Relatively prime inverse domination of a graph

Abstract

Let \(G\) be non-trivial graph. A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called a dominating set of \(G\) if every vertex in \(\mathrm{V}\) - \(\mathrm{D}\) is adjacent to a vertex in \(\mathrm{D}\). The minimum cardinality of a dominating set is called the domination number and is denoted by \(\gamma(G)\). If \(\mathrm{V}\) - D contains a dominating set \(\mathrm{S}\) of \(\mathrm{G}\), then \(\mathrm{S}\) is called an inverse dominating set with respect to \(D\). In an inverse dominating set \(S\), every pair of vertices \(u\) and \(v\) in \(S\) such that (deg \(u\), deg \(v\) ) =1, then \(S\) is called relatively prime inverse dominating set. The minimum cardinality of a relatively prime inverse dominating set is called relatively prime inverse dominating number and is denoted by \(\gamma_{r p}^{-1}(\mathrm{G})\). In this paper we find relatively prime inverse dominating number of some graphs.