) is taken as a multiset, as its elements may occur more than once. An element of V (G) is a vertex and an element of E(G) is an edge. We write uv if vertices u and v form an edge, and say that u and v are adjacent. For consistency, we will use the former term only. We say u and v are incident with the edge uv, and that u and v are endpoints of the edge uv. The order of a graph G is |V (G)|, and its size is |E(G)|; when these are finite, then we say G is finite. Graphs are often depicted by their drawings; see Figure 1.4. Vertices are usually represented as solid black circles, although they can empty circles containing text as annotations. We will consider simple graphs with no loops (that is, edges of the form uu for a vertex u) or multiple edges, unless otherwise stated. Further, unless we say differently, we only consider finite graphs. Given a graph G with vertex v, the degree of v, denoted by degG (v), is the number of edges incident with v. The subscript G may be omitted when there is no risk of confusion. The neighbor set of v, denoted NG (v), is {w ∈ V (G) : vw ∈ E(G)}; any w ∈ NG (v)

1.2. Graph theory

7

Figure 1.4. A drawing of the Heawood graph.

is called a neighbor of v. Notice that degG v = |NG (v)|. The closed neighbor set of v, denoted NG [v], is NG (v) ∪ {v}. We drop the subscripts from NG (v) and NG [v] if G is clear from context. For a nonnegative integer m, if deg(v) = m for all v ∈ V (G), then G is called m-regular. The number δ(G) = minv∈V (G) deg(v) is the minimum degree of G, and the number ∆(G) = maxv∈V (G) deg(v) is the maximum degree of G. A vertex that has degree zero is called an isolated vertex, while a vertex adjacent to all others is called universal. We now introduce our first theorem, called the First Theorem of Graph Theory (also called the Handshaking Lemma), which provides an elementary but important relationship between the degrees and the size of a graph. Theorem 1.1. If G is a graph, then X deg(u) = 2|E(G)|. u∈V (G)

Proof. In the sum, each edge is counted exactly twice.



A corollary of Theorem 1.1 is the following, whose proof is left as Exercise 1.2. Corollary 1.2. In a graph, the number of vertices of odd degree is even.

8

1. Introduction

A walk in a graph G from vertex u to vertex v is a sequence of vertices W = (u = v0 , v1 , . . . , vk = v) if vi vi+1 ∈ E(G) for 0 ≤ i < k. The length of a walk W is the number of vertices in W minus 1 (that is, the number of edges). A walk is closed if v0 = vk . A path is a walk without repeated vertices. The path of order n is denoted by Pn . A cycle is a closed path of length at least 3. We use the notation Cn for a cycle of order n. A Hamiltonian cycle is one that includes each vertex of G exactly once. The length of a shortest path in G between u and v is called the distance between u and v, denoted by dG (u, v). We drop the subscript G and write d(u, v) if G is clear from context. The diameter of G is diam(G) = max{dG (v, w) : v, w ∈ V (G)}. See Figure 1.5. A shortest path between two vertices may not exist, in which case their distance is ∞.

Figure 1.5. A graph of diameter 4.

For a nonnegative integer r and a vertex u in a graph G, define Nr (u) to be the set of vertices in G of distance r to a vertex u. Note that N1 (u) = N (u) and N0 (u) = {u}. We may refer to vertices in Nr (u) as rth neighbors of u. If G is a graph and v is a vertex of G, then the eccentricity of v is defined as max{d(v, u) : u ∈ V (G)}. The radius of G, written rad(G), is the minimum eccentricity over the set of all vertices in G. The

1.2. Graph theory

9

center of G consists of the vertices in G with minimum eccentricity; vertices in the center are called central. A graph H is a subgraph of a graph G, written H ⊆ G, if V (H) ⊆ V (G) and E(H) ⊆ E(G). The graph H is a spanning subgraph if V (H) = V (G). If S ⊆ V (G), then the subgraph of G induced by S, denoted by G[S], has vertices S and edges are those of G with endpoints in S. The subgraph of G obtained by removing a subset S from V (G) or E(G) is denoted G − S. When S contains a single vertex or edge, say x, we write G − x. A graph class is a set of graphs; usually, graph classes are defined by certain common characteristics. We introduce several graph classes that will be used throughout the book. A complete graph of order n, written Kn , contains all edges between its vertices. The complement of a graph G, written G, is the graph with vertices V (G) and edges {xy : xy ∈ / E(G)}. A null graph of order n, written Kn , is the complement of Kn . A set of vertices that is pairwise nonadjacent is an independent set. For a positive integer k, a graph G is k-partite if V (G) can be partitioned into k independent sets called parts. If k = 2, then we say G is bipartite; if all edges are present between the parts, then the graph is complete bipartite. For positive integers i and j, if the parts of a complete bipartite graph have cardinality i and j, then we write this graph as Ki,j . A wheel consists of a cycle of length at least 3 and a universal vertex, and is written Wn . ∼ H, if there is a Graphs G and H are isomorphic, written G = bijection f : V (G) → V (H) such that uv ∈ E(G) if and only if f (u)f (v) ∈ E(H). The bijection f is called an isomorphism. Graphs that are isomorphic are considered “the same.” We may think of isomorphic graphs as different drawings of the same graph. See Figure 1.6 for isomorphic graphs. Two vertices u and v in a graph G are connected if G contains a path between u and v. A graph is connected if every pair of distinct vertices is connected, and disconnected, otherwise. The components of a graph G are the maximal connected induced subgraphs of G, with respect to set inclusion. For a positive integer n, an n-bit binary string is a sequence of length n consisting of zeros and ones. The n-dimensional hypercube

10

1. Introduction

Figure 1.6. Two isomorphic graphs. The depicted graph is named the Petersen graph.

graph, written Qn , has vertices the n-bit binary strings, such that xy ∈ E(Qn ) if and only if x and y differ by exactly one bit. We define Q0 to be the graph with one vertex. See Figure 1.7. Note that Qn has 2n vertices and is n-regular.

Figure 1.7. From left to right, the hypercube graphs Q0 , Q1 , Q2 , and Q3 .

A connected graph with no cycle is a tree. Trees have the property that every pair of vertices are connected by a unique path. See Figure 1.5 for an example of a tree. In a tree G, we have that |E(G)| = |V (G)| − 1; see Exercise 1.8. A subgraph of a tree that is also a tree is called a subtree. Note subtrees are always induced subgraphs. A forest is a graph with no cycle. Hence, a tree is a connected forest. An end-vertex (or leaf ) is a vertex of degree 1. Every tree of order at least 2 has at least two end-vertices; see Exercise 1.8. A tree has either one or two vertices in its center; see Exercise 1.5.

1.2. Graph theory

11

A spanning tree is a spanning subgraph that is a tree. The graph Pn and an n-vertex star K1,n−1 are trees. We call S ⊆ V (G) a dominating set for G if for all v ∈ / S, there exists w ∈ S such that vw ∈ E(G). The minimum cardinality of a dominating set in G is denoted γ(G), and is called the domination number of G. A coloring of a graph is an assignment of labels or colors to its vertices. A proper coloring is achieved when no two neighboring vertices have the same color. The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors required to achieve a proper coloring of G. If G can be colored using at most k colors, then we say that G is k-colorable. We can assign a direction to each edge of a graph G. A directed graph (or digraph) G is a pair consisting of a vertex set V (G) and an edge set E(G) ⊆ {(x, y) : x, y ∈ V (G)}. See Figure 1.8. By default, we will consider undirected graphs, but on occasion directed graphs will be discussed. We use the arrow notation to depict an edge pointing from vertex to vertex. We refer to (x, y) as arcs. A digraph

Figure 1.8. An example of a digraph.

is simple if it contains no directed loops or multiple edges.

12

1. Introduction

The in-degree of a vertex v, written deg− (v), is the number of in-neighbors of v; that is, deg− (v) = |{u ∈ V : (u, v) ∈ E(G)}|. Similarly, the out-degree of a vertex v, written deg+ (v), is the number of out-neighbors of v; that is, deg+ (v) = |{u ∈ V : (v, u) ∈ E(G)}|. A directed graph is oriented if each pair of distinct vertices is in at most one arc. We finish with asymptotic notation. Let f (n) and g(n) be two functions whose domain is some fixed subset of R, and assume that g(n) > 0 for all n. We say that f is of order at most g, written f (n) = O(g(n)), if there exist constants A > 0 and N > 0 such that for all n > N, we have that |f (n)| ≤ A|g(n)|. Observe that f (n) could be negative or even oscillate between negative and positive values (for example, 5 cos(2n)n3 = O(n3 )). We say that f is of order at least g, written f (n) = Ω(g(n)), if there exist constants A > 0 and N > 0 such that for all n > N , f (n) ≥ Ag(n). We say that f is of order g, written f (n) = Θ(g(n)), if f (n) = O(g(n)) and f (n) = Ω(g(n)). For a polynomial p(n) of degree k, p(n) = Θ(nk ). We say that f is of order smaller than g, written f (n) = o(g(n)), if lim

n→∞

f (n) = 0. g(n)

Finally, f is asymptotically equal to g, written f (n) ∼ g(n) or f (n) = (1 + o(1))g(n), if f (n) lim = 1. n→∞ g(n) Some useful properties of asymptotic notation are outlined in Exercise 1.26.

1.3. Exercises

13

1.3. Exercises (1.1) Show that in every graph with order at least two, there are at least two vertices of the same degree. (1.2) Prove Corollary 1.2: In a graph, the number of vertices of odd degree is even. (1.3) Suppose that G is a graph of order n and is isomorphic to its complement. Determine the number of edges of G. (1.4) Show that three cops can win Cops and Robbers played on the dodecahedron graph in Figure 1.1. (1.5) Show that the center of a tree consists of either one or two vertices. Show that this property is false for general graphs. (1.6) Prove that every tree of order n ≥ 2 always has at least two end-vertices. (1.7) Prove that every forest G has at least ∆(G) end-vertices. (1.8) Show that if G is a tree, then |E(G)| = |V (G)| − 1. (1.9) If G is a tree of order n ≥ 2, then show that G has at least ∆(G) end-vertices. (1.10) Draw all the spanning trees of K4 . (1.11) A graph is chordal if each of its cycles of four or more vertices has a chord : an edge joining two vertices that are not adjacent in the cycle. A simplicial vertex has its neighbor set inducing a complete subgraph. Prove that every chordal graph of order at least 2 has at least two simplicial vertices. (1.12) Determine the domination number and chromatic number of the Heawood graph depicted in Figure 1.4. (1.13) Determine the domination number and chromatic number of the Petersen graph depicted in Figure 1.6. (1.14) (a) Show that if G is a graph with no isolated vertices with a minimum cardinality dominating set S, then V (G) \ S is also a dominating set. (b) Prove that for a connected graph G of order n, we have that γ(G) ≤ n2 .

14

1. Introduction

(1.15) Show that if diam(G) = 8, then the complement of G contains a complete subgraph of order 5. (1.16) (a) If G is a graph with the property that δ(G) ≥ b|V (G)/2c, then prove G is connected. (b) Find examples of graphs G with minimum degree smaller than b|V (G)/2c that are disconnected. (1.17) (a) Prove that the hypercube graphs Qn , where n ≥ 1, are bipartite. (b) Prove that a graph is bipartite if and only if it has no cycles of odd length. (1.18) Show that if a bipartite graph G has a Hamiltonian cycle, then each part of G has the same cardinality. (1.19) Show that in a tree, a vertex of maximum eccentricity is an end-vertex. (1.20) (a) Prove that for every graph G, rad(G) ≤ diam(G) ≤ 2rad(G). (b) Provide examples of graphs where the diameter is twice its radius, and examples of graphs where the radius and diameter are equal. . Find a for(c) Show that for every tree G, rad(G) ≤ diam(G) 2 mula for the diameter of G in terms of its radius. (1.21) Prove Mantel’s Theorem: If a graph G of order n contains no 2 triangle, then it contains at most n4 edges. (Hint: Use induction on n and consider a pair of adjacent vertices.) (1.22) If G is a connected graph of order n, then prove that G contains a path of length at least min{2δ(G), n − 1}. (1.23) Consider the class of graphs formed by the following process indexed by positive integers t. At time-step t = 1, we have K1 . At time-step t ≥ 2, we add one end-vertex to the existing graph. (a) Prove that all graphs formed by this process are trees. (b) Prove that every tree results from a graph in this process.

1.3. Exercises

15

(1.24) Prove the First Theorem of Digraphs: for a digraph G, we have that X X deg− (v) = deg+ (v). v∈V (G)

v∈V (G)

(1.25) A tournament is an oriented directed graph where each pair of distinct vertices is in an arc. Show that every tournament has a Hamiltonian path: a directed path visiting each vertex exactly once. (1.26) Prove the following statements, where n is a positive integer. (a) nr = O(an ) for positive real numbers r, a with a > 1. (b) log2 n = O(nr ) for a positive real number r > 0. (c) 2n = o(n!). (d) n! = o(nn ).
k (e) If k is a fixed positive integer, then show that nk ∼ nk! .

