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.