Strong domination number of a modified graph

TL;DR

This paper investigates how the strong domination number of a graph changes when the graph is modified through various vertex and edge operations, providing insights into its structural properties.

Contribution

It introduces the concept of strong domination number and analyzes its behavior under different graph modifications, which is a novel approach in domination theory.

Findings

Characterizes the impact of vertex operations on $oldsymbol{ ext{strong domination number}}$

Analyzes the effect of edge modifications on $oldsymbol{ ext{strong domination number}}$

Provides bounds and conditions for changes in $oldsymbol{ ext{strong domination number}}$

Abstract

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. In this paper, we study the effects on $\gamma_{st}(G)$ when $G$ is modified by operations on vertex and edge of $G$.