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A better bound for the cop number of general graphs
 Journal of Graph Theory
, 2008
"... Abstract: In this note,we prove that the cop number of any nvertex graph G, denoted by c(G), is at most O ( nlgn. Meyniel conjectured c(G) = O(√n). It appears that the best previously known sublinear upperbound is due to ..."
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Cited by 22 (2 self)
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Abstract: In this note,we prove that the cop number of any nvertex graph G, denoted by c(G), is at most O ( nlgn. Meyniel conjectured c(G) = O(√n). It appears that the best previously known sublinear upperbound is due to
COPS AND ROBBERS IN A RANDOM GRAPH
, 2008
"... We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices O(n 1 2) cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n, p) if p is not very small, and this ..."
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Cited by 18 (0 self)
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We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices O(n 1 2) cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n, p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the are a defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient.
On Meyniel’s conjecture of the cop number
, 2009
"... Meyniel conjectured the cop number c(G) of any connected graph G on n vertices is at most C √ n for some constant C. In this paper, we prove Meyniel’s conjecture in special cases that G has diameter at most 2 or G is a bipartite graph with diameter at most 3. For general connected graphs, n we prove ..."
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Cited by 17 (0 self)
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Meyniel conjectured the cop number c(G) of any connected graph G on n vertices is at most C √ n for some constant C. In this paper, we prove Meyniel’s conjecture in special cases that G has diameter at most 2 or G is a bipartite graph with diameter at most 3. For general connected graphs, n we prove c(G) = O (), improving the best previously known upperbound O ( n ln n 1
Meyniel’s conjecture on the cop number: a survey
 JOURNAL OF COMBINATORICS
, 2012
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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.
Fast robber in planar graphs
, 2008
"... In the cops and robber game, two players play alternately by moving their tokens along the edges of a graph. The first one plays with the cops and the second one with one robber. The cops aim at capturing the robber, while the robber tries to infinitely evade the cops. The main problem consists in m ..."
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Cited by 9 (6 self)
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In the cops and robber game, two players play alternately by moving their tokens along the edges of a graph. The first one plays with the cops and the second one with one robber. The cops aim at capturing the robber, while the robber tries to infinitely evade the cops. The main problem consists in minimizing the number of cops used to capture the robber in a graph. This minimum number is called the copnumber of the graph. If the cops and the robber have the same velocity, g cops are sufficient to capture one robber in any graph with genus g (Schröder, 2001). In the particular case of a grid, 2 cops are sufficient. We investigate the game in which the robber is slightly faster than the cops. In this setting, we prove that the copnumber of planar graphs becomes unbounded. More precisely, we prove that Ω ( √ log n) cops are necessary to capture a fast robber in the n × n squaregrid. This proof consists in designing an elegant evasionstrategy for the robber. Then, it is interesting to ask whether a high value of the copnumber of a planar graph H is related to a large grid G somehow contained in H. We prove that it is not the case when the notion of containment is related to the classical transformations of edge removal, vertex removal, and edge contraction. For instance, we prove that there are graphs with copnumber at most 2 and that are subdivisions of arbitrary large grid. On the positive side, we prove that, if H planar contains a large grid as an induced subgraph, then H has large copnumber. Note that, generally, the copnumber of a graph H is not closed by taking induced subgraphs G, even if H is planar and G is an distancehereditary inducedsubgraph.
CATCH ME IF YOU CAN: COPS AND ROBBERS ON GRAPHS
"... Vertex pursuit games are widely studied by both graph theorists and computer scientists. Cops and Robbers is a vertex pursuit game played on a graph, where some set of agents (or Cops) attempts to capture a robber. The cop number is the minimum number of cops needed to win. While cop number of a gra ..."
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Cited by 8 (8 self)
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Vertex pursuit games are widely studied by both graph theorists and computer scientists. Cops and Robbers is a vertex pursuit game played on a graph, where some set of agents (or Cops) attempts to capture a robber. The cop number is the minimum number of cops needed to win. While cop number of a graph has been studied for over 25 years, it is not well understood, and has few connections with existing graph parameters. In this survey, we highlight some of the main results on bounding the cop number, and discuss Meyniel’s conjecture on an upper bound for the cop number of connected graphs. We include a new proof of the fact that outerplanar graphs have cop number at most two.