Lazy Cops and Robbers played on random graphs and graphs on surfaces

Bal, Deepak; Bonato, Anthony; Kinnersley, William B.; Prałat, Paweł

doi:10.4310/joc.2016.v7.n4.a4

Cited by 10 publications

(18 citation statements)

References 19 publications

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“…The following bound given in [7] gives asymptotic upper bound on c L for graphs of genus g. It exploits the Gilbert, Hutchinson, and Tarjan separator theorem [22].…”

Section: Lazy Cops and Robbersmentioning

confidence: 99%

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Topological directions in Cops and Robbers

Bonato

Mohar

2020

Journal of Combinatorics

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We present the first survey of its kind focusing exclusively on results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface. After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus g, then its cop number is at most g + 3. While Schroeder's bound is known to hold for planar and toroidal graphs, the case for graphs with higher genus remains open. We consider the capture time of graphs on surfaces and examine results for embeddings of graphs on non-orientable surfaces. We present a conjecture by the second author, and in addition, we survey results for the lazy cop number, directed graphs, and Zombies and Survivors.2010 Mathematics Subject Classification. 05C10,05C57.

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“…The following bound given in [7] gives asymptotic upper bound on c L for graphs of genus g. It exploits the Gilbert, Hutchinson, and Tarjan separator theorem [22].…”

Section: Lazy Cops and Robbersmentioning

confidence: 99%

“…Is the lazy cop number bounded by a constant on the class of all planar graphs? See[7] (9). Determine the zombie number of the toroidal grid.…”

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confidence: 99%

Topological directions in Cops and Robbers

Bonato

Mohar

2020

Journal of Combinatorics

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“…There exists an α > 0 such that, with probability 1 − e −Ω(n) , αn cops do not suffice to catch the robber on the graph 3 are independent uniformly chosen random perfect matchings.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The game of cops and robbers was first introduced and popularised in the 1980's by Aigner and Fromme [1], Nowakowski and Winkler [15] and Quilliot [16]. Since then, many variants of the game have been studied, for example where cops can catch robbers from larger distances ( [10]), the robber is allowed to move at different speeds ( [2,13]), or the cops are lazy, meaning that in each turn only one cop can move ( [3,4]).…”

Section: Introductionmentioning

confidence: 99%

Even Flying Cops Should Think Ahead

Martinsson

Florian

Schnider

et al. 2018

Lecture Notes in Computer Science

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Abstract. We study the entanglement game, which is a version of cops and robbers, on sparse graphs. While the minimum degree of a graph G is a lower bound for the number of cops needed to catch a robber in G, we show that the required number of cops can be much larger, even for graphs with small maximum degree. In particular, we show that there are 3-regular graphs where a linear number of cops are needed.

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“…In addition, due to the relative simplicity and naturalness of the cops and robbers game, it has served as a model for studying problems in areas of applied computer science such as artificial intelligence, robotics and the theory of optimal search [8,11,15,21]. This paper examines a variant of the cops and robbers game, known alternately as the oneactive-cop game [17], lazy cops and robbers game [3,4,22] or the one-cop-moves game [25]. The corresponding cop number of a graph G in this game variant is called the one-cop-moves cop number of G, and is denoted by c 1 (G).…”

Section: Introductionmentioning

confidence: 99%

The one-cop-moves game on planar graphs

Gao

Yang

2019

J Comb Optim

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Cops and Robbers is a vertex-pursuit game played on graphs. In this game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph's edges with perfect information about each other's positions. If a cop eventually occupies the same vertex as the robber, then the cops win; the robber wins if she can indefinitely evade capture. Aigner and Fromme established that in every connected planar graph, three cops are sufficient to capture a single robber. In this paper, we consider a recently studied variant of the cops and robbers game, alternately called the one-active-cop game, one-cop-moves game or the lazy cops and robbers game, where at most one cop can move during any round. We show that Aigner and Fromme's result does not generalize to this game variant by constructing a connected planar graph on which a robber can indefinitely evade three cops in the one-cop-moves game.

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Lazy Cops and Robbers played on random graphs and graphs on surfaces

Cited by 10 publications

References 19 publications

Topological directions in Cops and Robbers

Topological directions in Cops and Robbers

Even Flying Cops Should Think Ahead

The one-cop-moves game on planar graphs

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