Complexity results for a cops and robber game on directed graphs

TL;DR

This paper studies a cops and robber game on directed graphs, analyzing the computational complexity of determining the minimum number of cops needed for guaranteed capture, with results varying across different graph classes.

Contribution

It establishes NP-hardness for general digraphs, polynomial-time decidability for tournaments, and fixed parameter tractability results based on graph parameters.

Findings

Deciding cop number 1 is NP-hard for digraphs.

Polynomial-time decision for cop number in tournaments.

Fixed parameter tractability results for general digraphs and tournaments.

Abstract

We investigate a cops and robber game on directed graphs, where the robber moves along the arcs of the graph, while the cops can select any position at each time step. Our main focus is on the cop number: the minimum number of cops required to guarantee the capture of the robber. We prove that deciding whether the cop number of a digraph is equal to 1 is NP-hard, whereas this is decidable in polynomial time for tournaments. Furthermore, we show that computing the cop number for general digraphs is fixed parameter tractable when parameterized by a generalization of vertex cover. However, for tournaments, tractability is achieved with respect to the minimum size of a feedback vertex set. Among our findings, we prove that the cop number of a digraph is equal to that of its reverse digraph, and we draw connections to the matrix mortality problem.