Cops and Robbers from a distance

Abstract

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 c k ( G ) , equals the minimum number of cops needed to win at a given distance k . We study the parameter c k from algorithmic, structural, and probabilistic perspectives. We supply a classification result for graphs with bounded c k ( G ) values and develop an O ( n 2 s + 3 ) algorithm for determining if c k ( G ) ≤ s for s fixed. We prove that if s is not fixed, then computing c k ( G ) is NP-hard. Upper and lower bounds are found for c k ( G ) in terms of the order of G . We prove that ( n k ) 1 / 2 + o ( 1 ) ≤ c k ( n ) = O ( n log ( 2 n k + 1 ) log ( k + 2 ) k + 1 ) , where c k ( n ) is the maximum of c k ( G ) over all n -vertex connected graphs. The parameter c k ( 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 c k ( G ) as a function of the average degree d ( n ) = p n forms an intriguing zigzag shape.