Guard Games on Graphs: Keep the Intruder Out!

A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the agents and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is a subgraph of the given graph. We investigate the algorithmic aspects of finding the minimum number of guards sufficient to patrol the area. We show that this problem is PSPACE-hard in general and proceed to investigate a variant of the game where the intruder must reach the guarded area in a single step in order to win. We show that this case approximates the general problem, and that for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the one-step guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is

W

[1]-hard parameterized by the treewidth of the input graph. We conclude the study of the one-step guarding problem in bounded treewidth graphs by showing that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including planar graphs and graphs of bounded genus, namely graphs excluding some fixed apex graph as a minor. We prove that the problem is FPT and give a PTAS on apex-minor-free graphs.