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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.
Chasing a Fast Robber on Planar Graphs and Random Graphs
"... We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this va ..."
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We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. We show that if G is planar then c∞(G) = Θ(tw(G)), and there is a constantfactor approximation algorithm for computing c∞(G). We also determine, up to constant factors, the value of c∞(G) of the ErdősRényi random graph G = G(n, p) for all admissible values of p, and show that when the average degree is ω(1), c∞(G) is typically asymptotic to the domination number.
Lower bounds for the cop number when the robber is fast
 Combinatorics, Probability and Computing
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Cops and robber game with a fast robber
, 2011
"... 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 ..."
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Cited by 1 (1 self)
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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 wellknown 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( 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 nvertex graph can be as large as Ω(ns/s+1). This improves the Ω(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(ns/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 ∆, we prove the cop number is between tw+1 ∆+1 and tw + 1. Using this we show that the cop number of the
COPS AND ROBBER WITH CONSTRAINTS
, 2012
"... Cops & Robber is a classical pursuitevasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem’s complexity and combinatorial properties with constraining the following natural ..."
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Cops & Robber is a classical pursuitevasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem’s complexity and combinatorial properties with constraining the following natural game parameters • Fuel: The number of steps each cop can make; • Cost: The total sum of steps along edges all cops can make;
CHASING ROBBERS ON RANDOM GEOMETRIC GRAPHS  AN ALTERNATIVE APPROACH
"... We study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. We focus on Gd(n, r), a random geometric graph in which n vertices are chosen unifo ..."
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We study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. We focus on Gd(n, r), a random geometric graph in which n vertices are chosen uniformly at random and independently from [0, 1]d, and two vertices are adjacent if the Euclidean distance between them is at most r. The main result is that if r3d−1> cd lognn then the cop number is 1 with probability that tends to 1 as n tends to infinity. The case d = 2 was proved earlier and independently in [4], using a different approach. Our method provides a tight O(1/r2) upper bound for the number of rounds needed to catch the robber.
FINDING A PRINCESS IN A PALACE: A PURSUIT–EVASION PROBLEM
"... Abstract. This paper solves a pursuit–evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied ‘Cops and Robbers Game’, the prince has no knowledge of the position of the princess; ..."
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Abstract. This paper solves a pursuit–evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied ‘Cops and Robbers Game’, the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess. 1.