Cops, a fast robber and defensive domination on interval graphs

Abstract

The game of Cops and ∞-fast Robber is played by two players, one controlling c cops, the other one robber. The players alternate in turns: all the cops move at once to distance at most one each, the robber moves along any cop-free path. Cops win by sharing a vertex with the robber, the robber by avoiding capture indefinitely.The game was proposed with bounded robber speed by Fomin et al. in “Pursuing a fast robber on a graph”, generalizing a well-known game of Cops and Robber which has robber speed 1. We answer their open question about the computational complexity of the game on interval graphs with ∞-fast robber, showing it to be polynomially decidable.We also generalize the concept of k-defensive domination introduced by Farley and Proskurowski in “Defensive Domination” to A-defensive domination and use it as a main tool in our proof. The generalization allows specifying arbitrary attacks and limiting the number of defenders of each vertex. While this problem is NP-complete even for split graphs, we show that A-defensive domination is decidable in polynomial time on interval graphs.