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Cop-win graphs with maximum capture-time

2010
*
Discrete Mathematics
*

We present an upper bound n − 4 for the

doi:10.1016/j.disc.2010.01.015
fatcat:sj73zkbfijcmtiglugbulnv6ge
*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. ...##
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Variations of the cop and robber game on graphs
[article]

2017
*
arXiv
*
pre-print

We prove that

arXiv:1710.11352v1
fatcat:3hwzj5mdlzbxfcsslzh76v7bde
*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. ...##
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Cops and Robbers ordinals of cop-win trees
[article]

2017
*
arXiv
*
pre-print

For general infinite

arXiv:1603.04266v2
fatcat:6fbg6bjx6nhkfdko4xephefn7e
*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. ...##
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Cops and Robbers ordinals of cop-win trees

2017
*
Discrete Mathematics
*

We finish

doi:10.1016/j.disc.2016.12.019
fatcat:r2tkjnut3ngz5bbtxjx6lmts4q
*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. ...##
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Topological directions in Cops and Robbers
[article]

2018
*
arXiv
*
pre-print

We consider the

arXiv:1709.09050v2
fatcat:nug57vbzc5ezxf75bvryhfni7m
*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. ...##
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The role of information in the cop-robber game

2008
*
Theoretical Computer Science
*

On the positive side,

doi:10.1016/j.tcs.2008.02.041
fatcat:72u2kfpsgfbulihadd2hym77ca
*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. ...##
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A study of cops and robbers in oriented graphs
[article]

2019
*
arXiv
*
pre-print

Finally, we consider some the aspects of optimal play, in particular the

arXiv:1811.06155v2
fatcat:czfi54ptjzdgvgnllsd5mokklm
*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 ...##
###
WHAT IS...Cop Number?

2012
*
Notices of the American Mathematical Society
*

By considering the second to last move of the

doi:10.1090/noti885
fatcat:dyzgcyabw5a2jgm427sjxfbqom
*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. ...##
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4-cop-win graphs have at least 19 vertices
[article]

2021
*
arXiv
*
pre-print

We also find all 3-

arXiv:2006.02998v2
fatcat:ask75apbijhspd23s5illqhvwu
*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 ...##
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The Game of Cops and Eternal Robbers
[article]

2020
*
arXiv
*
pre-print

A positive integer $t$ is fixed, and the

arXiv:2003.03791v2
fatcat:wfujalroezfmbbcbijhr4q5wra
*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. ...##
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Catching a Fast Robber on Interval Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

This gives a restricted game equivalent to the original one together

doi:10.1007/978-3-642-20877-5_35
fatcat:wvhy7yi7d5ckhcqhj6svupbaoe
*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. ...##
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Containment: A Variation of Cops and Robbers
[article]

2019
*
arXiv
*
pre-print

an edge occupied by a

arXiv:1405.3330v2
fatcat:ear3r3frtzcshl6bpahqeaku7u
*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. ...##
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Cop and Robber Games When the Robber Can Hide and Ride

2011
*
SIAM Journal on Discrete Mathematics
*

The

doi:10.1137/100784035
fatcat:j7hm3mnjwnerpfr6jeydniiuzm
*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. ...##
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Cops, robbers, and burning bridges
[article]

2018
*
arXiv
*
pre-print

We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the

arXiv:1812.09955v1
fatcat:zstyl44sofbjtotmkzgwpumqwi
*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*. ...##
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Bounds on the length of a game of Cops and Robbers

2018
*
Discrete Mathematics
*

It has long been known that the

doi:10.1016/j.disc.2018.05.025
fatcat:3mxj3ir6hzcjfpjixwpgkjhmk4
*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. ...
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