Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).

Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).

Réalisé avec le support financier du Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) et du Fonds de Recherche du Québec – Nature et technologies (FRQNT). / Ce mémoire étudie le jeu de policiers-voleur et contient trois articles, chacun portant sur une classe de graphes spécifique. Dans le premier chapitre, la notation et les définitions de base de la théorie de graphe qui nous serons utiles sont introduites. Bien que chaque article comporte une introduction citant les concepts et résultats pertinents, le premier chapitre de ce mémoire contient aussi une introduction générale au jeu de policiers-voleur et présente certains des résultats majeurs sur ce jeu. Le deuxième chapitre contient l’article écrit avec Seyyed Aliasghar Hosseini et Peter Bradshaw portant sur le jeu de policiers-voleurs sur les graphes de Cayley abéliens. Nous améliorons la borne supérieure sur le cop number de ces graphes en raffinant les méthodes utilisées précédemment par Hamidoune, Frankl et Bradshaw. Le troisième chapitre présente l’article concernant le cop number des graphes 2K2-libres. Plus précisément, il est prouvé que 2 policiers peuvent toujours capturer le voleur sur ces graphes, prouvant ainsi la conjecture de Sivaraman et Testa. Finalement, le quatrième chapitre est l’article écrit avec Samuel Yvon et porte sur les graphes qui ont cop number 4. Nous montrons que tous ces graphes ont au moins 19 sommets. En d’autres mots, 3 policiers peuvent toujours capturer le voleur sur tout graphe avec au plus 18 sommets, ce qui répond par la négative à une question de Andreae formulée en 1986. Un pan important de la preuve est faite par ordinateur; ce mémoire contient donc une annexe comprenant le code utilisé. / This thesis studies the game of cops and robbers and consists of three articles, each considering a specific class of graphs. In the first chapter, notation and basic definitions of graph theory are introduced. Al- though each article has an introduction citing the relevant concepts and results, the first chapter of this thesis also contains a general introduction to the game of cops and robbers and presents some of its major results. The second chapter contains the paper written with Seyyed Aliasghar Hosseini and Peter Bradshaw on the game of cops and robbers on abelian Cayley graphs. We improve the upper bound on the cop number of these graphs by refining the methods used previously by Hamidoune, Frankl and Bradshaw. The third chapter presents the paper concerning the cop number of 2K2-free graphs. More precisely, it is proved that 2 cops can always catch the robber on these graphs, proving a conjecture of Sivaraman and Testa. Finally, the fourth chapter is the paper written with Samuel Yvon which deals with graphs of cop number 4. We show that such graphs have at least 19 vertices. In other words, 3 cops can always catch the robber on any graph with at most 18 vertices, which answers in the negative a question by Andreae from 1986. An important part of the proof is by computer; this thesis thus has an appendix containing the code used.