On the Number of k-Dominating Independent Sets

Abstract

We study the existence and the number of k-dominating independent sets in certain graph families. While the case k=1 namely the case of maximal independent sets-which is originated from Erdos and Moser-is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k-dominating independent sets in n-vertex graphs is between ck·22kn and ck'·2k+1n if k≥2, moreover the maximum number of 2-dominating independent sets in n-vertex graphs is between c·1.22n and c'·1.246n. Graph constructions containing a large number of k-dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes. © 2016 Wiley Periodicals, Inc