Chapter 2

Cops and Robbers

You guide a digital character through a winding maze, chased by a tireless pursuer in an on-line game. One wrong move and the game is over. Unfortunately, you end up in a dead end alley represented by a path with five vertices, with your pursuer close behind you. See Figure 2.1. The pursuer enters the path on the middle vertex. They

C

R

C

R

C=R

Figure 2.1. The pursuer chases the evader on P5 , capturing them after two moves.

17

18

2. Cops and Robbers

move towards you as in the second figure, and the only options are to move closer to them or stay put instead. Moving towards the pursuer results in a quicker endgame, so you stay put. As in the third image in the figure, the pursuer moves closer and captures you. The game is over. While the gameplay is elementary on P5 , it becomes decidedly more complex for general graphs. How do we play on graphs? We restrict the evader and pursuer to play on vertices, so in every round, each player may only occupy a single vertex. The pursuer or cop, goes first. Players move along edges to adjacent vertices, or remain at their current vertex. The evader, which we call the robber, chooses a vertex next. The game’s goal for the cops is to land on the vertex containing the robber as in Figure 2.1. We call this a capture. The game of Cops and Robbers is a game played on a graph, akin to moving checkers or chess pieces around a board. The rules are straightforward, intuitive, and deterministic: there is no randomness involved in either players moves. Notice also that both players can see each other’s moves throughout the game. Quilliot first introduced Cops and Robbers in his 1978 doctoral thesis [170, 171], and independently, the game was introduced in 1983 by Nowakowski and Winkler in [162]. In both cases, there was only one cop. The game was introduced with multiple cops by Aigner and Fromme [3]. In the next section, we formally introduce the game and ramp-up to discuss its many properties and main parameter called the cop number of a graph. There is an ongoing discussion about the name “Cops and Robbers,” and calls to change it to one that is more culturally appropriate. For example, suggestions have been made to refer to the game as “Chasers and Runners,” to deemphasize a connotation of policing within the name and maintaining the letters C and R in the game description. We will continue to use “Cops and Robbers” throughout the book, in keeping with the references on the topic. Readers may opt to use Chasers and Runners or another name as would fit their personal preferences.

2.1. Introduction to the game of Cops and Robbers

19

2.0.1. Graph theory concepts. In this chapter, we will cover Cops and Robbers and the cop number. Some of the main concepts are the following: (1) The cop number of a graph. (2) Cop-win and dismantlable graphs. (3) Retracts and graph homomorphisms. (4) Girth, minimum degree, and the cop number. (5) Guarding isometric paths. (6) The treewidth of a graph. (7) Planar graphs, outerplanar graphs, and graph genus. (8) Meyniel’s conjecture. (9) Projective planes and their incidence graphs. (10) The Moore bound. (11) Matchings and Hall’s condition. (12) The capture time of a graph.

2.1. Introduction to the game of Cops and Robbers Cops and Robbers is a game played on a reflexive graph G, where vertices each have a loop. We can also play on graphs with no loops, and modify the rules to allow for players to not move during gameplay. We keep loops in the version we describe as these are helpful when discussing retracts, introduced in Section 2.2. One detail is that if we allow loops, then for a vertex u with a loop, N (u) = N [u]. To simplify notation, we assume in this setting that N (u) does not contain u. As we did in Figure 2.1, loops are omitted from figures for simplicity. We sometimes refer to G as the board, in analogy with games such as checkers or chess. Multiple edges are allowed, but make no difference to the gameplay, so we always assume there is exactly one edge between adjacent vertices. As we discussed above, there are two players consisting of a set of cops and a single robber. To keep the players gender-neutral,

20

2. Cops and Robbers

we stick to the convention of naming them explicitly as a cop or robber, or switching to singular “they” or “them” when referencing the players. The game is played over a sequence of discrete time-steps or rounds indexed by nonnegative integers, with the cops going first in round 0. The cops and robber occupy vertices; for simplicity, we often identify the player with the vertex they occupy. We refer to the set of cops as C and the robber as R. When a player is ready to move in a round, they must move to a neighboring vertex. With the loops available, players can pass, or remain on their own vertex. Any subset of C may move in a given round. The reader will note that here, Cops and Robbers diverges from games such as chess, where only one piece can move on the board at a time. The variant of Lazy Cops and Robbers, introduced in Chapter 8, will consider the game when one cop alone moves at a time. The cops win the game if, after a finite number of rounds, one of them can occupy the same vertex as the robber. This situation is called a capture. The robber wins if they can evade capture indefinitely. We take a moment to unpack what capture means for the robber. For the robber to win, the game never ends. In a finite graph, that means that the robber will eventually pass or return to a previously occupied vertex. However, for infinite graphs, the robber may visit a new vertex in each round. An example is given in Figure 2.2, which consists of an infinite, one-way path. The robber chooses a vertex to the right of the cop, at least distance two away, and keeps moving right throughout the game. As stated in the introduction, unless otherwise stated, we will assume the board G is finite. A winning strategy for a player is a set of rules that, if followed, results in a win for that player. In the path P5 depicted in Figure 2.1, the sole cop always moves directly towards the robber. We see that this is a winning strategy as no matter what the robber does, they will eventually be captured; for example, they may be cornered at an end-vertex and have no options for escape. There may be many winning strategies for the cop or robber, depending on the graph.

2.1. Introduction to the game of Cops and Robbers

21

... Figure 2.2. An infinite one-way path.

We can flood G with cops, adding many more than are needed. For example, if we place a cop at each vertex of the graph G, then the cops are guaranteed to win. Hence, we have an optimization problem to minimize the number of cops needed to capture the robber. The minimum number of cops required to win is a well-defined positive integer, called the cop number of G. We write c(G) for the cop number of a graph G. We may think of the cop number informally as a measure of how difficult it is for the robber to evade capture. The lower it is, the harder it is for the robber to evade capture. For example, in a complete graph, the cop number is 1, since the robber is always adjacent to the cop. If c(G) = k, then we say G is k-cop-win; in particular, G is copwin if k = 1. For example, a complete graph is cop-win. The following lemma establishes a connection between the cop number and the domination number of a graph. Lemma 2.1. If G is a graph, then c(G) ≤ γ(G). Proof. Place γ(G)-many cops on a minimum cardinality dominating set in G in the first round. No matter where the robber moves, they will be adjacent to at least one cop and so lose in the next round.  While Lemma 2.1 is straightforward to prove, we might call it somewhat whimsically The First Theorem of Cops and Robbers. Not only is it our first theorem on the cop number in the book, it is a useful one, giving tight bounds on the cop number for certain graphs. We may think of the domination number as a kind of first-order approximation of the cop number. That being said, the domination

22

2. Cops and Robbers

number can be much larger than the cop number, as our next results show. The following lemma gives the value of the cop number for cycles and paths. The disjoint union of graph G and H, written G + H, is the graph formed by taking the disjoint union of the vertices of G and H and the disjoint union of the edges of G and H. Lemma 2.2. Let n be a positive integer. (1) If n ≥ 1, then c(Pn ) = c(Kn ) = 1. (2) If n ≥ 4, then c(Cn ) = 2. (3) For graphs G and H, we have that c(G + H) = c(G) + c(H). Proof. For item (1) and the case of Pn , we place the cop on a central vertex, and move them towards the robber in each subsequent round. To evade capture, the robber must eventually occupy an end-vertex. After at most b n2 c rounds, the cop will capture the robber. The case for Kn follows by Lemma 2.1. The proof of item (2) has a similar flavor as for paths. On a cycle, place one cop on a fixed vertex and have them stay there for the remainder of the gameplay. The remaining cop then chases the robber towards the fixed cop to capture. This winning strategy for the cops shows that c(G) ≤ 2. A more efficient winning strategy for the cops is to place them maximum distance apart and have them both move towards to the robber. To show that Cn is not cop-win, place the robber distance at least two away from the cop. In the next round, no matter the move of the cop, the distance between them is at least one. The robber may then move away, and be distance at least two from the cop. We may repeat this process to give a winning strategy for the robber. For the final item, we place c(G) cops in G and c(H) cops in H. We execute the winning strategies for the cops in G and H, so no matter the moves of the robber, they are captured in whichever component they occupy. We then have that c(G + H) ≤ c(G) + c(H). If we have fewer than (c(G) + c(H))-many cops, then at least one of G or H has too few cops to capture the robber. We place the robber in that component, and they may evade capture. 

2.2. The structure of cop-win graphs

23

By Lemma 2.2 (3), the cop number is additive on components; that is, the cop number of a disconnected graph is the sum of the cop number of its components. Hence, the only interesting arena to study the cop number is for connected graphs. Hence, for the remainder of the chapter, we will only consider connected graphs. Following up on our discussion on the domination number, it can be shown that the domination number of Pn is b n3 c; see Exercise 2.2. Hence, the difference between the parameters c(G) and γ(G) may tend to infinity as a function of n.

2.2. The structure of cop-win graphs When considering the cop number, one approach is to think about the situation where a single cop has a winning strategy. Arguably, this is one of the simplest settings for the game. Historically, having only one cop is how the game was first considered in [162, 170, 171]. Trees have an elegant recursive structure, based on end-vertices. Every tree with order at least 2 has at least two end-vertices. Deleting an end-vertex from a tree gives another tree; hence, we can recursively delete vertices to ultimately arrive at a single vertex. We can also go in reverse, as every tree results by adding end-vertices to K1 . Trees are also an important, early example of cop-win graphs. The following result is a part of folklore. Theorem 2.3. The cop number of a tree is 1. Proof. The cop occupies a central vertex in the first round. Whatever vertex the robber occupies, there is a unique path between the cop C and the robber R. The cop always moves towards the robber along the unique path. To see that this is a winning strategy, note that the distance between C and R is monotonically decreasing. The robber will end up on an end-vertex, and the distance between them is strictly decreasing, resulting in an inevitable capture.  Theorem 2.3 gives a greedy winning strategy for the cops of always trying to minimize the distance to the robber. Further, to avoid capture as long as possible, the robber must occupy an end-vertex. Call the unique vertex adjacent to an end-vertex u its parent. In a

24

2. Cops and Robbers

certain sense, end-vertices are useless to the robber, as the cop will move to its parent and ensure capture in the next round. Trees form a microcosm of cop-win graphs, which have an analogous structure replacing end-vertices with so-called corner vertices. As we will learn in Theorem 2.7, cop-win graphs may be fully characterized via this kind of iterative deletion of corners. Before we can delve deeper into the structure of cop-win graphs, we need the notion of graph homomorphisms and retracts. A homomorphism f from graphs G to H is a function f : V (G) → V (H) which preserves edges; that is, if xy ∈ E(G), then f (x)f (y) ∈ E(H). We simply write f : G → H. Unlike an isomorphism, a homomorphism does not need to be injective or surjective. For example, there is a homomorphism from a C6 into K2 . To see this, we alternately label the vertices of C6 with either a 1 or 2. We can define a homomorphism by mapping all the vertices labeled 1 to a fixed vertex of K2 and map the rest to the remaining vertex. The same works for a bipartite graph; see Exercise 2.6. Reflexive graphs have the interesting property that we can homomorphically map the entire graph to a fixed single vertex. Let H be an induced subgraph of G. We say that H is a retract of G if there is a homomorphism f : G → H such that f (x) = x for x ∈ V (H); that is, f is the identity on H. The map f is called a retraction. See Figure 2.3 for an example. While retracts are induced subgraphs, not every induced subgraph is a retract. For example, if we consider a wheel W4 , then deleting the central vertex u results in a H = C4 that is not a retract. To see this, note that however we map u to a vertex v of H, v will not be adjacent to all other vertices, so the mapping is not a homomorphism. Retracts play an essential role in the characterization of cop-win graphs. The next theorem, due to Berarducci and Intrigila [15], shows that the cop number of a retract never increases. The proof uses the interesting method of parallelism, playing two interacting games simultaneously. Theorem 2.4 ([15]). If H is a retract of G, then c(H) ≤ c(G).

2.2. The structure of cop-win graphs

25

1

2

3

4

Figure 2.3. The subgraph induced by {1, 2} is a retract, with the retraction mapping 3 to 1, 4 to 2, and fixing 1 and 2.

