A cops and robber game and the meeting time of synchronous directed walks

Abstract

AbstractIn a previous work, the authors showed that the maximum number of infinitely long synchronous directed walks that never meet is equal to the dimension of the no‐meet matroid, namely the largest order of a collection of vertex‐disjoint cycles. Given , we want to compute the meeting time of walks: the first time step such that, given any set of walks, at least two of them must meet no later than . We precisely prove that the meeting time is at most , where is the number of vertices. A connection is established with a cops and robber game on directed graphs with helicopter cops and an invisible slow robber. The meeting time of walks equals the capture time in this game, when at most capture attempts are allowed. While this capture time can be computed in polynomial time, we show that it is NP‐hard to compute the minimum number of cops needed to catch the robber. More insights are also given on the number and its relation to pathwidth and other graph parameters. Finally we analyze these game measures on digraph tensor products.