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The Cop Number of the One-Cop-Moves Game on Planar Graphs [article]

Ziyuan Gao, Boting Yang
2018 arXiv   pre-print
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  ...  We show that Aigner and Frommer's result does not generalise 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.  ...  The Cops and Robbers Game Versus the One-Cop-Moves Game on Planar Graphs Before presenting the main result, we show that for planar graphs, the one-cop-moves cop number can in general be larger than the  ... 
arXiv:1705.11184v3 fatcat:4yrowkyqdjbs3n3xyyw4643ut4

Topological directions in Cops and Robbers [article]

Anthony Bonato, Bojan Mohar
2018 arXiv   pre-print
We survey 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.  ...  The authors gratefully acknowledge support from NSERC. The second author is also supported in part by the Canada Research Chairs program, and by the Research Grant P1-0297 of ARRS (Slovenia).  ... 
arXiv:1709.09050v2 fatcat:nug57vbzc5ezxf75bvryhfni7m

Cops and Robbers on Planar-Directed Graphs

Po-Shen Loh, Siyoung Oh
2017 Journal of Graph Theory  
Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit  ...  We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar directed graph.  ...  and sharp result that every planar graph has cop number at most three.  ... 
doi:10.1002/jgt.22129 fatcat:w47hioykkzfxbccldtaw3fytgm

Cops and robbers on planar directed graphs [article]

Po-Shen Loh, Siyoung Oh
2015 arXiv   pre-print
Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit  ...  We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar directed graph.  ...  and sharp result that every planar graph has cop number at most three.  ... 
arXiv:1507.01023v1 fatcat:bpot54sppbdhviznba737z4oea

Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs [article]

Abbas Mehrabian
2011 arXiv   pre-print
We study this game on interval graphs, chordal graphs, planar graphs, and hypercube graphs. Let c_∞(G) denote the number of cops needed to capture the robber in graph G in this variant.  ...  We consider a variant of the Cops and Robber game, introduced by Fomin, Golovach, Kratochvil, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not  ...  Planar Graphs In one of the first papers on the original Cops and Robber game, Aigner and Fromme [1] proved that three cops can capture the robber in any planar graph.  ... 
arXiv:1008.4210v2 fatcat:sirfpxn2qjfb7iaufghf6hfklm

Chasing a Fast Robber on Planar Graphs and Random Graphs

Noga Alon, Abbas Mehrabian
2014 Journal of Graph Theory  
Let c ∞ (G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G.  ...  We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied  ...  The second author is grateful to Nick Wormald for continuous support and lots of fruitful discussions.  ... 
doi:10.1002/jgt.21791 fatcat:vo4hxedabzc5lklpszr5crmofa

Confining the Robber on Cographs [article]

Masood Masjoody
2020 arXiv   pre-print
We conclude by posing two conjectures concerning the confining cop number of P_5-free graphs and the smallest planar graph of confining cop number of three.  ...  Utilizing such conditions, we show that for planar cographs and planar P_5-free graphs the confining cop number is at most one and two, respectively.  ...  Here, we pose the counterpart of this conjecture in terms of the confining cop number: Conjecture. For every connected planar graph G on at most 19 vertices one has ccn(G) ≤ 2.  ... 
arXiv:2006.08941v3 fatcat:d44kwzfgzfbcxhg3mbng7efcwe

Conjectures on Cops and Robbers [chapter]

Anthony Bonato
2016 Problem Books in Mathematics  
We consider some of the most important conjectures in the study of the game of Cops and Robbers and the cop number of a graph.  ...  The conjectures touch on diverse areas such as algorithmic, topological, and structural graph theory. 1991 Mathematics Subject Classification. 05C57, 05C85, 05C10.  ...  Graph genus With the analogy of the chromatic number in mind, what can be said on bounds on the cop number in planar graphs? This was settled early on by Aigner and Fromme [1] . Theorem 5 ([1]).  ... 
doi:10.1007/978-3-319-31940-7_3 fatcat:sqe3lm2zivhsnppsvjgxzol7em

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 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.  ...  We study the minimum number of cops needed to capture a robber on a graph G, written cc(G).  ...  Isometric paths play an important role in several key theorems in the game of Cops and Robbers, such as the cop number of planar graphs (see Chapter 4 of [8] ).  ... 
doi:10.1007/978-81-322-1680-3_1 fatcat:lqyeq446gjffjf3d5vxbmwnofe

Surrounding cops and robbers on graphs of bounded genus [article]

Peter Bradshaw, Seyyed Aliasghar Hosseini
2019 arXiv   pre-print
We obtain bounds on the number of cops required to surround a robber on planar graphs, toroidal graphs, and outerplanar graphs.  ...  We consider a surrounding variant of cops and robbers on graphs of bounded genus.  ...  Finally, the authors thank Bojan Mohar for several suggestions that improved the quality of the paper.  ... 
arXiv:1909.09916v5 fatcat:67bh5eq72nfjnnufpcrlvkwn5y

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 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.  ...  We study the minimum number of cops needed to capture a robber on a graph G, written cc(G).  ...  Isometric paths play an important role in several key theorems in the game of Cops and Robbers, such as the cop number of planar graphs (see Chapter 4 of [8] ).  ... 
arXiv:1308.2843v1 fatcat:tql2g4fr6fgglgdkq6edpon5w4

Hyperopic Cops and Robbers

A. Bonato, N.E. Clarke, D. Cox, S. Finbow, F. Mc Inerney, M.E. Messinger
2019 Theoretical Computer Science  
We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop.  ...  We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number.  ...  Acknowledgements The authors would like to thank the referees for suggestions that improved the paper.  ... 
doi:10.1016/j.tcs.2018.10.004 fatcat:f72u5bwwmnas3o3cqtsab72zwe

Cops and Robbers playing on edges

Andrzej Dudek, Przemysław Gordinowicz, Paweł Prałat
2014 Journal of Combinatorics  
In the game of cops and robbers, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G.  ...  In this paper, we consider the variant of the game in which both players play on edges instead of vertices, yielding the edge cop number.  ...  In particular, the edge cop number of incidence graphs of the projective plane is studied. Random graphs are considered in Section 5 and planar graphs in Section 6.  ... 
doi:10.4310/joc.2014.v5.n1.a6 fatcat:smnqiefy6jagve7vyzukeqedvi

Hyperopic Cops and Robbers [article]

A. Bonato, N.E. Clarke, D. Cox, S. Finbow, F. Mc Inerney, M.E. Messinger
2017 arXiv   pre-print
We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop.  ...  We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number.  ...  Planar graphs In this section, we analyze the game of Hyperopic Cops and Robbers played on planar graphs.  ... 
arXiv:1710.10112v1 fatcat:7autklypgbgfplpgdeghh2xtyi

The Cops & Robber game on series-parallel graphs [article]

Dirk Oliver Theis
2008 arXiv   pre-print
The Cops and Robber game is played on undirected finite graphs. k cops and one robber are positioned on vertices and take turn in moving along edges.  ...  It is a well-known trick in the literature of cops & robber games to define variants of the game which impose restrictions on the possible strategies of the cops (see Clarke, 2002).  ...  Proof of Theorem 1. We prove by induction on the number of blocks of the graph.  ... 
arXiv:0712.2908v2 fatcat:zgeu4sqmifcopanrud3l3s5vbm
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