Some results on the super domination number of a graph

TL;DR

This paper investigates the super domination number in graphs, providing bounds and analyzing specific graph classes to deepen understanding of this graph invariant.

Contribution

It introduces new bounds for the super domination number and explores its properties across various graph classes.

Findings

Sharp bounds for super domination number under graph operations

Characterization of super domination number in specific graph classes

Comparison with classical domination number

Abstract

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V-S$, there exists $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a smallest super dominating set of $G$, denoted by $\gamma_{sp}(G)$, is the super domination number of $G$. In this paper, we study super domination number of some graph classes and present sharp bounds for some graph operations.