Proof. Suppose that k cops have a winning strategy in G, and let f : G → H be a retraction. We consider two parallel Cops and Robbers games: one played in G and one in H. The game in H may be considered as being played in G, since H is an induced subgraph of G. The strategy of the cops in G may not be sufficient to capture the robber in H. For example, the robber may need to leave H to be captured in G. We therefore consider the following shadow strategy. Let the cops in G play as usual. In H, the cops play as the images under f of the cops in G. For simplicity, we label the images of the cops, or shadows, as f (C). That is, if a cop C moves from vertex u to v, then a cop f (C) moves from f (u) to f (v). These moves are possible as f is a homomorphism. We claim the shadow strategy is a winning one for f (C). Let the cops play in G with R restricted to H. Suppose the cops are about to win in G. It must be that R and each of its neighbors v in H (as

26

2. Cops and Robbers

well as its neighbors in G − H) are adjacent to some cop. However, the edge RC becomes Rf (C) under the retraction, and vC becomes vf (C). Therefore, N [R] ⊆ N [f (C)] in H, and the robber loses in the game played in H in the next round. Hence, c(H) ≤ k, and the proof follows.  We will see shadows appear later in various contexts, such as when we guard isometric paths in Theorem 2.11. An immediate consequence of the previous theorem is the following corollary. Corollary 2.5. If G is cop-win, then so is each retract of G. Proof. As c(G) = 1, and c(H) ≤ c(G) by Theorem 2.4, we have that c(H) = 1.  A corner in a graph G is a vertex u such that N [u] ⊆ N [v] for some v ∈ V (G). We write u → v; in analogy with trees, we say v is the parent of u. For example, an end-vertex is a corner. Hence, every tree with order at least two contains at least two corners. Corners are useless for the robber: if they stay there long enough, a cop will move to a parent and capture them. For cop-win graphs, we can always discover at least one corner. Lemma 2.6 ([162, 170, 171]). If G is a cop-win graph, then G contains at least one corner. Proof. Consider the second to last move of the cop C. The robber R could pass, so C must be adjacent to R. The robber could move to a neighboring vertex, so C is adjacent to each neighbor of R.  A graph is dismantlable if there exists some sequence of corners whose iterative deletion results in K1 , the graph with one vertex. For example, each tree is dismantlable: delete end-vertices repeatedly until a single vertex remains. With the help of retracts, we can now prove the following important theorem. Theorem 2.7 ([162, 170, 171]). If G is cop-win, then G is dismantlable.

2.2. The structure of cop-win graphs

27

Proof. We use induction on the order of G. The base case is straightforward as G ∼ = K1 . By Lemma 2.6, each cop-win graph contains a corner u with parent v. Form the retract G − u by retracting u to v. By Theorem 2.4, G − u is cop-win. As G − u is dismantlable by the induction hypothesis, and the proof follows.  Theorem 2.7 gives a necessary condition for a graph to be copwin. For example, in cycles of order at least 4, there are no corners, so we can immediately say they are not cop-win. Corner may exist in a graph G but G may not be cop-win. For example, a four-cycle with five end-vertices adjacent to a fixed vertex is not cop-win, as the reader may verify. The following theorem characterizing cop-win graphs, the main one of this section, was proved by Nowakowski and Winkler [162], and independently by Quilliot [170]. Theorem 2.8 ([162, 170, 171]). A graph is cop-win if and only if it is dismantlable. Proof. The forward direction is Theorem 2.7. We next suppose that G is dismantlable. We use induction on the order of G to show G is cop-win. The base case is immediate as G ∼ = K1 . Suppose that G is a dismantlable graph of order n + 1, where n ≥ 1 is fixed. We then have that G contains some corner u, with u → v for some vertex v, so that G − u is dismantlable. By the induction hypothesis, G − u is cop-win as it has order n. We use this fact to show that G is cop-win. The cop plays in G − u using their winning strategy on there, but so that if R moves to u, then C moves as though R moves to v. This situation is possible as u → v. Now C eventually captures the image of the robber f (R) with their winning strategy on G − u. Either R = f (R) in which case the robber is captured in G − u, or R is on u with C on v. In the latter case, R is on a corner so the cop wins in the next round.  We observe that Theorem 2.8 gives us a way of viewing cop-win graphs in terms of certain graph labelings of vertices. On the surface, this is remarkable as the game has nothing explicitly to do with graph labelings. See Figure 2.4 for a graph with a cop-win labeling.

28

2. Cops and Robbers

5

3

6 1

4

2

Figure 2.4. A cop-win labeling of a cop-win graph.

Given Theorem 2.8, cop-win graphs G are precisely those with a so-called cop-win labeling: a labeling of the vertices of G by positive integers 1, 2, . . . , n in such a way that for each i < n, the vertex i is a corner in the subgraph induced by {i, i+1, . . . , n}. For example, given a complete graph of order n, if distinct vertices receive different labels, then we have a cop-win labeling. The result is (n!)-many distinct such cop-win labelings. We invite the reader to find a cop-win labeling of the graph in Figure 2.5, thereby showing that it is cop-win by Theorem 2.8. The graph G is an interesting example of a cop-win graph as it only contains a single corner vertex.

2.3. Bounds on the cop number and guarding As with any graph parameter, having both upper and lower bounds is useful. An interesting bound sets up a connection between the cop number and the minimum degree of a graph. To motivate the bound, suppose the robber is on a vertex u with degree ten, and nine cops

2.3. Bounds on the cop number and guarding

29

Figure 2.5. A cop-win graph G.

are chasing them. There is a chance the robber can evade capture, so long as the cops do not dominate N (u). For example, a single cop may be adjacent to two or more neighbors of the robber. However, if we have no 4-cycles, this situation never arises. Theorem 2.9 ([3]). If G is a graph with girth at least 5, then we have that c(G) ≥ δ(G). Proof. Let δ(G) = d and suppose that (d − 1)-many cops play the game. For the robber R to survive the first round, we must show that there is a vertex of G not adjacent to a vertex of the set of cops C. Suppose otherwise; that is, suppose to the contrary that C is a dominating set. Let u be a vertex outside of C. Suppose that u is adjacent to x ≥ 1 many vertices in C, and y many vertices not in C. Let X be the set of vertices in C adjacent to u, and let Y be the set of vertices not in C adjacent to u. Note that x + y ≥ d. As C is a dominating set by our assumption, each vertex of Y is adjacent to some vertex of C.

30

2. Cops and Robbers

As there are neither three nor four cycles, no vertex of Y is adjacent to a vertex of X, and no two distinct vertices in Y share a common neighbor in C. Hence, each vertex of Y is adjacent to a unique vertex of C \ X. It follows that d − 1 = |C| ≥ x + y ≥ d, which is a contradiction. Thus, some vertex is not adjacent to C, and R may choose that vertex in round 0. The remainder of the proof follows by induction. Suppose that we are in round t ≥ 0, and the robber has arranged things so they occupy a vertex u which is not adjacent to a vertex in C. As there are no four cycles, each cop is adjacent to at most one neighbor of u. Hence, there is at least one neighbor not adjacent to a cop, and R can move there to be safe for the next round t + 1. In this way, the robber may avoid capture indefinitely.  Notice that the bulk of the proof of Theorem 2.9 is devoted to showing that the robber may choose a “safe” vertex in the first round. Once we achieve that, the remainder of the proof follows by an induction; further, at this stage, we only use that four cycles are missing. If we relax the hypothesis to having no cycles of length four but possibly having cycles of length three, then we have a weaker bound on the cop number. A graph G is H-free if it does not contain H as a subgraph. With a weaker hypothesis than in Theorem 2.9, we obtain another lower bound on the cop number. Theorem 2.10 ([24]). Let t ≥ 1 be an integer. If G is K2,t -free, then c(G) ≥ δ(G)/t. Proof. Suppose there are k cops playing, where k < δ(G)/t. We show first that the robber may choose a vertex that is not adjacent to a cop in the first round. Let S be the set of vertices occupied by the cops, and let u be a vertex not in S. If no vertex of S is adjacent to u, then the robber can pick vertex u. Suppose that N (u) ∩ S 6= ∅. Since G is K2,t -free, a cop C which is adjacent to u can be adjacent to at most t − 1 other neighbors of u. Hence, C is equal or adjacent to at most t neighbors of u. Similarly, a cop not adjacent to u can be

2.3. Bounds on the cop number and guarding

31

adjacent to at most t − 1 neighbors of u. It follows that the number of neighbors of u that are equal or adjacent to a cop is at most kt. Since kt < δ(G), this means that there must be some neighbor v of u which is not adjacent to a cop, and the robber can begin the game on v. We proceed now in a fashion similar to the last part of the proof of Theorem 2.9. Suppose that in round t ≥ 0 of the game, the cops and robber have moved and the robber is on some vertex u such that no cop is adjacent to u. Suppose in round t+1 the cop moves to some vertex of their neighbor set. A similar argument to that in the initial round shows that there is a neighbor v of u which is not adjacent to a cop, so the robber can move to v in round t + 1 and avoid capture for another round.  As an immediate consequence of Theorem 2.10, we have that if there are no four cycles in G, then c(G) ≥ δ(G)/2. We next introduce the key notion of guarding subgraphs. The goal here is to show that the cops can arrange things so parts of the graph are off-limits to the robber: entry into those parts is bad news for the robber. These ideas will be useful when we discuss planar graphs in the next section. An induced subgraph H of G is k-guardable if after finitely many moves, k cops can arrange themselves in H so that the robber is immediately captured upon entering H. We say that H is guarded and that the cops guard H. For example, a complete graph is 1guardable. A path P in G is isometric if for all x, y ∈ V (P ), distP (x, y) = distG (x, y). Hence, a path is isometric if the shortest path between its vertices is on the path itself. The following theorem shows that one cop can guard an isometric path, which will be important when we discuss the cop number of planar graphs. Theorem 2.11 ([3]). Isometric paths are 1-guardable.

32

2. Cops and Robbers

Proof. Suppose P is an isometric path in a graph G, with V (P ) = {v0 , v1 , . . . , vk }. If we let Di = {x ∈ V (G) : distG (x, v0 ) = i}, then we see that vi ∈ Di for i = 0, 1, . . . , k. The cop C is restricted to P , and plays as if the robber R is on vj whenever R is on some vertex of Dj for j = 0, 1, . . . , k − 1, and also as if R is on vk for j ≥ k. Thus, C chases an “image” of the robber called the shadow of R and denoted R0 (we reuse the term “shadow” here as first introduced in the proof of Theorem 2.4). If R is on a vertex of Dj , then R can only move to Dj−1 , Dj , or Dj+1 , so R0 can only move to vj−1 , vj , or vj+1 . Since P is a path, C will eventually capture R0 on P . Therefore, if R enters P , then R will take the place of R0 and be captured.  As referenced in the proof of Theorems 2.4 and 2.11, the cop captures the robber’s shadow, or image under the retraction. The concept of a shadow is a central one in Cops and Robbers, and will be useful later in the chapter. Given Theorem 2.11, we may think of isometric paths as beats that the cops patrol. One cop can keep a beat robber-free indefinitely, after some initial ramp up. The idea of covering beats with a single cop play an important role in giving a general upper bound on the cop number of connected graphs; see Section 2.5. We finish this section by considering an upper bound for the cop number in terms of treewidth. In a tree decomposition, each vertex of the graph is represented by a subtree, such that vertices are adjacent only when the corresponding subtrees intersect. Formally, given a graph G, a tree decomposition is a pair (X, T ), where X = {B1 , B1 , . . . , Bn } is a set of subsets of V (G) called bags, and T is a tree whose vertices are the subsets Bi , satisfying the following three properties. Sn (1) V (G) = i=1 Bi . That is, each graph vertex is associated with at least one tree vertex. (2) For every edge (v, w) in the graph, there is a subset Bi that contains both v and w. That is, vertices are adjacent in G only when the corresponding subtrees have a vertex in common.

2.3. Bounds on the cop number and guarding

33

(3) If Bi , Bj and Bk are vertices, and Bk is on the path from Bi to Bj , then Bi ∩ Bj ⊆ Bk . Item (3) is equivalent to the fact that for each vertex x, the bags containing x form a subtree of T. The width of a tree decomposition is the cardinality of its largest bag Bi minus one. The treewidth of a graph G, written tw(G), is the minimum width among all possible tree decompositions of G. A tree decomposition whose width is the treewidth is called optimal. For more on treewidth, see [83]. We may think of treewidth as a measure of how far a graph is from being a tree. If a graph has treewidth 1, then it is a forest; this is the reason why we subtract 1 in the definition of treewidth. At the other extreme, if a graph of order n has treewidth n − 1 exactly when it is a complete graph. See Exercise 2.12. A graph may have several different tree decompositions, and each graph has a tree decomposition where each bag has cardinality 1. Note also that bags are multi-sets of vertices. See Figure 2.6.

5 356

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3

12

234

346 467

1

2

4

7

Figure 2.6. A graph along with an optimal tree decomposition.

