Cops and Robbers from a distance

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(n) 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-vertex connected graphs. The parameter ck(G) is investigated asymptotically in random graphs G(n, p) for a wide range of p = p(n). For each k ≥ 0, it is shown that ck(G) as a function of the average degree d(n) = pn forms an intriguing zigzag shape.