Some results on the super domination number of a graph II

TL;DR

This paper investigates the super domination number of graphs, providing new results on how it behaves under vertex modifications and establishing sharp bounds for specific graph structures.

Contribution

It introduces novel results on the super domination number affected by vertex operations and derives sharp bounds for chain and bouquet graph configurations.

Findings

New bounds for super domination number of graphs

Results on super domination number under vertex modifications

Sharp bounds for chain and bouquet of graphs

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 obtain more results on the super domination number of graphs which is modified by an operation on vertices. Also, we present some sharp bounds for super domination number of chain and bouquet of pairwise disjoint connected graphs.