Cops and Robbers from a distance

13 years 2 months ago

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Cops and Robbers is a pursuit and evasion game played on graphs that has received much attention. We consider an extension of Cops and Robbers, distance k Cops and Robbers, where the cops win if at least one of them is of distance at most k from the robber in G. The cop number of a graph G is the minimum number of cops needed to capture the robber in G. The distance k analogue of the cop number, written ck(G), equals the minimum number of cops needed to win at a given distance k. We study the parameter ck from algorithmic, structural, and probabilistic perspectives. We supply a classiﬁcation result for graphs with bounded ck(G) values and develop an O(n2s+3 ) algorithm for determining if ck(G) ≤ s for s ﬁxed. We prove that if s is not ﬁxed, then computing ck(G) is NP-hard. Upper and lower bounds are found for ck(G) in terms of the order of G. We prove that n k 1/2+o(1) ≤ ck(n) = O   n log 2n k+1 log(k + 2) k + 1   , where ck(n) is the maximum of ck(G) over all n-v...

Anthony Bonato, Ehsan Chiniforooshan, Pawel Pralat

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