Elsevier

Theoretical Computer Science

Volume 794, 19 November 2019, Pages 47-58
Theoretical Computer Science

Cops, a fast robber and defensive domination on interval graphs

https://doi.org/10.1016/j.tcs.2018.09.031Get rights and content
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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.

Keywords

Cops and robber
Pursuit-evasion
Combinatorial game
Interval graph
Defensive domination

Cited by (0)

This is a significantly revised and extended version of a paper presented at TAMC 2011 [12].

1

Partially supported by CE-ITI project GAČR P202/12/606.

2

Partially supported by National Science Centre (Poland) grant number 2015/17/B/ST6/01887.