30 research outputs found

    Le jeu de policiers-voleur sur différentes classes de graphes

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    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

    Cops and robber on subclasses of P5P_5-free graphs

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    The game of cops and robber is a turn based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say the team of cops win the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in each component of a graph is called the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp. 2306-2307] conjectured that for every t≥5t\geq 5, the cop number of a connected PtP_t-free graph is at most t−3t-3, where PtP_t denotes a path on tt~vertices. Turcotte [Discrete Math. 345 (2022), pp. 112660] showed that the cop number of any 2K22K_2-free graph is at most 22, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is 2K22K_2-free, then it is also P5P_5-free. Liu showed that the cop number of a connected (PtP_t, HH)-free graph is at most t−3t-3, where HH is a cycle of length at most tt or a claw. So the conjecture of Sivaraman is true for (P5P_5, HH)-free graphs, where HH is a cycle of length at most 55 or a claw. In this paper, we show that the cop number of a connected (P5,HP_5,H)-free graph is at most 22, where H∈{C4H\in \{C_4, C5C_5, diamond, paw, K4K_4, 2K1∪K22K_1\cup K_2, K3∪K1K_3\cup K_1, P3∪P1}P_3\cup P_1\}

    The kk-visibility Localization Game

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    We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the kk-visibility localization number ζk\zeta_k, where kk is a non-negative integer. We give bounds on kk-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all kk, we give a family of trees with unbounded ζk\zeta_k values. Extending results known for the localization number, we show that for k≥2k\geq 2, every tree contains a subdivision with ζk=1\zeta_k = 1. For many nn, we give the exact value of ζk\zeta_k for the n×nn \times n Cartesian grid graphs, with the remaining cases being one of two values as long as nn is sufficiently large. These examples also illustrate that ζi≠ζj\zeta_i \neq \zeta_j for all distinct choices of ii and $j.

    On Euler Tours in Streaming Models and some Games on Graphs

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    In this thesis, we take a look at several graph theoretical problems. We present two streaming algorithms for finding an Euler tour in a graph, prove a tight bound on the capture time in the Bridge-Burning Cops and Robbers Game and solve an open problem for the Extreme Vertex Destruction Model. In Chapter 2, we consider the classical Euler tour problem and take a modern look at this problem in the context of the graph streaming model. Here, RAM is of size O(n polylog(n)), where n is the number of nodes, and the graph is given as a stream of its edges. With this restricted memory space, we give a one-pass algorithm for finding Euler tours in the graph streaming model. In Chapter 3, we regard a lesser-known streaming model, the so-called StrSort model, to tackle a downside of our algorithm mentioned above. The algorithm stores an Euler tour on an output tape in form of a successor function. The order of the edges is given, but the edges are not actually sorted in the order of the Euler tour. Therefore, further processing the tour with another streaming algorithm might become difficult. We give an algorithm for sorting the edges of a graph according to a found Euler tour, that has a preparation step in the graph streaming model and a processing step in the StrSort model. The so-called Bridge-Burning Cops and Robbers Game is the topic of Chapter 4. Here, every time the robber traverses an edge, this edge is deleted afterwards. We study winning strategies of a single cop and make statements on the maximum number of turns of such strategies. In Chapter 5, we study networks formed by selfish agents. When a node is ‘destroyed’, i.e. the adjacent edges are deleted, the network is damaged and some players lose the connection to each other. In the Extreme Vertex Destruction Model, we observe the impact of such a deletion on swap equilibrium graphs and make statements on the maximum amount of damage that can cause here

    A note on Cops and Robbers, independence number, domination number and diameter

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    We study relations between diameter D(G)D(G), domination number γ(G)\gamma(G), independence number α(G)\alpha(G) and cop number c(G)c(G) of a connected graph GG, showing (i.) c(G)≤α(G)−⌊D(G)−32⌋c(G) \leq \alpha(G)-\lfloor \frac{D(G)-3}{2} \rfloor, and (ii.) c(G)≤γ(G)−D(G)3+O(D(G))c(G) \leq \gamma (G) - \frac{D(G)}{3} + O (\sqrt{D(G)})

    Pursuing a fast robber on a graph

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    AbstractThe Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s=1 but is NP-hard if s=2. We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for s≤2. Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops
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