Cop number and treewidth are connected by the following theorem. Theorem 2.12 ([122]). If G is a graph, then   tw(G) + 1. c(G) ≤ 2 Proof. The idea of the proof is straightforward: we guard bags (or, more precisely, the subgraph induced by bags), and show that we may

34

2. Cops and Robbers

move the cops to other bags to guard them, without letting the robber into previously guarded ones. If we can accomplish these tasks, then we can chase the robber into a bag that is an end-vertex in the tree decomposition and capture them, since they cannot escape that bag. In this fashion, we will have a winning strategy for the cops. l m  tw(G) First, we may guard a fixed bag B using + 1 -many 2 cops. To see this, list the vertices of B as b1 , b2 , . . . , bk , where k ≤ tw(G) + 1. By reordering the indices if necessary, we assign one cop to guard a shortest (and hence, isometric) path connecting bi to bi+1 , for all 1 ≤ i ≤ k. If k is odd, then we place an additional cop to sit on bk . As every vertex is in at least one of these guarded paths or equals bk , the cops may guard all of B in this fashion. The next step is to show that the cops can move between adjacent bags B and B 0 in the tree decomposition, so that if B is guarded, then B 0 becomes guarded, and the robber cannot enter B without being captured. The latter condition is important, since otherwise, the robber may find an entry point into B, which would defeat the cops’ strategy. For this, we carefully move the cops from guarding isometric paths in B to those in B 0 , breaking this into cases. In the first case, if the cop is guarding a path in B ∩ B 0 , then there is nothing to do, and the cops can stay put. In the next case, if the cops guard a path with endpoints in B \ B 0 , then we move them to guard two unguarded vertices in B 0 . As before, they may guard at the last unguarded vertex in B 0 , if it exists. In the final case, the cops guard an isometric path with one endpoint bi in B \ B 0 and the other being bj in B ∩ B 0 . When moving to B 0 , we must ensure the robber cannot enter B through bj . For this, the cop moves to bj if they are not already there. That cop then guards a path connecting bj to an arbitrary as yet unguarded vertex of B 0 . In all three cases, in no round may the robber move to B ∩ B 0 without being captured. We have shown that the cops may move between bags, while disallowing the robber to move to previously guarded bags. The proof follows.  If each bag induces a complete subgraph, then the idea of the proof of Theorem 2.12 shows that the graph is cop-win. To see this,

2.4. The cop number of planar graphs

35

note that one cop can guard a bag B and that cop can move to B ∩B 0 without concern that the robber will enter B. This fact gives one proof that chordal graphs (introduced in Exercise 1.11) are cop-win, as such graphs have tree decompositions with bags inducing complete subgraphs (see Proposition 12.3.11 of [83], for example).

2.4. The cop number of planar graphs Imagine playing Cops and Robbers in a maze, where your moves are limited by the walls surrounding you. We may view this as a graph by taking junctions where your path splits or changes directions as vertices, and corridors you traverse as edges. See Figure 2.7. There are no short cuts in the maze, of course: you cannot jump over the walls, which makes these puzzles fun and challenging.

Figure 2.7. A maze and its corresponding graph.

We may also imagine playing Cops and Robbers on a map, where you can move between countries if they share a border. Playing Cops and Robbers on your favorite maze or map is equivalent to playing the game on a planar graph. A graph is planar if it may be drawn in the plane, so edges only intersect at vertices. In fancier language, planar graphs are those embeddable in a nonorientable surface of genus 0. We will revisit the idea of graph genus later in this section.

36

2. Cops and Robbers

Planar graphs are famous in graph theory owing to the Four Color Theorem, which states that the chromatic number of a planar graph is at most 4. Planar graphs may be characterized in the following way. A graph H formed from G by first taking a subgraph (that is, removing some vertices and edges from G) and then contracting some of the remaining edges is said to be a minor of G. Kuratowski’s Theorem says that a graph is planar if and only if it has neither K5 nor K3,3 as a minor; see [83]. Aigner and Fromme [3] proved the first and arguably most influential result on the cop number of planar graphs. Theorem 2.13 ([3]). If G is a planar graph, then c(G) ≤ 3. The proof of Theorem 2.13 is elementary but nontrivial. It makes essential use of guarding isometric paths, as discussed in the previous section. The idea of the proof of Theorem 2.13 is to increase the cop territory; that is, a set S of vertices such that if the robber moved to S, then they would be caught. The cops may begin on a single vertex, and that vertex trivially is cop territory. If the territory can always be increased, then the number of vertices the robber can move to without being caught is eventually reduced to the empty set, and so the robber is captured. The cop does so by ensuring the cop territory is always one of three types. For each type, the cops inductively use isometric paths to enlarge the cop territory and remain among the three types. For more details, the reader is directed to the proof presented in [47] or the original one in [3]. It is interesting to note that the proof only uses implicitly the Jordan Curve Theorem and no other deeper properties of planar graphs. We focus our attention in this section on a special kind of planar graph, where proofs on the cop number become more manageable. A graph is outerplanar if it can be embedded in the plane so that the following properties hold: (1) Every vertex lies on a circle. (2) Every edge of G either joins two consecutive vertices around the circle or is a chord across the circle. (3) If two chords intersect, then they do so at a vertex.

2.4. The cop number of planar graphs

37

There is always a drawing of an outerplanar graph in the plane so all its vertices are on the outer face. See Figure 2.8 for an example of an outerplanar graph.

Figure 2.8. An outerplanar graph.

Outerplanar graphs are precisely those which do not contain a K2,3 and K4 as a minor; see [83]. The following result was first proven by Nancy Clarke in her doctoral thesis. Theorem 2.14 ([70]). If G is an outerplanar graph, then c(G) ≤ 2. Note that Theorem 2.14 follows directly from Theorem 2.12, since outerplanar graphs are known to have treewidth at most 2; see Exercise 2.14. We give a direct proof of Theorem 2.14, and one that is simpler than the ones for Theorem 2.13. A cut-vertex v in a graph G is a vertex such that G − v is disconnected. A graph is 2-connected if it contains no cut-vertex. The proof breaks into the case when G is 2-connected or not. In the 2-connected case, one first observes that the boundary of the outer face is a Hamiltonian cycle, and the two cops successively enlarge their territory by moving along the cycle towards the robber. We focus on the 2-connected case.

38

2. Cops and Robbers

For the case where there are cut-vertices, you first take a retraction of blocks to cut-vertices, and then using the winning strategy from the 2-connected case to chase the robber into a block where the robber cannot escape. See Exercise 2.21. Proof of Theorem 2.14. As stated before the proof, we focus on the case that G is 2-connected, so G has no cut-vertices. Note that this condition implies that there are no end-vertices. We label the vertices clockwise on the circle containing the vertices of the outerplanar graph by vi , where 0 ≤ i ≤ n − 1. We first claim that we obtain a Hamiltonian cycle such that vi is adjacent to both vi−1 and vi+1 , with subscripts taken modulo n. With that, the circle along the outer face becomes a cycle. For simplicity, we call this the cycle of G. Too see this, suppose that for contradiction that for a given i that vi is not adjacent to vi+1 . We can renumber the subscripts so that i = 0. Since G is connected and the degree of v0 is at least two, then let vj be the vertex of least index which is adjacent to v0 . The edge v0 vj prevents a vertex in {v1 , v2 , . . . , vj−1 } from being adjacent to a vertex of {vj+1 , vj+2 , . . . , vn−1 }, and no vertex of {v1 , v2 , . . . , vj−1 } is adjacent to v0 . Hence, v0 is a cut-vertex, which is a contradiction. If each vertex is of degree two, then G itself is a cycle, and the proof follows by Lemma 2.2. Hence, there are vertices of degree three appearing in the cycle. Let x0 , x1 , . . . , xk be the vertices of degree at least three in the cycle, ordered clockwise. Between every two vertices xi and xi+1 , we have a unique path, and so if the robber is trapped by cops at those vertices, they will be captured. Place the cops C1 and C2 on x0 and x1 . The robber would not choose a vertex on the path connecting these vertices, or they would lose as there are no chords to use for an escape. We can think of the path connecting x0 and x1 as an off-limits zone for the robber. This discussion is the beginning of an inductive argument that seeks to increase this off-limits zone for the robber. We now assume more generally that C1 and C2 are on xi and xj for some i < j, respectively, and that the robber is not between or on

2.4. The cop number of planar graphs

39

a vertex from xi to xj . See Figure 2.9 for the case considering x1 and x2 .

Figure 2.9. The path between x1 and x2 .

This region of the graph is then the cop territory, which is our offlimits zone for the robber. We show that eventually, all of G becomes cop territory, and so the robber is captured. Suppose first there is no chord from either xi or xj to outside the cop territory. We then slide C2 over to xj+1 . There is no way for the robber to enter the path connecting xj and xj+1 , so the cop may do this and ensure the robber cannot enter the cop territory. In this way, we have enlarged the cop territory from xi to xj+1 . Next, suppose one of xi or xj has a chord to a vertex outside the cop-territory. Without loss of generality, we assume that such a chord is incident with xi . Such a chord is of the form xi xm for some index m distinct from i. We choose m such that xm is closest to xj . If the robber is on the path of the cycle connecting xi and xm not including xj , then the robber cannot enter into the path connecting xm with xj . We may then move C2 to xm and increase the copterritory. If the robber is on the path between xj and xm , then C1 moves from xi to xm and also increases the cop territory. See Figure 2.10.

40

2. Cops and Robbers

xi

C

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 1xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

C2

xj

R

xm Figure 2.10. Expanding the cop territory in the final case. The shaded vertices are cop territory.

In each of the three cases, we may increase the cop territory. The proof follows by induction.  While a formal discussion of surfaces and graphs on surfaces would take us too far a field, we instead direct the reader to [154] or [46] for a discussion. Examples of surfaces are the plane or the torus. The genus of a surface roughly is its number of “holes.” The plane has genus 0, while the torus with one hole has genus 1. The genus of a graph G is the minimum g such that G is embeddable in a surface of genus g. For example, the genus of K5 is 1; see Figure 2.11. Much less is known about the cop number of graphs with positive genus. Schr¨ oder conjectured that if G is a graph of genus g, then c(G) ≤ g +3. By Theorem 2.13, Schr¨oder’s conjecture holds for g = 0.

2.5. Meyniel’s conjecture on the cop number

41

Figure 2.11. An embedding of K5 on the torus.

In the case g = 1, Lehner proved in [136] that c(G) ≤ 3. Recent work by Bowler, Erde, Lehner, and Pitz [55] gives the best-known bound on the cop number for graphs with a given genus. Theorem 2.15 ([55]). If G is a graph of genus g, then   4g 10 c(G) ≤ + . 3 3

2.5. Meyniel’s conjecture on the cop number We considered cop-win graphs in Section 2.2, where there was only a single cop playing the game. Cop-win graphs are characterized by a simple recursive structure. An inverse problem comes from considering graphs that need a large number of cops to capture the robber. If G is not connected and order n, then the cop number may be as high as n as we see from a complement of a complete graph. To make the problem of finding graphs with large cop number more interesting, we restrict our attention to connected graphs. In √ that case, the cop number can reach order n. To see this, we introduce incidence graphs of projective planes. A projective plane consists of a set of points and lines satisfying the following axioms: (1) There is exactly one line incident with every pair of distinct points.

42

2. Cops and Robbers (2) There is exactly one point incident with every pair of distinct lines. (3) There are four points such that no line is incident with more than two of them.

We only consider finite projective planes. The axioms do not tell us how many points are in a line, or the number of lines through a point. It can be shown that projective planes have q 2 + q + 1 many points and lines for an integer q, each point is on (q + 1)-many lines, and each line contains (q + 1)-many points. The only orders where projective planes are known to exist are prime powers; indeed, this is a deep conjecture in finite geometry. For these items and further background on projective planes, see [64]. For a given projective plane P , define G(P ) to be the bipartite graph with red vertices the points of P, and the blue vertices represent the lines. Vertices of different colors are adjacent if they are incident. We call this the incidence graph of P. See Figure 2.12 for G(P ), where P is the Fano plane, the projective plane of order 2. We note

1

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4 345

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Figure 2.12. The Fano plane and its incidence graph. Vertices corresponding to lines are represented by triples.

the incidence graph of the Fano plane is isomorphic to the Heawood graph.

2.5. Meyniel’s conjecture on the cop number

43

The graphs G(P ) have no odd cycles, and axioms (2) and (3) tell us that there are no four cycles. For example, if there was a four cycle, it would consist of two points and two lines. Since two points are incident with a unique line, this gives a contradiction. In particular, the girth of G(P ) is at least 6. Hence, as G(P ) is (q + 1)-regular, Theorem 2.9. gives us that c(G(P )) ≥ q + 1. As G(P ) has order √ 2q 2 +2q+2, this gives that c(G(P )) = O( n). In fact, c(G(P )) = q+1; see Exercise 2.22. In 1985, Henri Meyniel conjectured that projective planes are examples of connected graphs with the asymptotically largest cop number. The following conjecture appeared first in Frankl’s paper [99] as a personal communication. Meyniel’s conjecture: If G is a connected graph of order n, then (2.1)

√ c(G) = O( n).

Hence, (2.1) tells us that for n sufficiently large there is a constant d > 0 such that √ c(G) ≤ d n. Meyniel’s conjecture appears incredibly difficult to resolve. One can argue it is the deepest problem on the cop number. For many years, the best-known upper bound was the one proved by Frankl [99]. For a positive integer n, let c(n) be the maximum cop number of a connected graph of order n. Theorem 2.16 ([99]). For n a positive integer, we have that 

log log n c(n) = O n log n

 .

