On the number of isolate dominating sets of certain graphs

Abstract

Let [Formula: see text] be a simple graph. A dominating set of [Formula: see text] is a subset [Formula: see text] such that every vertex not in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The cardinality of a smallest dominating set of [Formula: see text], denoted by [Formula: see text], is the domination number of [Formula: see text]. A dominating set [Formula: see text] is an isolate dominating set of [Formula: see text], if the induced subgraph [Formula: see text] has at least one isolated vertex. The isolate domination number, [Formula: see text], is the minimum cardinality of an isolate dominating set of [Formula: see text]. In this paper, we count the number of isolate dominating sets of some specific graphs.