Eternal Distance-k Domination on Graphs

Abstract

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-k domination, guards initially occupy the vertices of a distance-k dominating set. After a vertex is attacked, guards “defend” by each moving up to distance k to form a distance-k dominating set, such that some guard occupies the attacked vertex. The eternal distance-k domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the situation where $$k=1$$ . We introduce eternal distance-k domination for $$k > 1$$ . Determining whether a given set is an eternal distance-k domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we use decomposition arguments to bound the eternal distance-k domination numbers, and solve the problem entirely in the case of perfect m-ary trees.