For the proof of Theorem 2.16, we use the following inequality called the Moore bound, connecting the order of a graph, its maximum degree, and diameter. The bound was named after Edward Forrest Moore, as referenced in [118].

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2. Cops and Robbers

Theorem 2.17 (Moore bound). Let G be a graph of order n, with maximum degree ∆ > 2 and diameter D. We then have that   (∆ − 1)D − 1 (2.2) n≤1+∆ . ∆−2 Proof. Fix u a vertex of G. We then have that u has at most ∆ neighbors. If v is a neighbor of u, then v has at most ∆ − 1 neighbors not in N [u]. Hence, there at most ∆(∆ − 1) vertices at distance two from u. In general, for 1 ≤ k ≤ D, there are ∆(∆ − 1)k−1 vertices at distance k from u. As every vertex in G is some integer distance to u (recall that we are assuming that graphs in this chapter are connected), we then have that ! D−1 X k |V (G)| ≤ 1 + ∆ (∆ − 1) k=0

 =

1+∆

(∆ − 1)D − 1 ∆−2

 ,

where the equality follows by summing the geometric series. The proof follows.  We now turn to the proof of Theorem 2.16. Proof of Theorem 2.16. Each closed neighbor set of a vertex u of maximum degree ∆ is 1-guardable, and Theorem 2.11 provides that an isometric path is 1-guardable. Bythe Moore bound (2.2), both ∆  log n and D cannot be less than O log log n . In particular, there is a closed neighbor set or isometric path X of order at least f (n) = logloglogn n in G. Note that f (n) may not be integer-valued, but without loss we may assume it is by applying a floor to the fraction. We leave one cop on X to guard it. Delete X to form the graph G1 . Although graph G1 may be disconnected, the robber is confined to a component G0 of this graph. The cops then move to G0 . We then have that (2.3)

c(G) ≤ c(G0 ) + 1,

since X is 1-guardable. See Figure 2.13.

2.5. Meyniel’s conjecture on the cop number

45

C X

R

G'

Figure 2.13. A cop guards a star X, while the robber moves into a component G0 in G − X.

We can repeat this process on G1 to form analogously formed induced subgraphs G2 , G3 , and so on. Let c(n) be the maximum cop number of a connected graph of order n. For example, c(1) = c(2) = c(3) = 1 and c(4) = 2. With this notation, we must show that c(n) ≤ O(f (n)) to prove the theorem. By (2.3), we have that (2.4)

c(n) ≤ c(n − f (n)) + 1.

By (2.4) and since f (n/2) ≤ f (n), we have that for all x ≤ n/2, that (2.5)

c(n − x) ≤ c(n − x − f (n/2)) + 1.

Iteratively applying (2.5), we derive that c(n) ≤ c(n/2) +

n/2 . f (n/2)

46

2. Cops and Robbers

We then have that c(n) ≤ O(f (n)) by induction on n, and the proof follows.  The state-of-the-art of upper bounds for the cop number are given in Theorem 2.18. Theorem 2.18 ([100, 140, 175]). For n a positive integer and a graph of order n, we have that n √ c(G) ≤ . 2(1−o(1)) log2 n Although the statement of Theorem 2.18 is deterministic, interestingly, all the known proofs of the theorem use the probabilistic method. While a treatment of the probabilistic method is outside the scope of the book, we give a sketch of a proof of Theorem 2.18 found in [100] using expansion properties. Even as a sketch, the proof is nontrivial. First, we need some preliminaries. A matching in a graph G is a set of pairwise nonadjacent edges. A perfect matching in a graph G is a matching in G that is a spanning subgraph of G. The following well-known theorem of Hall that provides a sufficient and necessary condition for a bipartite graph to have a perfect matching. See Figure 2.14. If a bipartite graph G = (X ∪ Y, E) contains a perfect matching, then |X| = |Y |. For a set S of vertices, define N (S) to be the set of vertices in G that are adjacent to some vertex in S. The following property also holds, which is called Hall’s condition: every subset of X must have enough neighbors in Y ; that is, |N (S)| ≥ |S| for all S ⊆ X. Hall’s condition is both necessary and sufficient for the existence of a perfect matching. For an accessible proof, see [185]. Theorem 2.19 ([112]). (Hall’s Theorem) A bipartite graph G = (X ∪ Y, E) with |X| = |Y | contains a perfect matching if and only if |N (S)| ≥ |S| for all S ⊆ X. We also need the notion of the boundary of a set of vertices. For a set of vertices S in G, the boundary of S, written ∂S, is the set of vertices not in S but which are adjacent to some vertex in S.

2.5. Meyniel’s conjecture on the cop number

47

Figure 2.14. A perfect matching in the hypercube Q4 .

Proof of Theorem 2.18. We prove the theorem by a series of claims. Claim 1 : If G has a set of vertices S with |S| < n/2 and |∂S| ≤ p|S| for some 0 < p < 1, then c(G) ≤ p|S| + c(G0 ), where G0 is a connected graph with at most n − |S| vertices. To see this, place a cop on each vertex of ∂S. The robber is forced to remain within a single component X of G[V (G) \ ∂S]. Note that each such component is either a subgraph of G[S] or a subgraph of G[V (G) \ (S ∪ ∂S)]. By hypothesis, |S| < n/2 < n − |S|. Observe that all components in G[V (G) \ (S ∪ ∂S)] have cardinality at most n − |S| − |∂S| ≤ n − |S| as 0 < p. Let G0 = G[X]. The cops may then capture the robber in G0 with c(G0 )-many cops. Hence, c(G) ≤ |∂S| + c(G0 ), and claim follows.

48

2. Cops and Robbers

Claim 2 : If G is bipartite with parts A and B and k is a constant, then we may partition A as S ∪ T such that |N (S)| ≤ k|S| and for every subset U ⊆ T, we have that |N (U )| > k|U |, We describe a process where S increase in cardinality, while the cardinality of T decreases. The process starts with S = ∅ and T = A. If there exists U ⊆ T with |N (U )| ≤ k|U |, then add elements of U to S and delete them from T. Repeat this process so long as such a U exists. At the end of the process, in the graph G, we have that |N (S)| ≤ k|S| and for all U ⊆ T , we have that |N (U )| > k|U |. The claim follows. We omit the proof of the following claim, as it uses tools from the probabilistic method (such as the Chernoff bounds applied to binomial random graphs). It is interesting that this is the one part of the proof that is nondeterministic! For a positive integer r and a set S of vertices in a graph G, define Nr (S) to be the set of vertices in G of distance r to a vertex in S. Note that N1 (S) = N (S). Claim 3 : Fix p ∈ (0, 1), and let n and r be positive integers. For np sufficiently large, it is possible to place 2pn cops in G such that for every set S of vertices with |Nr (S)| ≥

16 |S| log n, p

there are at least |S| cops in Nr (S). The next claim plays a major role in the proof, as using it we can force the robber into a subgraph with expansion properties. We may iterate the application of Claim 4, finding new smaller induced subgraphs where the robber resides, at the cost of pn cops, where p is suitably chosen. Claim 4 : For a fixed p ∈ (0, 1), we have that c(G) ≤ pn + c(G0 ), where G0 is a connected graph with m ≤ n vertices and maximum degree at most 1/p. Further, G0 is a p-expander : every subset V (G) of at most m/2 vertices has |∂S| > p|S|. We iteratively remove vertices to find the desired G0 . If there is a vertex v of degree at least 1/p, then by placing one cop on v, we can reduce the number of vertices in G by at least 1/p + 1. By Claim

2.5. Meyniel’s conjecture on the cop number

49

1, if there is a set S of with |S| < n/2, but |∂S| ≤ p|S|, then we can reduce the number of vertices in G by at least |S| using p|S| cops. By repeating this process as long as possible we will end up with an induced subgraph G0 with the stated properties in the claim, such that c(G) ≤ p(n − m) + c(G0 ) ≤ pn + c(G0 ). The following claim describes the process of trapping the robber in a suitably sized ball around their initial vertex. Hall’s Theorem is a critical tool here for how the cops move between levels of the ball. √ Claim 5 : There is a function p = p(n) = 2−(1−o(1)) log2 n such that for all G with order m ≤ n, maximum degree less than p 1/p and expansion at least p, we can guard G with at most (1 + o(1)) log2 n · 2pn cops. Claim√5 will finish the proof of the theorem by choosing p = 2−(1−o(1)) log2 n and applying Claim 4. For the proof of Claim 5, we may assume, without loss of generality, that m > pn, or else the result follows. It can be shown (we there √ omit details here; see [100]) that p is a function p = 2−(1−o(1)) log2 n and a positive integer t ∼ log2 n such that if k = 16 p log n, then we have that:

(2.6)

t

k t+1 ≤ (1 + p)−2 np/2,

and

t

(1 + p)2 ≥ k.

We place the cops into (t + 1)-many groups C0 , C1 , . . . , Ct each of cardinality 2pn. We choose the initial positions of the cops in Ci by applying Claim 3 with r = 2i . Suppose now that the robber’s initial position is v. Let N0 = N (v). Hence, |N0 | = |N (v)| ≤ 1/p < k. Consider the auxiliary bipartite graph formed with parts A = N0 , B = V (G), and a ∈ A adjacent to b ∈ B if and only they are at distance at most one in G. Claim 2 then implies that we may partition N0 as S0 ∪ T0 such that in G, |N1 (S0 )| ≤ k|S0 |, and for every subset U ⊆ T0 we have that |N (U )| ≥ k|U |. Hence, by Hall’s Theorem, we may send a distinct cop from C0 to each vertex in T0 in the first move of the cops, thereby preventing the robber from moving to T0 .

50

2. Cops and Robbers

We therefore have that the robber, after their second move, must remain in M1 = N1 (S0 ). Further, |N1 (S0 )| ≤ k|S0 | ≤ k 2 . We may then partition M1 = S1 ∪ T1 , and have the cops in C1 guard T1 . Also, we have that |N2 (S1 )| ≤ k|S1 | ≤ k 3 . The cops have now restricted the robber to M2 = N2 (S1 ). We may now iterate. In each iteration of this procedure of restricting the possible positions of the robber, the balls double in radii. Thus, in their 2t -th move, the robber is restricted within a set Mt = N2t−1 (St−1 ) of cardinality at most k t+1 . In the final iteration, we must have that Mt = St ∪ Tt with St = ∅. Indeed, as G is a p-expander, every nonempty set S of vertices of cardinality at most (1 + p)−(r−1) m/2 > (1 + p)−(r−1) np/2 has |Nr (S)| ≥ (1 + p)r |S|. As the inequalities in (2.6) imply that t

|Mt | ≤ k t+1 ≤ (1 + p)−2 np/2 t

and that (1 + p)2 ≥ k, we derive that St is empty. Hence, the cops in Ct can cover all vertices of Nt in 2t moves. As Nt consists of all possible positions of the robber after their 2t -th move, the robber must eventually be captured. 

2.6. Capture time For our final topic on Cops and Robbers, we focus on capture time. If we are playing Cops and Robbers on a graph where we know that our cops can win, then another problem is to consider how long it takes them to do so. For a complete graph, or more generally, a situation where the cops occupy a dominating set in the first round, the game ends in one move of the cops. The length of a game depends heavily on where the players start the game. For example, consider the path P9 with a cop starting on the leftmost end-vertex. If the robber chooses the rightmost endvertex, then it will take the cop eight steps to capture them. However, if the cop starts at the middle vertex, it will only take four steps, which is as fast as they can do it. The same analysis holds for a path Pn , where the length of the game may double depending on the starting position of the cop. For another example, if we consider the 8-cycle,

2.6. Capture time

51

then two cops placed at distance four apart will capture the robber in two steps. If the cops start on the same vertex, then it will take them four rounds if they both move, and eight rounds if one stays put. The length of a Cops and Robbers game is the number of rounds it takes (not including the 0th round) the cops to capture the robber. Hence, the length counts the number of moves needed for the cops to win. The length is infinite if we have insufficient cops to win, so we always assume we have at least c(G)-many cops. To talk more meaningfully about the length of Cops and Robbers games, we need to make certain assumptions. First, we assume the cop plays optimally throughout; that is, they choose the winning strategy that minimize the length of the game. That includes choosing a starting vertex that minimizes the length of gameplay, as in our examples above. Optimality also implies that throughout the game, the cop makes choices that minimizes the number of moves needed for capture. Second, the robber also plays optimally: they move to maximize the length of the game. For example, a robber staying put on an end-vertex that is the maximum distance from a cop is optimal play, while running towards a cop is an example of non-optimal play. Optimal gameplay also makes sense if we have more cops than is required to win. For example, in P9 with vertices labeled i, where 1 ≤ i ≤ 9 from left to right, then with optimal play, the cops wins in two rounds. If k cops play on a graph with k ≥ c(G), we denote the graph parameter captk (G), which we call the k-capture time of G. In the case k = c(G), we just write capt(G) and refer to this as the capture time of G. Capture time was first introduced in [29], and will be our main focus in this section. If k ≥ n = |V (G)|, then captk (G) = 0. If γ(G) ≤ k < n, we have that captk (G) = 1. As k increases, then captk (G) either remains the same or decreases. Hence, we have a range of values for each graph for the length of the game, from near instant victories when k is the domination number, and the capture time when k is the cop number. The setting where k > c(G) is referred to as overprescribed Cops and Robbers in [48].

