Strong domination number of some operations on a graph

TL;DR

This paper investigates how the strong domination number of a graph changes when the graph undergoes various edge operations, providing insights into the structural properties affecting domination.

Contribution

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

Findings

Derived bounds for strong domination number after edge operations

Identified conditions under which the strong domination number increases or decreases

Provided examples illustrating the effects of specific edge modifications

Abstract

Let $G=(V(G),E(G))$ be a simple graph. A set $D\subseteq V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V(G)\setminus D$ 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 examine the effects on $\gamma_{st}(G)$ when $G$ is modified by operations on edge (or edges) of $G$.