The guarding game is E-complete

TL;DR

This paper proves that determining whether cops can guard a region in a graph against a robber is E-complete for arbitrary directed graphs, establishing the problem's computational complexity class.

Contribution

It establishes the E-completeness of the guarding game on arbitrary directed graphs, extending prior NP- and PSPACE-completeness results for special cases.

Findings

The guarding game is E-complete on arbitrary directed graphs.

Complexity results extend previous NP- and PSPACE-completeness findings.

The problem's computational complexity is fully characterized for general directed graphs.

Abstract

The guarding game is a game in which several cops try to guard a region in a (directed or undirected) graph against Robber. Robber and the cops are placed on the vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, Robber on the remaining vertices (the robber-region). The goal of Robber is to enter the guarded region at a vertex with no cop on it. The problem is to determine whether for a given graph and given number of cops the cops are able to prevent Robber from entering the guarded region. Fomin et al. [Fomin, Golovach, Hall, Mihalak, Vicari, Widmayer: How to Guard a Graph? Algorithmica 61(4), 839--856 (2011)] proved that the problem is NP-complete when the robber-region is restricted to a tree. Further they prove that is it PSPACE-complete when the robber-region is restricted to a directed acyclic graph, and they ask about the problem complexity for arbitrary graphs. In this paper we prove that the problem is E-complete for arbitrary directed graphs.