52

2. Cops and Robbers

For simplicity, we focus mainly on the capture time of cop-win graphs, and we consider trees first. For trees, the capture time is exactly the radius. Theorem 2.20 ([48]). For every tree T , we have that capt(T ) = rad(T ). Proof. As trees are cop-win, we play with one cop. For the lower bound we consider a robber strategy. We place the robber on a vertex u at a maximum distance from cop, and have them pass in every round. Observe that u must be an end-vertex. Whatever the strategy of the cop, the robber will evade capture for at least rad(T )-many rounds. For the upper bound on capt(T ), we give a winning strategy for the cop. The cop places themselves on a fixed central vertex of T . The cop moves along the unique path to the robber. The robber will be caught in at most rad(T ) rounds.  Consider the following graph named H7 in Figure 2.15. The graph

2 6 4 5

1

7 3 Figure 2.15. The cop-win graph H7 . Notice that vertex 1 is the unique corner.

2.6. Capture time

53

H7 is cop-win, as the labeling in the figure is a cop-win ordering. If the cop starts at 6, the robber must start at 1 or 3. If the robber is at 1, then the cop moves to 4, and the robber loses in the next round. If the robber is at 3, then the cop moves to 7. The robber must move to 1 and then loses in two rounds. Hence, capt(H7 ) ≤ 3. In fact, capt(H7 ) = 3; see Exercise 2.23. We now give a bound on the capture time of cop-win graphs. We observe that if a cop-win graph has order 7, then its capture time is at most 3; see [103]. This result then provides the base case of an induction. Theorem 2.21 ([103]). In a cop-win graph G of order n ≥ 7, the cop can capture the robber in at most n − 4 rounds.

Proof. The proof is by induction on n, with the case for n = 7 following by the discussion before the theorem. Suppose that the conclusion holds for all graphs of a fixed order n ≥ 7, and let G have n + 1 vertices. Hence, G contains a corner u with parent v and, since G − u is a retract of G, it is cop-win. By induction hypothesis, the cop can capture the robber in G − u in at most n − 4 rounds. The cop plays their winning strategy on G − u, and captures the shadow of R. If R is on u, then C plays as if they were on v. After n − 4 rounds, the robber is caught, or R is on u and C is on v. Hence, the robber is caught in at most (n + 1) − 4 moves in G. 

The bound in Theorem 2.21 is tight. We may attach a path of length n − 7 to vertex 7 to form the graph Hn . It was shown in [103] that Hn has capture time n − 4 for all n ≥ 7. We summarize a few other results for capture time. Define the Cartesian product of G and H, written GH, to have vertices V (G)× V (H), and vertices (a, b) and (c, d). If G and H are both paths of order n, then we have an n × n Cartesian grid, written Gn,n . In [146], it was proved that if G is the Cartesian product of two trees, then capt(G) = bdiam(G)/2c.

54

2. Cops and Robbers

The capture time of hypercubes was studied in [30], where the authors used the probabilistic method to prove that capt(Qn ) = Θ(n ln n). For planar graphs, a tight upper bound on the capture time is not known. It is known that if G is planar, then capt3 (G) ≤ 2|V (G)|; see [168]. If the number of cops is around the square root of the number of vertices, then an additional bound on the capture time is √ known. In [48] it was shown that if k ≥ 12 n and G is planar, then captk (G) ≤ 6rad(G) log |V (G)|.

2.7. Exercises (2.1) Show that the Petersen graph has cop number equaling 3. You may use Theorem 2.9 for the lower bound.   (2.2) Prove that for an integer n ≥ 3, that γ(Pn ) = n3 . (2.3) Give an example of a graph G with an induced subgraph H such that c(G) < c(H). (2.4) If n ≥ 2 is an integer, then the rooks graph is defined as Kn Kn . Determine c(Kn Kn ). (2.5) Find the cop numbers of the hypercubes Qi for i = 2, 3, 4. Make a conjecture about c(Qn ), where n ≥ 2. (2.6) Show that if a graph G is bipartite, then G admits a homomorphism to K2 . (2.7) [15] Prove that if H is a retract of G, then c(G) ≤ max{c(H), c(G − H) + 1}. (2.8) Prove that every cop-win graph that is m-regular for some integer m ≥ 1 is a complete graph. (2.9) Find all the cop-win graphs, up to isomorphism, of order at most 4. (2.10) Suppose that G has an induced subgraph isomorphic to a cycle with length at least 4, and at least one vertex of the cycle has degree at least 2. Show that G is not cop-win. (2.11) Characterize the cop-win graphs which do not contain a triangle.

2.7. Exercises

55

(2.12) For an integer n ≥ 2, prove that the treewidth of Kn is n − 1. (2.13) Prove that for a graph G, tw(G) = 1 if and only if G is a forest. (2.14) Prove that an outerplanar graph has treewidth at most 2. (2.15) Find a graph with girth 4 satisfying c(G) < δ(G). (2.16) Show that the graph in Figure 2.5 is cop-win by providing a cop-win ordering. (2.17) [29] Find infinitely many examples of cop-win graphs with exactly one corner. (2.18) In the active version of Cops and Robbers, at least one cop and the robber must move on their respective turns. Our definition of Cops and Robbers and the active version coincide on graphs where each vertex has a loop, but may differ on graphs with no loops. Define c0 (G) to be the cop number in the active version of the game. (a) Give an example of a graph G satisfying c0 (G) 6= c(G). (b) [158] Show for a graph G without loops that c(G) − 1 ≤ c0 (G) ≤ c(G). (2.19) Find the cop number of the graph G in Figure 2.16.

Figure 2.16. The graph G.

56

2. Cops and Robbers

(2.20) For a positive integer n, a latin square of order n is an n × n array of cells with each cell containing a symbol from a set S with |S| = n, such that each symbol occurs exactly once in each row and in each column. The latin square graph of a latin square L of order n, written as G(L), is the graph with n2 vertices labeled with the cells of the latin square, where distinct vertices are adjacent if they share a row, column, or symbol. See Figure 2.17 for the graph corresponding to the following latin square of order 3: 1 2 3 L3 = 2 3 1 . 3 1 2

E1

E2

E3

E4

E5

E6

E7

E8

E9

Figure 2.17. The latin square graph of L3 . Cells are labeled Ei , where 1 ≤ i ≤ 9.

(a) Show that c(G(L3 )) = 3. (b) For each positive integer n, give an example of an n × n latin square. (c) [2] Let L be an n × n latin square with n ≥ 4. Prove that c(G(L)) = 3. (2.21) Show that for an outerplanar graph G, we have that c(G) ≤ 2. (Hint: Apply the result for the 2-connected case in Theorem 2.14, and use a suitably defined retraction.)

2.8. Projects

57

(2.22) Prove that for a projective plane P of order q that c(G(P )) = q + 1. (2.23) Show that the capture time of the graph H7 in Figure 2.15 is 3.

2.8. Projects (2.1) For a positive integer k, what is the smallest order of a graph with cop number k? Investigate the minimum order graphs with a given cop number. Show that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. See [7]. (2.2) Study the proof that the cop number of planar graphs is at most 3. Follow up with the proof that a toroidal graph has cop number at most 4. See [3] and [174], respectively. A simplified proof for planar graphs may be found in [47]. (2.3) Consider the game of Cops and Fast Robbers played on Cartesian grids. In this variant of Cops and Robbers, for a fixed positive integer s, the robber can move from its current vertex to a vertex along a path of length s, so long that there is no cop on the path. The parameter corresponding to the cop number in this game is denoted by cs . Prove the result in [96, 160] that for a positive integer n, then c2 (Gn,n ) = Ω(log n). Analogous results for s > 2 were consider in [10]. (2.4) Capture time is less understood for higher values of the cop number. Investigate the capture time of the Cartesian product of two trees. Show that if G is the Cartesian product of two trees, then capt(G) = bdiam(G)/2c. See [146]. (2.5) Explore CR-ordinals of cop-win graphs. This project requires background on infinite sets and ordinals. Consider a graph G of order κ, where κ is a (possibly infinite) cardinal. Define the relations {≤α }α d. We then have that each of its neighbors must contain a searcher, or an edge incident with v will become recontaminated in the next round. Hence, s(G) ≥ d + 1 and the proof follows in this case. Next, consider the last round where the status of the game went from that there are no cleared vertices in G to a single cleared vertex v. By the previous paragraph, we must have that deg(v) = δ; we label the neighbors of v by x1 , x2 , . . . , xd . Suppose for a contradiction that d-many searchers win the game. Suppose that the edge vx1 was the last edge incident with v that was cleared, with vx2 being the second-to-last edge cleared. Right before vx1 is cleared, we must have that each of vxi is cleared, where 2 ≤ i ≤ d. Hence, there are searchers on the endpoints of each of these edges, which accounts for all of the d-many searchers. To clear vx1 , the searcher on v must slide from v to x1 , and so all other edges incident with x1 must be contaminated. Hence, the searcher at x1 can never move throughout the game, or vx1 will become recontaminated as d ≥ 3. See Figure 3.2. We consider the state of the game before vx2 is cleared. As each edge vxi is cleared, where 3 ≤ i ≤ d, there is a searcher on each neighbor xi , which accounts for d−1 searchers. The only unaccounted for searcher must therefore, be on one of v or x2 . If the searcher is on x2 and slides to v, then the edge vx2 will be recontaminated. We must therefore leave the searcher on x2 , and that searcher can never move as d ≥ 3. Let Si be the searchers on the vertices xi , where 3 ≤ i ≤ d. By considering three cases of adjacency of xi with x1 and x2 , we show that the searcher Si cannot move without recontaminating cleared edges. Once that is established, then none of the d-many searchers may move, which is a contradiction as there remain contaminated edges in G. In the first case where xi is adjacent to x1 and x2 , the edges xi x1 and xi x2 are contaminated, and the searcher Si cannot move. In the

66

3. Graph Searching

x1 x2 S v

xd Figure 3.2. The searcher S at v must slide to x1 and can never leave there.

second case, if xi is adjacent to exactly one of x1 or x2 , then since the minimum degree is d, we must have that xi is adjacent to another vertex z that is not adjacent to v. The edge xi z must be cleared and there is no searcher on z, so z is a cleared vertex. However, this contradicts that v is the first cleared vertex. We also find in this case that Si cannot move. In the final case, suppose that xi is not adjacent to either x1 or x2 . We then have that xi is adjacent to two vertices that are not neighbors of v. By a similar argument as in the previous case, neither of these two vertices are cleared, and so Si cannot move. Hence, we arrive at a round where each of the searchers cannot move and there are contaminated edges. This contradiction finishes the proof.  From Theorem 3.3 we derive the following corollary: Corollary 3.4 ([187]). If n ≥ 4 is an integer, then s(Kn ) = n. Proof. The upper bound follows from Lemma 3.1. The lower bound follows from Theorem 3.3. 

3.2. Classes and bounds

67

We finish the section by discussing how the search number is hereditary on minors. If uv is an edge in G, then contracting uv gives the graph formed by replacing u and v by a single vertex x, so that x is adjacent to each neighbor of u or v. For a graph G, a graph H is a minor if by either deleting or contracting edges of G we arrive at H. See Figure 3.3.

u v

x

Figure 3.3. Contracting the edge uv to a vertex x.

Theorem 3.5 ([187]). If H is a minor of G, then s(H) ≤ s(G). Proof. Consider a winning strategy in G with s(G)-many searchers. We show that this also gives rise to a winning strategy with s(G)many searchers in H. Suppose that the searchers clear an edge e in G. If e is also an edge in H, then we clear it in H with this strategy. If e was contracted in passing to H, then we do nothing in H. If e was deleted in passing to H, then we do nothing once again. Using this strategy, all of the edges of H are cleared.  We derive the following immediate corollary. For a graph G, we define its clique number, written ω(G), to be the maximum cardinality of a complete subgraph in G. Corollary 3.6 ([187]). If G is a graph, then ω(G) ≤ s(G).

