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Cop-win graphs with maximum capture-time

Tomáš Gavenčiak
2010 Discrete Mathematics  
We present an upper bound n − 4 for the maximum length of a cop and robber game (the capture-time) on a cop-win graph of order n. This bound matches the known lower bound.  ...  We analyze the structure of the class of all graphs attaining this maximum and describe an inductive construction of the entire class.  ...  Let M be the class of the graphs with the maximum capture-time among all the cop-win graphs of the same order.  ... 
doi:10.1016/j.disc.2010.01.015 fatcat:sj73zkbfijcmtiglugbulnv6ge

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.  ...  We prove that if $G$ is a connected graph with maximum degree $d$, then the cop can win with probability at least $\frac{\sqrt d}{1+\sqrt d}$ after learning the killer's distribution.  ...  We prove that if G is a connected graph with maximum degree d, then the cop can win with Introduction The game of cop and robber on a graph is a simple model of the process of pursuing an adversary.  ... 
arXiv:1710.11352v1 fatcat:3hwzj5mdlzbxfcsslzh76v7bde

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.  ... 
arXiv:1603.04266v2 fatcat:6fbg6bjx6nhkfdko4xephefn7e

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.  ... 
doi:10.1016/j.disc.2016.12.019 fatcat:r2tkjnut3ngz5bbtxjx6lmts4q

Topological directions in Cops and Robbers [article]

Anthony Bonato, Bojan Mohar
2018 arXiv   pre-print
We consider the capture time of graphs on surfaces and examine results for embeddings of graphs on non-orientable surfaces.  ...  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  ...  There is a growing literature on the capture time for graphs with a higher cop number. In [10] , the authors proved that if G is cop-win of order n ≥ 5, then capt(G) ≤ n − 3.  ... 
arXiv:1709.09050v2 fatcat:nug57vbzc5ezxf75bvryhfni7m

The role of information in the cop-robber game

Volkan Isler, Nikhil Karnad
2008 Theoretical Computer Science  
On the positive side, with a simple argument, we show that a cop with small or no visibility can capture the robber on any cop-win graph (even if the robber still has global visibility).  ...  A graph G is called cop-win if a single cop can capture the robber on G. We study the effect of reducing the cop's visibility.  ...  limited visibility can still win on a cop-win graph but the capture time may increase exponentially.  ... 
doi:10.1016/j.tcs.2008.02.041 fatcat:72u2kfpsgfbulihadd2hym77ca

A study of cops and robbers in oriented graphs [article]

Devvrit Khatri, Natasha Komarov, Aaron Krim-Yee, Nithish Kumar, Ben Seamone, Virgélot Virgile, AnQi Xu
2019 arXiv   pre-print
Finally, we consider some the aspects of optimal play, in particular the capture time of cop-win digraphs and properties of the relative positions of the cop(s) and robber.  ...  We also refute a conjecture on the structure of cop-win digraphs, study orientations of outerplanar graphs, and study the cop number of line digraphs.  ...  We give a construction of an n-vertex cop-win ograph for which the capture time is in Θ(n 2 ) (in contrast with the known result that all undirected cop win graphs have capture time at most linear in n  ... 
arXiv:1811.06155v2 fatcat:czfi54ptjzdgvgnllsd5mokklm

WHAT IS...Cop Number?

Anthony Bonato
2012 Notices of the American Mathematical Society  
By considering the second to last move of the cop before the cop captures the robber, it is evident that a cop-win graph must have at least one corner.  ...  This observation along with an induction proves that a graph is cop-win if and only if we may iteratively delete corners and end up with a single vertex.  ... 
doi:10.1090/noti885 fatcat:dyzgcyabw5a2jgm427sjxfbqom

4-cop-win graphs have at least 19 vertices [article]

Jérémie Turcotte, Samuel Yvon
2021 arXiv   pre-print
We also find all 3-cop-win graphs on 11 vertices, narrow down the possible 4-cop-win graphs on 19 vertices and make some progress on finding the minimum order of 3-cop-win planar graphs.  ...  We show that the cop number of any graph on 18 or fewer vertices is at most 3. This answers a question posed by Andreae in 1986, as well as more recently by Baird et al.  ...  We note that there are no 3-cop-win graphs with maximum degree 3 on 13 vertices (which can also be seen in Table 4 ), and that the 3-cop-win graphs with maximum degree 3 on 11 and 13 vertices are 3-regular  ... 
arXiv:2006.02998v2 fatcat:ask75apbijhspd23s5illqhvwu

The Game of Cops and Eternal Robbers [article]

Anthony Bonato, Melissa Huggan, Trent Marbach, Fionn Mc Inerney
2020 arXiv   pre-print
A positive integer $t$ is fixed, and the cops are required to capture the robber in at most $t$ time-steps in each play.  ...  We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game.  ...  In (1) , if the capture time on G with k cops is at most t 2, then c ∞ t (G) ≤ k.  ... 
arXiv:2003.03791v2 fatcat:wfujalroezfmbbcbijhr4q5wra

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.  ... 
doi:10.1007/978-3-642-20877-5_35 fatcat:wvhy7yi7d5ckhcqhj6svupbaoe

Containment: A Variation of Cops and Robbers [article]

Danny Crytser, Natasha Komarov, John Mackey
2019 arXiv   pre-print
an edge occupied by a cop), and the cops win by "containing" the robber---that is, by occupying all $\deg(v)$ of the edges incident with a vertex $v$ while the robber is at $v$.  ...  We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and note that there exist cubic graphs with $\xi(G) \geq \Omega(n^{1/6})$.  ...  Graphs on which a cop can win are called "cop-win." More precisely, a graph is cop-win if there is a vertex u such that for every vertex v, the cop beginning at u can capture a robber beginning at v.  ... 
arXiv:1405.3330v2 fatcat:ear3r3frtzcshl6bpahqeaku7u

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  
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.  ...  We also establish some connections between cop-win graphs for this game with s < s and Gromov's hyperbolicity.  ...  Any graph G = (V, E) of CWW(2) is 2-bidismantlable, however there exist 2bidismantlable graphs G with G / ∈ CWW(2). For any odd integer k ≥ 3, if a graph G is kbidismantlable, then G ∈ CWW(k). [1] J.  ... 
doi:10.1137/100784035 fatcat:j7hm3mnjwnerpfr6jeydniiuzm

Cops, robbers, and burning bridges [article]

William B. Kinnersley, Eric Peterson
2018 arXiv   pre-print
We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the capture time of a graph (the minimum number of rounds needed for a single cop to capture a robber on  ...  The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning cop number} of $G$ and denoted $c_b(G)$.  ...  We denote capture time in the bridge-burning model by capt b (G), and we aim to determine the maximum capture time of an n-vertex graph on which a single cop can win.  ... 
arXiv:1812.09955v1 fatcat:zstyl44sofbjtotmkzgwpumqwi

Bounds on the length of a game of Cops and Robbers

William B. Kinnersley
2018 Discrete Mathematics  
It has long been known that the capture time of an n-vertex graph with cop number k is O(n k+1 ).  ...  We also show that n-vertex strongly-connected directed graphs with cop number 1 can have capture time Ω(n 2 ), thereby showing that the result of Bonato et al. [3] does not extend to the directed setting  ...  The maximum capture time among n-vertex strongly-connected directed graphs with cop number 1 is Θ(n 2 ). Proof.  ... 
doi:10.1016/j.disc.2018.05.025 fatcat:3mxj3ir6hzcjfpjixwpgkjhmk4
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