Equality of Domination and Inverse Domination Numbers

Abstract

A set $D$ of vertices of a graph $G=(V,E)$ is a $\mathit{\text{dominating set}}$ if every vertex of $V-D$ is adjacent to at least one vertex in $D$. The $\mathit{\text{domination number}}$ $\gamma (G)$ is the minimum cardinality of a dominating set of $G$. A subset of $V-D$, which is also a dominating set of $G$, is called an $\mathit{\text{averse dominating set}}$ of $G$ with respect to $D$. The $\mathit{\text{inverse domination number}}$ ${\gamma }^{\prime }(G)$ equals the minimum cardinality of an inverse dominating set $D$. In this paper, we study classes of graphs whose domination and inverse domination numbers are equal.