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Cop and Robber Games when the Robber can Hide and Ride
, 2011
"... In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber ..."
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Cited by 9 (4 self)
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In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski and Winkler [Discrete Math., 43 (1983), pp. 235–239] and Quilliot [Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes, Thèsededoctorat d’état, Université de Paris VI, Paris, 1983] characterized the copwin graphs as graphs admitting a dismantling scheme. In this paper, we characterize in a similar way the class CWFR(s, s ′)ofcopwin graphs in the game in which the robber and the cop move at different speeds s and s ′ , s ′ ≤ s. We also establish some connections between copwin graphs for this game with s ′ <sand Gromov’s hyperbolicity. In the particular case s =2ands ′ = 1, we prove that the class of copwin graphs is exactly the wellknown class of dually chordal graphs. We show that all classes CWFR(s, 1), s ≥ 3, coincide, and we provide a structural characterization of these graphs. We also investigate several dismantling schemes necessary or sufficient for the copwin graphs in the game in which the robber is visible only every k moves for a fixed integer k>1. In particular, we characterize the graphs which are copwin for any value of k.
CONJECTURES ON COPS AND ROBBERS
"... Abstract. We consider some of the most important conjectures in the study of the game of Cops and Robbers and the cop number of a graph. The conjectures touch on diverse areas such as algorithmic, topological, and structural graph theory. 1. ..."
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Abstract. We consider some of the most important conjectures in the study of the game of Cops and Robbers and the cop number of a graph. The conjectures touch on diverse areas such as algorithmic, topological, and structural graph theory. 1.
LAZY COPS AND ROBBERS ON HYPERCUBES
"... Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojak ..."
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Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes. 1.
A PROBABILISTIC VERSION OF THE GAME OF ZOMBIES AND SURVIVORS ON GRAPHS
"... Abstract. We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the c ..."
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Abstract. We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is strictly greater than 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids. 1.
LAZY COPS AND ROBBERS PLAYED ON RANDOM GRAPHS AND GRAPHS ON SURFACES
"... We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. The lazy cop number is the analogue of the usual cop number for this game. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic up ..."
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We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. The lazy cop number is the analogue of the usual cop number for this game. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the analogue of the cop number of the hypercube. By investigating expansion properties, we provide asymptotically almost sure bounds on the lazy cop number of binomial random graphs G(n, p) for a wide range of p = p(n). We provide an upper bound for the lazy cop number of graphs with genus g by using the GilbertHutchinsonTarjan separator theorem.
LAZY COPS AND ROBBERS PLAYED ON GRAPHS
"... We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who p ..."
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We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By investigating expansion properties, we provide asymptotically almost sure bounds on the lazy cop number of binomial random graphs G(n, p) for a wide range of p = p(n). By coupling the probabilistic method with a potential function argument, we also improve on the existing lower bounds for the lazy cop number of hypercubes. Finally, we provide an upper bound for the lazy cop number of graphs with genus g by using the GilbertHutchinsonTarjan separator theorem.