# Inverse Domination in X-Trees and Sibling Trees

## Authors

• V. Shalini Vellore Institute of Technology, Chennai
• Indra Rajasingh

## Keywords:

Domination, Inverse domination, Connected Domination, Hypertrees

## 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 $D^{'}$ of $G$, then $D^{'}$ is called an inverse dominating set with respect to $D$.  The inverse domination number $\gamma^{'}(G)$ 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 $\langle D \rangle$ 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.

2024-04-30

## Section

Nonlinear Analysis

## How to Cite

Inverse Domination in X-Trees and Sibling Trees. (2024). European Journal of Pure and Applied Mathematics, 17(2), 1082-1093. https://doi.org/10.29020/nybg.ejpam.v17i2.5038

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