The Game of Cops and Robbers on Geometric Graphs

Mark van den Bergh

Master thesis

Supervisors: dr. T. M¨uller(UU), dr. F.M. Spieksma

Date master exam: September 21, 2017

Mathematisch Instituut, Universiteit Leiden The Game of Cops and Robbers on Geometric Graphs

Mark van den Bergh

September 8, 2017

Supervisors: dr. T. M¨uller (UU) dr. F.M. Spieksma

Mathematisch Instituut Universiteit Leiden

Abstract The Game of Cops and Robbers is played by two players on a graph, the one player controlling a set of cops and the other controlling a robber. The goal of the cops is to capture the robber in a finite number of moves, whereas the robber wants to evade the cops indefinitely. The central question we ask is: how many cops are needed on a given graph for them to be able to win the game? This amount is called the cop number of a graph. In this thesis, we give an extensive overview of results that give an upper bound on the cop number for graphs of a given type, like unit disk graphs and string graphs, as well as some examples of these graphs having a high cop number. In this overview, we will see that the results all greatly depend on the statement or proof method of one of two theorems published by Aigner and Fromme [1]. In addition to a survey of the existing work, we explore whether the two main theorems can be used to further sharpen these results. Leading to mostly negative answers, we conjecture that a different approach is necessary in order to book further progress. Preface

In 1983, a board game called Scotland Yard won the prestiguous Spiel des Jahres award for being the best board game of the year according to a panel of critics. In this game, named after the headquarters of the London police, an elusive criminal called Mr. X tries to avoid capture by a number of police officers who chase him across a map of London. The goal of the policemen is to capture Mr. X within a set number of turns, whereas Mr. X wins the game by evading the officers during this time. The mathematical field of extremal combinatorics deals with the question of how large or small some set or structure needs to be, if it is to adhere to certain requirements. One might for example wonder how many people can attend a party without having a set of three people who either all have met before or are complete strangers to each other. The answer to this question turns out to be five, a proof of which may be found on a page dedicated to Ramsey’s Theorem [3]. In this thesis, we will explore a game that ties the gameplay of Scotland Yard and the mathematical field of extremal combinatorics together, known as the Game of Cops and Robbers. In this game, a set of cops hunt a robber moving along the vertices of some given graph. The cops win the game by capturing the robber in finite time, while the robber wins if he can move without being captured forever. For a given graph, we then ask the question: how many cops are sufficient and necessary to always capture the robber whatever his strategy? By trying to find this cop number, we delve into the field of extremal combinatorics introduced above. Before this small preface comes to an end, some acknowledgements are in place. I would like to thank my kind supervisors Tobias Müllerand Floske Spieksma for allowing me great freedom in exploring the given subject, as well as for the fruitful yet always “gezel- lige” meetings, lacking a suitable English adjective. Furthermore, I would like to thank my loving parents, especially my father for listening to my endless ramblings on the subject and his aid in manufacturing a couple of the many figures in this text.

3 Contents

1 Introduction 5

2 Background 7 2.1 Notation and basic concepts ...... 7 2.2 Geometric graphs ...... 8 2.3 Rules of the game ...... 12 2.4 Some examples ...... 13 2.5 Dismantlability ...... 14 2.6 Graph products and k-cop-win graphs ...... 17

3 Computational aspects 22 3.1 Two algorithms for k ﬁxed ...... 22 3.2 Complexity for k part of inp