Inverse Domination in X-Trees and Sibling Trees

Abstract

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex not in $D$ is adjacent to at least one vertex in $D$. The minimum cardinality of a dominating set in $G$ is called the domination number and is denoted by $\gamma (G)$. Let $D$ be a minimum dominating set of $G$. If $V-D$ contains a dominating set say $D^{{}^{\prime }}$ of $G$, then $D^{{}^{\prime }}$ is called an inverse dominating set with respect to $D$.  The inverse domination number ${\gamma }^{{}^{\prime }}(G)$ is the cardinality of a minimum inverse dominating set of $G$. A dominating set $D$ is called a connected dominating set or an independent dominating set of $G$ according as the induced subgraph $⟨D⟩$ is connected or independent in $G$. The minimum of the cardinalities of the connected dominating sets of $G$ or the independent dominating sets of $G$ is called the connected domination number ${\gamma }_c(G)$ or the independent domination number ${\gamma }_i(G)$ respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included.