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### Capture-time Extremal Cop-Win Graphs [article]

David Offner, Kerry Ojakian
2019 arXiv   pre-print
The capture time of a cop-win graph is the minimum number of moves the cop needs to capture the robber.  ...  We consider graphs that are extremal with respect to capture time, i.e. their capture time is as large as possible given their order. We give a new characterization of the set of extremal graphs.  ...  Let capt(n) denote the capture time of a cop-win graph on n vertices with maximum capture time. Definition 1 . 2 . 12 Let G s n be the set of cop-win graphs with n vertices and capture time s.  ...

### Cops vs. Gambler [article]

Natasha Komarov, Peter Winkler
2013 arXiv   pre-print
The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously.  ...  We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected $n$-vertex graph is exactly $n$.  ...  Graphs on which a cop can win are called "cop-win."  ...

### Cop vs. Gambler

Natasha Komarov, Peter Winkler
2016 Discrete Mathematics
The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously.  ...  We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected n-vertex graph is exactly n.  ...  Graphs on which a cop can win are called "cop-win."  ...

### Capture-time extremal cop-win graphs

David Offner, Kerry Ojakian
2019 Discussiones Mathematicae Graph Theory
The capture time of a cop-win graph is the minimum number of moves the cop needs to capture the robber.  ...  We consider graphs that are extremal with respect to capture time, i.e., their capture time is as large as possible given their order. We give a new characterization of the set of extremal graphs.  ...  Let capt(n) denote the capture time of a cop-win graph on n vertices with maximum capture time.  ...

### Variations of the cop and robber game on graphs [article]

Espen Slettnes, Carl Joshua Quines, Shen-Fu Tsai, Jesse Geneson
2017 arXiv   pre-print
We prove that graphs with multiple cycles longer than triangles permit cop-win and killer-win graphs.  ...  For $\left(m,n\right)\neq\left(1,5\right)$ and $n\geq4$, we show that there are cop-win and killer-win graphs with $m$ $C_n$s. In addition, we identify game outcomes on specific graph products.  ...  We prove that graphs with multiple cycles longer than triangles permit cop-win and killer-win graphs. For (m, n) = (1, 5) and n ≥ 4, we show that there are cop-win and killer-win graphs with m Cns.  ...

### Cops and Robbers ordinals of cop-win trees [article]

Anthony Bonato, Przemysław Gordinowicz, Gena Hahn
2017 arXiv   pre-print
For general infinite cop-win graphs, we provide an example whose CR-ordinal is not of this form. We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs.  ...  A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers.  ...  Our family of cop-win graphs derived from the Polat graph supports the assertion that Λ = Υ. We leave this as an open problem. Some ordinals do not seem possible to attain as a CRordinal.  ...

### Cops and Robbers ordinals of cop-win trees

Anthony Bonato, Przemysław Gordinowicz, Geňa Hahn
2017 Discrete Mathematics
We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs. 1991 Mathematics Subject Classification. 05C63,05C57.  ...  A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers.  ...  Our family of cop-win graphs derived from the Polat graph supports the assertion that Λ = Υ. We leave this as an open problem. Some ordinals do not seem possible to attain as a CRordinal.  ...

### Digraph measures: Kelly decompositions, games, and orderings

Paul Hunter, Stephan Kreutzer
2008 Theoretical Computer Science
We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, k-trees and cops and robber games and study their natural translations to digraphs.  ...  We demonstrate its usefulness by exhibiting potential applications including polynomial-time algorithms for NP-complete problems on graphs of bounded Kelly-width, and complexity analysis of asymmetric  ...  For otherwise, let r be the maximal index such that the robber can reach v r (with cops on v i and v j ) and from v r can reach v i (with a cop on v j ) and v j (with a cop on v i ).  ...

### Catching a Fast Robber on Interval Graphs [chapter]

Tomáš Gavenčiak
2011 Lecture Notes in Computer Science
This gives a restricted game equivalent to the original one together with a winning strategy computable in polynomial time.  ...  The players alternate in turns, all the cops move at once to distance at most one, the robber moves along any cop-free path. Cops win by capturing the robber, the robber by avoiding capture.  ...  Should a cop be at or move to the robber's vertex, the cops immediately win. The robber wins by avoiding the capture indefinitely.  ...

### 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  ...

### Large classes of infinite k-cop-win graphs

Anthony Bonato, Geňa Hahn, Claude Tardif
2010 Journal of Graph Theory
While finite cop-win finite graphs possess a good structural characterization, none is known for infinite cop-win graphs.  ...  As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number.  ...  The class of cop-win, vertex-transitive graphs with the property that the cop can win in two moves is large.  ...

### Cop-Win Graphs: Optimal Strategies and Corner Rank [article]

David Offner, Kerry Ojakian
2017 arXiv   pre-print
From the corner rank we can determine the capture time of a graph, i.e. the number of turns the cop needs to win.  ...  We investigate the game of cops and robber, played on a finite graph, between one cop and one robber. If the cop can force a win on a graph, the graph is called cop-win.  ...  We also have a submitted paper [7] in which we use corner rank to characterize graphs of a given size with maximal capture time.  ...

### Hunter & Mole [article]

Natasha Komarov, Peter Winkler
2014 arXiv   pre-print
We characterize the graphs---which we will call "hunter-win"---on which the hunter can guarantee capture of the mole in bounded time.  ...  We also define an optimal hunter strategy (and consequently an upper bound on maximum game time on hunter-win graphs) and note that an optimal hunter strategy need not take advantage of the hunter's unconstrained  ...  Graphs on which capture will occur, under optimal play by both parties, are called "cop-win."  ...

### General Cops and Robbers Games with randomness [article]

Frédéric Simard, Josée Desharnais, François Laviolette
2020 arXiv   pre-print
As in general pursuit evasion games, pursuers (cops) seek to capture evaders (robbers); however, players move in turn and are constrained to move on a discrete structure, usually a graph, and know the  ...  In this paper we present a novel model with stochastic elements that we call a Generalized Probabilistic Cops and Robbers game (GPCR).  ...  A given graph is copwin if the cop has a winning strategy: for any possible move the robber makes, the cop has an answer that leads him to eventually catch the robber (in finite time).  ...

### Entanglement and the complexity of directed graphs

Dietmar Berwanger, Erich Grädel, Łukasz Kaiser, Roman Rabinovich
2012 Theoretical Computer Science
It has been proved [21] that this restriction is necessary: there exist families of graphs where the difference between the DAG-width and the number of cops that can capture the robber with a non-monotone  ...  DAG-width can also be characterised by a variant of a graph searching game (the directed cops and visible robber game), but with the somewhat unsatisfactory restriction that the cops are only allowed to  ...  k, whether a given graph G with n nodes has ent(G) ≤ k can be solved in time O(n k+1 ).  ...
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