On minimum intersections of certain secondary dominating sets in graphs

Abstract

In this paper we consider secondary dominating sets, also named as $(1,k)$-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the $(1,1)$-dominating sets and proper $(1,2)$-dominating sets. We introduce $(1,\overline{2})$-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.