Pursuit and Evasion from a Distance: Algorithms and Bounds

Cops and Robber is a pursuit and evasion game played on graphs that has received much attention. We consider an extension of Cops and Robber, distance k Cops and Robber, where the cops win if they are 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 supply a classification result for graphs with bounded ck(G) values and develop an O(n2s+3) algorithm for determining if ck(G) ≤ s. In the case k = 0, our algorithm is faster than previously known algorithms. Upper and lower bounds are found for ck(G) in terms of the order of G. We prove that [EQUATION] where ck(n) is the maximum of ck(G) over all n-node connected graphs.