Small k-Dominating Sets of Regular Graphs

Abstract

A k-dominating set of a graph, G, is a set of vertices, D⊆ V(G), such that for every vertex v ∈ V(G), either v ∈ D or there exists a vertex u ∈ D that is at distance at most k from v in G. We are interested in finding k-dominating sets of small cardinality. In this paper we consider a simple, yet efficient, randomised greedy algorithm for finding a small k-dominating set of regular graphs. We analyse the average-case performance of this heuristic by analysing its performance on random regular graphs using differential equations. This, in turn, proves an upper bound on the size of a minimum k-dominating set of random regular graphs.