On the Cop Number of a Graph

Berarducci, Alessandro; Intrigila, Benedetto

doi:10.1006/aama.1993.1019

Cited by 99 publications

(85 citation statements)

References 9 publications

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“…If the cops all remain in place, then so does the robber. None of the preceding cases benefit the cops; the only remaining possibility is that the cop on c 2 moves to s 2 , to which the robber responds by moving to w , 3 .…”

Section: 2mentioning

confidence: 99%

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Cops and Robbers is EXPTIME-complete

Kinnersley

2015

Journal of Combinatorial Theory, Series B

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Section: 2mentioning

confidence: 99%

“…(This observation also implies that the problem is solvable in polynomial time when k is fixed in advance, rather than being considered part of the input; see also [2,3,5,9].) Goldstein and Reingold [8] showed that the generalization of C&R in which G may be directed is EXPTIME-complete.…”

Section: Introductionmentioning

confidence: 99%

Cops and Robbers is EXPTIME-complete

Kinnersley

2015

Journal of Combinatorial Theory, Series B

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“…There is the following simple generalization of Lemma 4.1. It was stated in [3] (in a different form) and we believe that this statement should be more known.…”

Section: Guarding Shortest Paths and Curves In String Graphsmentioning

confidence: 69%

“…The class 3-GRID (and thus also 3-BOX) contains G (3) for all graphs G: the vertices are represented by long parallel segments, say, in the direction of the z-axis, having pairwise different x and y coordinates. Each edge is represented by an L-shape (consisting of two segments), connecting the parallel segments representing the corresponding vertices.…”

Section: Unbounded Cop Number Of Intersection Graphs Of Disconnected mentioning

confidence: 99%

Cops and Robbers on Intersection Graphs

Gavenčiak

Jelínek

Klavík

et al. 2013

Algorithms and Computation

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The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs:• The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4.• The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

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“…In [5], an algorithm to determine if K pursuers are sufficient to capture an evader is presented. Aigner and Fromme [13] proved that in a planar graph G, 3 pursuers are sufficient for the pursuers to win the game.…”

Section: Related Workmentioning

confidence: 99%

Scalable and Practical Pursuit-Evasion

Vieira¹,

Govindan²,

Sukhatme³

2009

Proceedings of the 2nd International Conference on Robotic Communication and Coordination

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Abstract-In this paper, we consider the design and implementation of practical, yet near-optimal, pursuit-evasion games. In prior work, we developed, using the theory of zero-sum games, minimal completion-time strategies for pursuit-evasion. Unfortunately, those strategies do not scale beyond a small number of robots. In this paper, we design and implement a partition strategy where pursuers capture evaders by decomposing the game into multiple multi-pursuer single-evader games. Our algorithm terminates, has bounded capture time, is robust, and is scalable in the number of robots. In our implementation, a sensor network provides sensing-at-a-distance, as well as a communication backbone that enables tighter coordination between pursuers. Our experiments in a challenging office environment suggest that this approach is near-optimal, at least for the configurations we have evaluated. Overall, our work illustrates an innovative interplay between robotics and communication.

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On the Cop Number of a Graph

Cited by 99 publications

References 9 publications

Cops and Robbers is EXPTIME-complete

Cops and Robbers is EXPTIME-complete

Cops and Robbers on Intersection Graphs

Scalable and Practical Pursuit-Evasion

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