Eternal dominating sets on digraphs and orientations of graphs

Abstract

Eternal domination is a problem that asks the following question: Can we eternally defend a graph ? The principle is to defend against an attacked vertex, that changes every turn, by moving guards along the edges of the graph. In the classical version of the game (Burger et al., 2004), only one guard can move at a time, but in the m-eternal version (Goddard et al., 2005), any number of guards can move in a single turn. This problem led to the introduction of two parameters: the eternal and m-eternal domination numbers of the graph, that are the minimum numbers of guards necessary to defend against any infinite sequence of attacks.In this paper, we first study the (m-)eternal domination number on digraphs. We show that a lot of results proven for undirected graphs can be generalized to digraphs. Then, we introduce the problem of oriented (m-)eternal domination, that consists in finding an orientation of a graph that minimizes the eternal domination number. We prove that computing the oriented eternal domination number is a coNP-hard problem, and we give a complete characterization of the graphs for which the oriented m-eternal domination number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, and different kinds of grids and products of graphs.