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12
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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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
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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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
A Graph Search Algorithm for Indoor Pursuit / Evasion
, 2008
"... Using concepts from both robotics and graph theory, we formulate the problem of indoor pursuit / evasion in terms of searching a graph for a mobile evader. We present an offline, greedy, iterative algorithm which performs guaranteed search, i.e. no matter how the evader moves, it will eventually be ..."
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Cited by 9 (2 self)
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Using concepts from both robotics and graph theory, we formulate the problem of indoor pursuit / evasion in terms of searching a graph for a mobile evader. We present an offline, greedy, iterative algorithm which performs guaranteed search, i.e. no matter how the evader moves, it will eventually be captured; in fact our algorithm can be applied to either an adversarial (actively trying to avoid capture) or randomly moving evader. Furthermore the algorithm produces an internal search (the searchers move only along the edges of the graph, “teleporting” is not used) and can accommodate “extended” (across nodes) visibility and finite or infinite evader speed. We present search experiments for several indoor environments, some of them quite complicated, in all of which the algorithm succeeds in clearing the graph (i.e. capturing the evader).
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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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.
Catch me if you can: Pursuit and capture in polygonal environments with obstacles
 In Proc. of 26th Conference on Artificial Intelligence
, 2012
"... We resolve a severalyears old open question in visibilitybased pursuit evasion: how many pursuers are needed to capture an evader in an arbitrary polygonal environment with obstacles? The evader is assumed to be adversarial, moves with the same maximum speed as pursuers, and is “sensed” by a pursue ..."
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Cited by 5 (2 self)
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We resolve a severalyears old open question in visibilitybased pursuit evasion: how many pursuers are needed to capture an evader in an arbitrary polygonal environment with obstacles? The evader is assumed to be adversarial, moves with the same maximum speed as pursuers, and is “sensed” by a pursuer only when it lies in lineofsight of that pursuer. The players move in discrete time steps, and the capture occurs when a pursuer reaches the position of the evader on its move. Our main result is that O ( √ h + log n) pursuers can always win the game with a deterministic search strategy in any polygon with n vertices and h obstacles (holes). In order to achieve this bound, however, we argue that the environment must satisfy a minimum feature size property, which essentially requires the minimum distance between any two vertices to be of the same order as the speed of the players. Without the minimum feature size assumption, we show that Ω ( √ n / log n) pursuers are needed in the worstcase even for simplyconnected (holefree) polygons of n vertices! This reveals an unexpected subtlety that seems to have been overlooked in previous work claiming that O(log n) pursuers can always win in simplyconnected ngons. Our lower bound also shows that capturing an evader is inherently more difficult than just “seeing ” it because O(log n) pursuers are provably sufficient for lineofsight detection even against an arbitrarily fast evader in simple ngons.
Seepage in directed acyclic graphs
 AUSTRALASIAN JOURNAL OF COMBINATORICS VOLUME 43 (2009), PAGES 91–102
, 2009
"... In the firefighting and the graph searching problems, a contaminate spreads relatively quickly. We introduce a new model, on directed acyclic graphs, in which the contamination spreads slowly. The model was inspired by the efforts to stem the lava flow from the Eldfell volcano in Iceland. The contam ..."
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Cited by 3 (0 self)
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In the firefighting and the graph searching problems, a contaminate spreads relatively quickly. We introduce a new model, on directed acyclic graphs, in which the contamination spreads slowly. The model was inspired by the efforts to stem the lava flow from the Eldfell volcano in Iceland. The contamination starts at a source, only one vertex at a time is contaminated and for some fixed k, k vertices are protected. The slowness is indicated by the name ‘seepage’. The object is to protect the sinks of the graph. We show that if a sink of the graph can be contaminated then at most one directed path need be contaminated. We also investigate the Cartesian product of directed paths. We show that for the product of 3 directed paths that is truncated to only vertices up to a distance of d from the source, if d ≥ 9, then only one vertex need be protected on each turn to protect the sinks. We also present bounds for the Cartesian product of more than 3 paths.
Directed Searching Digraphs: Monotonicity and Complexity
, 2006
"... In this paper, we introduce and study two new search models on digraphs: the directed searching and mixed directed searching. In these two search models, both searchers and intruders must follow the edge directions when they move along edges. We prove the monotonicity of both search models, and we ..."
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In this paper, we introduce and study two new search models on digraphs: the directed searching and mixed directed searching. In these two search models, both searchers and intruders must follow the edge directions when they move along edges. We prove the monotonicity of both search models, and we show that both directed and mixed directed search problems are NPcomplete