68

3. Graph Searching

3.3. Monotonicity When searching a path or cycle as we did in the proof of Theorem 3.2, edges never become recontaminated. As we will explore in this section, such a scenario occurs with every graph. We say that a search strategy is monotone if no edge ever become recontaminated. One of the cornerstones of graph searching is that there always exist monotone winning strategies, first discovered by LaPaugh [134] and proved in a different way by Bienstock and Seymour [20]. Theorem 3.7 ([20, 134]). If G is a graph, then there exists a monotone winning strategy with s(G)-many searchers. Theorem 3.7 together with Lemma 3.1 imply that there always exist winning strategies that require at most n rounds. In other words, there are winning strategies that run in linear time. This fact is not obvious a priori. As another nice application of Theorem 3.7, we find an alternative proof that for n ≥ 4 that s(Kn ) = n that does not use Theorem 3.3. Suppose to the contrary that s(Kn ) < n. By Theorem 3.7, there is a monotone winning strategy with at most n − 1 searchers. As there are fewer than n searchers, in some round, a clear edge e will not have both endpoints guarded, and e will be incident with a contaminated edge. We then have that e will be recontaminated, which contradicts monotonicity. The proof we present closely follows the one in [20]. Interestingly, the latter used a variant of searching to prove monotonicity. While the proof remains technical, it is more straightforward than the one in [134]. In the mixed search game, the searchers clear edges as in the search game, but may also clear an edge by placing two searchers on its endpoints. That is, either sliding or placing searchers at endpoints will clear an edge. We define the mixed search number, written ms(G), to be the minimum number of searchers needed to have a winning strategy. See Figure 3.4. Lemma 3.8 ([20, 129]). For a graph G, we have that ms(G) ≤ s(G) ≤ ms(G) + 1.

3.3. Monotonicity

69

C C

C

Figure 3.4. In mixed search, an edge is cleared by either sliding a searcher along it or by placing searchers on its endpoints.

Proof. As a winning strategy for the search game is also a winning mixed-search strategy, we have that ms(G) ≤ s(G). For the inequality s(G) ≤ ms(G) + 1, we use one additional searcher C in a mixedsearch winning strategy, so that each time a pair of searchers clear an edge e by placing themselves on the endpoints of e, the searcher C slides along e. In this way, we have a winning strategy for the search game.  Form the graph Ge be replacing each edges with a path of length 2; that is, each edge of G is subdivided (see Section 3.5 below for more on subdivisions). We state the following lemma from [20] without proof that reduces monotone search strategies to the existence of monotone mixed-search strategies. Theorem 3.9 ([20]). Let G be a graph. A monotone winning mixedsearch strategy for Ge with k searchers may be converted to a monotone winning strategy for G with k searchers.

70

3. Graph Searching

Hence, to prove Theorem 3.7, by Theorem 3.9 we must show that monotone winning mixed-search strategies exist in graphs. For this, we need more machinery. If X ⊆ E(G), then we let b(X) be the set of vertices that are endpoints of an edge in X and also of an edge in E(G) \ X. See Figure 3.5. We collect some elementary facts on b(X).

Figure 3.5. The set X consists of the bold edges, and b(X) are the gray vertices.

Lemma 3.10 ([20]). For a graph G and X, Y ⊆ E(G), we have the following: (1) b(X) = b(E(G) − X). (2) |b(X ∩ Y )| + |b(X ∪ Y )| ≤ |b(X)| + |b(Y )|. Proof. Item (1) follows directly from the definition of the b(X). For item (2), suppose that v is a vertex counted in b(X ∩ Y )| + |b(X ∪Y )|. We then have that v is also counted at least as many times in |b(X)| + |b(Y )|, and so the inequality holds.  A crusade in G is a sequence (X0 , X1 , . . . , Xn ) of edges with the following properties: (1) X0 = ∅. (2) Xn = E(G).

3.3. Monotonicity

71

(3) For all 1 ≤ i ≤ n, |Xi − Xi−1 | ≤ 1. The crusade uses at most k searchers if b(Xi ) ≤ k, for all 0 ≤ i ≤ n. Intuitively, we think of a crusade as the succession of cleared edges over the course of the game. Lemma 3.11 ([20]). Let k be a positive integer. For a graph G, if ms(G) ≤ k, then there is a crusade in G using at most k searchers. Proof. For a nonnegative integer i, let Ci be the set of vertices occupied by searchers immediately before the (i + 1)th round, and let Xi is the set of clear edges in round i. We then have that b(Xi ) ⊆ Ci , and so b(Xi ) ≤ k. It follows that (X0 , X1 , . . . , Xn ) is a crusade in G using at most k searchers.  We say that a crusade (X0 , X1 , . . . , Xn ) is progressive if for all 0 ≤ i ≤ n−1, Xi ⊆ Xi+1 and for all 1 ≤ i ≤ n, |Xi −Xi−1 | = 1. While the following lemma is critical in the proof of monotonicity and has an elementary proof, it is on the technical side and can be skipped without losing comprehension of the proof of Theorem 3.7. Lemma 3.12 ([20]). Let k be a positive integer. For a graph G, if there is a crusade in G using at most k searchers, then there is a progressive crusade in G using at most k searchers. Proof. We choose a crusade (X0 , X1 , . . . , Xn ) using at most k searchers subject to the following conditions: P (1) 0≤i≤n (|b(Xi )| + 1) is a minimum. P (2) Subject to (1), we have that 0≤i≤n |Xi | is a minimum. Note that a crusade satisfying (1) and (2) exists as there are only finitely many choices for (X0 , X1 , . . . , Xn ). The claim is that such a crusade satisfying (1) and (2) is progressive. For a fixed 1 ≤ i ≤ n, by hypothesis, we have that |Xi −Xi−1 | ≤ 1. If we have that Xi ⊆ Xi−1 , then we may eliminate Xi and find that (X0 , X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ) is a crusade, which contradicts (1). Similarly, we must have that (3.1)

|b(Xi−1 ∪ Xi )| ≥ b(Xi )|.

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If not, then (X0 , X1 , . . . , Xi−1 , Xi−1 ∪ Xi , Xi+1 , . . . , Xn ) is a crusade, which contradicts (1). By Lemma 3.10 (2), we derive that |b(Xi−1 ∩ Xi )| + |b(Xi−1 ∪ Xi )| ≤ |b(Xi−1 )| + |b(Xi )|. From (3.1), we have that |b(Xi−1 ∩Xi )| ≤ |b(Xi−1 )|. We then find that (X0 , X1 , . . . , Xi−2 , Xi−1 ∩ Xi , Xi , Xi+1 , . . . , Xn ) is a crusade using at most k searchers. By item (2), we must have that |Xi−1 ∩ Xi | ≥ |Xi |, which in turn implies that Xi−1 ⊆ Xi . As i was arbitrary, the claim follows, and we deduce that (X0 , X1 , . . . , Xn ) is a progressive crusade using at most k searchers.



We next prove the following theorem, which acts as a kind of converse to Lemma 3.11. Theorem 3.13 ([20]). Let G be a graph with minimum degree at least 2, and let k and n be positive integers. Suppose that (X0 , X1 , . . . , Xn ) is a progressive crusade using at most k searchers so that for all 1 ≤ i ≤ n, Xi \ Xi−1 = {ei }. There is then a monotone mixed search strategy for G using at most k guards so that the edges of G are searched in the order e1 , e2 , . . . , en . Proof. We proceed by induction on n ≥ 0. The base case is immediate. Suppose that for a fixed integer i ≥ 0, that we have cleared e1 , e2 , . . . , ei−1 in order. Without loss of generality, we assume that no other edges have been cleared yet. Let Y be the set of vertices u such that every edge incident with u is in Xi−1 . Every vertex in b(Xi−1 ) is occupied by a searcher, as it is incident with both a clear and a contaminated edge. We then remove all other searchers, and note that no recontamination occurs. As ei 6∈ Xi−1 , it has no endpoint in Y. Let S be the set of endpoints of ei . If |S ∪ b(Xi−1 )| ≤ k, then we may place additional searchers on the endpoints of ei and the edge is cleared. Therefore, we assume that |S ∪ b(Xi−1 )| > k. As |b(Xi−1 )| ≤ k, we have that S * b(Xi−1 ). Fix v ∈ S \ b(Xi−1 ). We

3.4. Node search and pathwidth

73

note that S \ b(Xi−1 ) ⊆ b(Xi ), since each vertex of G has degree at least two. Thus, b(Xi−1 ) * b(Xi ). Fix w ∈ b(Xi−1 ) \ b(Xi ). We then have that w ∈ S and ei has endpoints v, w. Further, v is incident with no edge in E(G) \ Xi−1 except ei . Hence, we may clear ei by sliding a searcher at v along ei to w. Continuing in this way, every edge is cleared and no edge is recontaminated. The proof follows.  We finish the section with a proof of the main monotonicity result. Proof of Theorem 3.7: We form G0 by adding a loop to each vertex. This step ensures that every vertex has degree at least two. Note that the mixed-search and search games proceed in the exact same way for reflexive graphs. It is straightforward to see that ms(G0 ) = ms(G) ≤ k. By Lemmas 3.11 and 3.12, there is a progressive crusade in G0 using at k searchers. By Theorem 3.13, there is a monotone mixed-search strategy in G0 using at most k searchers, and so the same holds for G. Now, we apply the previous paragraph to the graph Ge , so that G has a monotone mixed-search strategy with at most k searchers. By Theorem 3.9, there is monotone strategy using at most k searchers in G.  e

3.4. Node search and pathwidth The search number has important connections with several graph parameters. One of the principle of these is pathwidth, which we describe shortly, which is exactly equal to the node search number minus 1. Pathwidth is an extremely influential parameter in graph minor theory, graph algorithms, and other topics within graph theory. We also explore connections between the search number and the vertex separation number and interval thickness of a graph. We come to the final variant of graph searching that we discuss in this chapter (outside of the exercises). Node search is a kind of hybrid of search and mixed-search. In the node search game, the rules are as in mixed-search, except the searchers do not slide across edges to clear them. Hence, the only way to clear an edge is to have a searcher

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at each of its endpoints. We define the node search number of a graph G, written ns(G), to be the minimum number of searchers needed to have a winning strategy. For example, on K2 , we need two searchers to clear the single edge. Hence, ns(K2 ) = 2. See Figure 3.6 for another example. The node search number is well-defined, since placing a searcher on each vertex results in a winning strategy.

Figure 3.6. We have that ns(K3,3 ) = 4, while s(K3,3 ) = 5.

The search number, node search number, and mixed search number differ only by a few values. Lemma 3.14 ([20, 129]). For a graph G, we have the following: (1) ns(G) − 1 ≤ s(G) ≤ ns(G) + 1. (2) ms(G) ≤ ns(G) ≤ ms(G) + 1. Proof. For item (1), consider the upper bound. In addition to the ns(G) searchers used in a winning node search strategy, we play with an additional searcher S. Each time a pair of searchers clears an edge e in the node search game, we have S slide along e to clear it in the search game. This fact gives a winning search strategy. For the lower bound, if an edge e = uv is cleared in the search game by sliding a searcher from u to v, we play with one additional

3.4. Node search and pathwidth

75

searcher who slides from v to u. In this way, we have searchers at the endpoints of e, and so the edge is cleared in the search game. Hence, ns(G) ≤ s(G) + 1 and the lower bound follows. For item (2), the lower bound is immediate, as the searchers have fewer possible moves in the node search game when compared to the mixed-search game. For the upper bound, as in (1), whenever a searcher slides along an edge to clear it in the mixed search game, we have an additional searcher slide along the edge to the endpoint without a searcher. Hence, two searchers will occupy the endpoints of each cleared edge of the mixed search game so it is cleared in the node search game. It follows that ns(G) ≤ ms(G) + 1.  In particular, Lemma 3.14 tells us that s(G) is exactly one of ns(G) − 1, ns(G), or ns(G) + 1. Each of these possibilities occur. For example, s(K2 ) = 1, while ns(K2 ) = 2, s(K1,3 ) = ns(K1,3 ) = 2, and s(K3,3 ) = 5, while ns(K3,3 ) = 4. Similarly, s(G) is exactly one of ms(G) or ms(G) + 1. Node search always has monotonic winning strategies, as in the search game. For a graph G, we form G3 by replacing each edge by three parallel ones. The following theorem reduces node search to searching; we omit the proof here. Hence, monotonic winning strategies for searching give monotonic winning ones for node searching. Theorem 3.15 ([129]). For a graph G, ns(G) = s(G3 ) − 1. The node search number is essentially pathwidth, as we prove next. First, recall the definition of treewidth from Chapter 1. Given a graph G, a tree decomposition is a pair (X, T ), where X = {B1 , B2 , . . . , Bn } is a set of subsets of V (G) called bags, and T is a tree whose vertices are the subsets Bi , satisfying the following three properties: Sn (1) V (G) = i=1 Bi . That is, each graph vertex is associated with at least one tree vertex. (2) For every edge (v, w) in the graph, there is a bag Bi that contains both v and w. That is, vertices are adjacent in G only when the corresponding subtrees have a vertex in common.

