Distributed algorithms for finding the unique minimum distance dominating set in directed split-stars

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum distance-k dominating set for k = 1, 2 in a directed split-star, which has recently been developed as a new model of the interconnection network for parallel and distributed computing systems.