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Cops and Robbers on Intersection Graphs [chapter]

Tomás Gavenčiak, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2013 Lecture Notes in Computer Science
The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph G, one controlling k cops and the other one robber, all positioned on VG.  ...  In this paper we show an upper bound on the cop-number of string graphs and sharp bounds on the cop-number of interval filament graphs, circular graphs, circular arc graphs and function graphs.  ...  (Unlike in interval graphs where edges only on intersections of [a, b] and [c, d].)  ...

Cops and Robbers on Intersection Graphs [article]

Tomáš Gavenčiak, Przemysław Gordinowicz, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2016 arXiv   pre-print
The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$.  ...  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  ...  We note that it is proved in  that the maximum cop number of unit disk graphs is at most 9. Their strategy is similar to our strategy for string graphs, by applying Lemma 4.3.  ...

Cop and Robber Games When the Robber Can Hide and Ride

Jérémie Chalopin, Victor Chepoi, Nicolas Nisse, Yann Vaxès
2011 SIAM Journal on Discrete Mathematics
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.  ...  Vaxès, Cop and robber games when the robber can hide and ride, arXiv:1001.4457. inria-00482117, version 1 -9 May 2010 Author manuscript, published in "8th French Combinatorial Conference (2010)"  ...

Most Generalized Petersen graphs of girth 8 have cop number 4 [article]

Joy Morris, Tigana Runte, Adrian Skelton
2020 arXiv   pre-print
Even more generally, in a graph of girth at least $9$ and minimum valency $\delta$, the cop number is at least $\delta+1$.  ...  A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$ vertices (the parameter $k$ is used to define some of the edges).  ...  Cops and robbers is a pursuit-evasion game played on graphs [13, 14] by two players on a simple graph G.  ...

Pursuit-evasion games on latin square graphs [article]

Shreya Ahirwar, Anthony Bonato, Leanna Gittins, Alice Huang, Trent G. Marbach, Tomer Zaidman
2021 arXiv   pre-print
We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number.  ...  The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given.  ...  The fifth author was supported by funds from NSERC and The Fields Institute for Research in Mathematical Sciences.  ...

Localization game on geometric and planar graphs [article]

Bartłomiej Bosek, Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, Małgorzata Śleszyńska-Nowak
2017 arXiv   pre-print
The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph.  ...  We provide upper bounds on the related graph invariant $\zeta (G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs  ...  If these cycles intersect in only one point, this is the location of the robber and the Cop-player wins.  ...

Cops, Robber and Medianwidth Parameters [article]

Konstantinos Stavropoulos
2016 arXiv   pre-print
We study a generalisation of the classical Cops and Robber game, where the robber plays against not just one, but $i$ cop players.  ...  We characterise the $i$-latticewidth of a graph in terms of maximal intersections of bags of $i$ path decompositions of the graph.  ...  The game is played on a finite, undirected graph G by the cop player, who controls k cops, and the robber player.  ...

The localization number and metric dimension of graphs of diameter 2 [article]

Anthony Bonato, Melissa A. Huggan, Trent Marbach
2020 arXiv   pre-print
We consider the localization number and metric dimension of certain graphs of diameter $2$, focusing on families of Kneser graphs and graphs without 4-cycles.  ...  We determine bounds on the localization number and metric dimension of Moore graphs of diameter $2$ and polarity graphs.  ...  For more on polarity graphs, see  . Polarity graphs were studied for the game of Cops and Robbers in  , where bounds were given on the cop number.  ...

Vertex-to-vertex pursuit in a graph

Richard Nowakowski, Peter Winkler
1983 Discrete Mathematics
We characterize the graphs on which the cop has a winning strategy, and connect the problem with the structure theory of graphs based on products and retracts.  ...  A graph G is given and two players, a cop and a robber, play the folioking game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop.  ...  The cop plays his winning strategy on each Gi ; once he catches the robber on one graph, he can stay with his quarry until he has won all n games and is now on the sa.me vertex as the robber on the product  ...

A note on bounds for the cop number using tree decompositions [article]

Anthony Bonato, N.E. Clarke, S. Finbow, S. Fitzpatrick, M.E. Messinger
2013 arXiv   pre-print
In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth.  ...  For additional background on Cops and Robbers and Meyniel's conjecture, see the recent book  .  ...  The cops win if after some finite number of rounds, one of them can occupy the same vertex as the robber (in a reflexive graph, this is equivalent to the cop landing on the robber).  ...

The Robber Strikes Back [chapter]

Anthony Bonato, Stephen Finbow, Przemysław Gordinowicz, Ali Haidar, William B. Kinnersley, Dieter Mitsche, Paweł Prałat, Ladislav Stacho
2013 Advances in Intelligent Systems and Computing
We study the minimum number of cops needed to capture a robber on a graph G, written cc(G).  ...  We give bounds on cc(G) in terms of the cop number of G in the classes of bipartite graphs and diameter two, K1,m-free graphs.  ...  The cops and robber occupy vertices, and more than one cop may occupy a vertex.  ...

The robber strikes back [article]

Anthony Bonato, Stephen Finbow, Przemyslaw Gordinowicz, Ali Haidar, William B. Kinnersley, Dieter Mitsche, Pawel Pralat, Ladislav Stacho
2013 arXiv   pre-print
We study the minimum number of cops needed to capture a robber on a graph $G$, written $cc(G)$.  ...  We give bounds on $cc(G)$ in terms of the cop number of $G$ in the classes of bipartite graphs and diameter two, $K_{1,m}$-free graphs.  ...  The cops and robber occupy vertices, and more than one cop may occupy a vertex.  ...

A Timecop's Work Is Harder Than You Think

Nils Morawietz, Carolin Rehs, Mathias Weller, Javier Esparza, Daniel Kráľ
2020 International Symposium on Mathematical Foundations of Computer Science
We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately.  ...  As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W-hard when parameterized  ...  Our hardness results are based on intuitive algebraic problems called Periodic Character Alignment and Periodic Full Character Alignment, asking whether a given set of periodic sequences over {0, 1} is  ...

Visibility graphs, dismantlability, and the cops and robbers game

Anna Lubiw, Jack Snoeyink, Hamideh Vosoughpour
2017 Computational geometry
We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides.  ...  In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes.  ...  Acknowledgements The cops and robbers problem for points inside a region with a curved boundary was initially suggested by Vinayak Pathak and was posed in the Open Problem Session of CCCG 2013 by the third  ...

Fast Robber in Planar Graphs [chapter]

Nicolas Nisse, Karol Suchan
2008 Lecture Notes in Computer Science
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.  ...  If the cops and the robber have the same velocity, 3 + 3 2 g cops are sufficient to capture one robber in any graph with genus g (Schröder, 2001).  ...  Add a universal vertex u to G and denote the resulting graph by H. Catching the robber in C 4 takes 2 cops, whereas just one cop placed on u in H does the job in one move.  ...
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