76

3. Graph Searching (3) If Bk is on the path from Bi to Bj in T, then Bi ∩ Bj ⊆ Bk .

The width of a tree decomposition is the cardinality of its largest bag Bi minus one. The treewidth of a graph G, written tw(G), is the minimum width among all possible tree decompositions of G. If we restrict T to be a path, then the resulting parameter is called the pathwidth of G, written pw(G). Note that tw(G) ≤ pw(G). We refer to the tree decomposition in this case as a path decomposition. We may also restate (3) equivalently as: for all 1 ≤ i ≤ k ≤ j ≤ n, Bi ∩ Bj ⊆ Bk . See Figure 3.7 for an example.

1

2 3 4

9

5 6 7

123

134

145

156

8

167

178

189

Figure 3.7. A graph with pathwidth 2 and its corresponding path decomposition.

We use the following fact about pathwidth, which also applies to treewidth.

3.4. Node search and pathwidth

77

Lemma 3.16 ([172]). Suppose that {B1 , B2 , . . . , Bn } is a path decomposition of G. For all 1 ≤ i ≤ n, removing Bi ∩ Bj disconnects S G. In particular, 1≤j≤i Xj \ Xi+1 is in a different component than S i i}. The vertex separation of G with respect to L, denoted by vsL (G), is defined as vsL (G) = max{|VL (i)| : 1 ≤ i ≤ |V (G)|}. The vertex separation of G is defined as vs(G) = min{vsL (G) : L is a layout of G}.

Figure 3.8. Two layouts of K3,3 . The top one has vertex separation 4, while the bottom one has vertex separation 3.

The following was proved by Nancy Kinnersley [127]. Interestingly, she was the mother of William Kinnersley, who is an active researcher in pursuit-evasion games. Theorem 3.18 ([127]). If G is a graph, then ns(G) = vs(G) + 1. Kirousis and Papadimitriou [128] discovered a connection between node search and interval graphs. An interval graph has its vertices associated to nonempty intervals of the real line, one for each

3.4. Node search and pathwidth

79

vertex, such that two intervals intersect if and only if the corresponding vertices are adjacent. See Figure 3.9.

2

4

1

6

3 1

5

3

6 4

2

5

Figure 3.9. An interval graph representation of a graph. Overlapping intervals correspond to adjacent vertices.

As interval graphs contain complete graphs of every order, a given graph is a subgraph of some interval graph. The interval thickness of a graph G, denoted by θ(G), is the smallest maximum order complete graph over all interval graphs containing G as a subgraph. Theorem 3.19 ([128]). For a graph G, ns(G) = θ(G). Theorems 3.17, 3.18, and 3.19 tell us that searching is among only a few possible values of one of the pathwidth, vertex separation number, and interval thickness. Corollary 3.20. If G is a graph, then the following hold: (1) s(G) ∈ {pw(G), pw(G) + 1, pw(G) + 2}. (2) s(G) ∈ {vs(G), vs(G) + 1, vs(G) + 2}.

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3. Graph Searching (3) s(G) ∈ {θ(G) − 1, θ(G), θ(G) + 1}.

3.5. The search number of trees We know a good deal about the search number of trees, which were studied early on by Parsons [165]. Central to the results on searching trees is what is now called Parsons’ Lemma, which characterizes trees with a given search number. The lemma leads to a tight bound on the search number of trees. While it is technical and a little below the surface, it is important so we give a full proof here. Given a tree T, a vertex v and a component S of T − v, let S + v be the subtree induced by V (S) ∪ {v}. See Figure 3.10. For a positive integer k, we say that a vertex v in a tree T is k-special if there exist at least three components T1 , T2 , and T3 of T − v such that for 1 ≤ i ≤ 3, s(Ti + v) ≥ k.

v

S

Figure 3.10. The subtree S + v is the subgraph induced by V (S) and v.

Parsons’ Lemma is the following. Lemma 3.21 ([165]). For a tree T and a positive integer k, we have that s(T ) ≥ k + 1 if and only if there exists a k-special vertex in T.

3.5. The search number of trees

81

Proof. The reverse direction is the shorter one. Suppose there exists a k-special vertex v. For a contradiction, suppose that s(T ) = k. There is a winning monotone search strategy with k searchers by Theorem 3.7. By hypothesis, all n searchers are needed to clear Ti + v, where 1 ≤ i ≤ 3. By monotonicity, we assume that each of T1 + v, T2 + v, and T3 + v have all of their edges cleared. Without loss of generality, we assume that they are cleared in increasing order of their indices, with T1 + v cleared first. To clear all edges in T2 +v, in some round, all k searchers must be located on vertices of T2 . In that situation, the contaminated edges of T3 have a path without searchers to T2 through v. Hence, T2 + v is recontaminated, which violates monotonicity. This contradiction implies that s(T ) ≥ k + 1, and one direction of the proof follows. For the forward direction, suppose that s(T ) ≥ k + 1. We must find a k-special vertex. Let S be a subtree of T that is of minimum order satisfying s(T ) ≥ k + 1. In this case, if S has a k-special vertex, then so does T and we are done. Without loss of generality, we may assume that S = T. If u is a vertex of T that is not an end-vertex, then each subtree with root u has search number at most k, by our assumption on the minimum order of T. If none of these subtrees have search number equaling k, then this contradicts that s(T ) ≥ k + 1. Let u1 be a vertex in T that is not an end-vertex, whose subtree T1 rooted at u1 has minimum size and satisfies s(T1 ) = k. Let w1 be the unique vertex of T1 adjacent to u1 . Each subtree of T at w1 , except for the one containing u1 , has search number at most k − 1. One searcher remains at w1 , while the remaining searchers clear all subtrees not containing u1 , and then return to w1 . The searchers then slide along the edge w1 u1 from w1 to u1 . If there exist another two subtrees rooted at u1 with search number k, then the proof follows. If not, then u1 must have exactly one other subtree T2 with search number k. We perform a similar analysis of T2 as we did with T1 . Let u2 be the vertex of T2 adjacent to u1 . We place one searcher on u1 while

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the rest of the searchers clear all subtrees of T at u1 not containing w1 or u2 , and then move back to u1 . The searchers then slide from u1 to u2 . Consider the subtree T1 (u2 ) of T at u2 containing u1 . As T1 is a proper subtree of T1 (u2 ), we have that s(T1 (u2 )) ≥ k. If there are two other subtrees rooted at u2 , then the proof follows. Hence, there is exactly one subtree T2 (u2 ) distinct from T1 (u2 ) with search number equaling k. We let u3 be the vertex of T2 (u2 ) adjacent to u2 . In this fashion, we can find vertices u1 , u2 , u3 , . . . and subtrees T1 , T1 (u2 ), T1 (u3 ), . . . so that T1 is a proper subtree of T1 (u2 ), and for all i ≥ 1, T1 (ui ) is a proper subtree of T1 (ui+1 ). Further, we have searchers at um that have cleared T1 (um ). As T is finite and satisfies s(T ) ≥ k + 1, we do not have that T1 (um ) = T. Therefore, the sequence is finite and finishes at some um , which is a k-special vertex.  The proof of Lemma 3.21 gives rise to the notion of an avenue, which is a path in a tree T so that the subtrees rooted to the path have search number smaller than s(T ). The intuitive idea is that the searchers slide along an avenue, pausing to clear the subtrees rooted at each vertex of the path. As a corollary of Lemma 3.21, we may provide a complete characterization of trees with a given search number. Define T1 to be K2 . Assuming that Tk is defined for a fixed k ≥ 1, we form trees Tk+1 from the following process: (1) Choose S1 , S2 , and S3 to be trees in Tk , which are not necessarily distinct. (2) For a given 1 ≤ i ≤ 3, choose vertices vi in Si such that for each i, either vi is an end-vertex in Si or vi is not an end-vertex and not adjacent to an end-vertex of Si . (3) Let u be a vertex not in S1 , S2 , or S3 . (4) Form a tree T by identifying u with vi if vi is an end-vertex in Sj , or by adding an edge uvi if vi is not an end-vertex of Si . See Figure 3.11.

3.5. The search number of trees

T1

83

T3

T2

Figure 3.11. The trees in T1 , T2 , and T3 .

The subdivision of some edge uv replaces it by a path of length two; in particular, a new vertex x is added along with edges ux and xv. A subdivision of a graph G is a graph resulting from the subdivision of some number of edges in G. A graph G is homeomorphic to H if there is a graph isomorphism from some subdivision of G to some subdivision of H. Note that s(T ) = 1 if and only if the tree is homeomorphic to K1 or K2 . See Exercise 3.1. We have the following. Corollary 3.22 ([165]). If k ≥ 2 is an integer, then a tree T satisfies s(T ) = k if and only if T has a subtree homeomorphic to a tree in Tk but has no subtree homeomorphic to one in Tk+1 . By Lemma 3.21, if a tree has search number k, then its size satisfies 3k−1 ≤ |E(T )|. To see this, note that the existence of a k-special vertex gives rise to 3 other vertices. Iterating the use of the lemma gives the lower bound. Hence, since |E(T )| = |V (T )| − 1, we derive the following. Corollary 3.23 ([165]). If T is a tree of order n, then s(T ) ≤ 1 + log3 (n − 1).

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Note that the bound in Corollary 3.23 is tight, as we can see by considering a rooted tree where all internal vertices have degree 3 and all leaves are at the same distance from the root. See Exercise 3.18. More can be said about the search number of trees. For example, in [145] using Lemma 3.21 and other analysis, it is shown that the search number of a tree can be computed in time linear in the order of the tree.

3.6. Helicopter Cops and Robbers Seymour and Thomas [178] introduced the pursuit-evasion game Helicopter Cops and Robbers, which has elements in common with graph searching, especially node search. As in Cops and Robbers, there is a set of cops and a single robber, with players moving in alternate turns, over a series of discrete rounds. Further, the players are on vertices not edges, and the robber is visible. The cops are in helicopters, and announce to the robber the vertex they land on. The cops do not play on the graph, so may move to any vertex on their turn. In each round including the initial one, the robber moves first, occupying a vertex, followed by a move of the cops. Given the robber’s knowledge of where the cops land, they will choose a vertex (if possible) not containing a cop. On their turn, the robber may move from their current vertex to any vertex along a path not containing a cop. This setting is analogous to recontamination in searching. The cops capture the robber if they land on a vertex occupied by the robber. Note that this requires that each neighbor of the robber contains a cop; otherwise, the robber would have had an opportunity to move away. The robber wins if they can indefinitely avoid capture. The helicopter cop number, written hc(G), is the minimum number of cops needed to guarantee that the robber is captured. Note that as placing the cops on each vertex of G results in a capture, the parameter is well-defined. See Figure 3.12. In fact, the strategy described in Figure 3.12 works for any forest. Theorem 3.24 ([178]). For a forest T, hc(T ) ≤ 2.

3.6. Helicopter Cops and Robbers

C1 C2

T

85

v

u

R

Figure 3.12. In the tree T, one cop C1 stays at a fixed vertex v that acts like a root, while C2 moves to the subtree of T − v containing R. In this way, the robber is captured.

Proof. The cops see the tree component T where the robber resides. Fix a vertex v of T and place a cop on v. We think of v as the root, with subtrees of T adjacent to the root. The robber must be on some subtree of T , say T 0 , which has a unique vertex u adjacent to v. A second cop then lands on u, followed by the first cop moving to u from v. Note that in these moves, the robber cannot leave T 0 because of the cop on u. In this way, we have contained the robber on a subtree T 0 with smaller order than T. We may repeat this process choosing u as the new root. In this way, after finitely many rounds, the robber will be captured on an endvertex.  In a disconnected graph, the robber is free to choose their component, but may never leave it. Hence, the helicopter cop number of a graph G is the maximum of the helicopter cop number of the components of G. Helicopter cop number is actually treewidth in disguise.

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Theorem 3.25 ([178]). If G is a graph, then hc(G) = tw(G) + 1. Proof. We will just prove the upper bound, leaving the more involved proof of the lower bound to Project 3.1. Suppose that we have tw(G) + 1 cops at our disposal. Fix a bag B1 of G, and move the cops to occupy each vertex of B1 . The cops see the subtree T 0 that the robber resides in the tree decomposition. The cops then move to the unique bag B2 adjacent to B1 in the tree decomposition. Note that the robber cannot leave T 0 by Lemma 3.16. Proceeding in this way, the robber is chased into a bag that is an end-vertex in the tree decomposition, and is captured there.  The winning strategy for the cops in the proof of Theorem 3.25 given in [178] is monotone, and we can always find such a winning monotone strategy when playing the game. We present another way of characterizing graphs with a given helicopter cop number. For a set X of vertices in a graph G, the vertex set of a component of G − X is called an X-flap. If the cops are at X, then an X-flap tells us where the robber resides: as they can move to any vertex, all that matters is the component of G − X containing them. Two vertex sets X, Y ⊆ V (G) touch if either X ∩ Y 6= ∅ or some vertex in X has a neighbor in Y . Havens correspond to good winning strategies for the robber. Let [V ]