Stability of the Strong Domination Number of Graphs

TL;DR

This paper investigates the stability of the strong domination number in graphs, providing exact values for key classes, bounds, and behavior under graph operations, advancing understanding of domination stability in graph theory.

Contribution

It introduces the stability of the strong domination number, determines exact values for fundamental graph classes, and explores its behavior under various graph operations.

Findings

Exact stability values for paths, cycles, and other classes

Bounds and inequalities for the stability parameter

Analysis of stability under graph operations

Abstract

This paper introduces and studies the stability of the strong domination number of a graph, denoted $\operatorname{st}_{\gamma_{st}}(G)$, defined as the minimum number of vertices whose removal changes the strong domination number $\gamma_{st}(G)$. We determine exact values of this stability parameter for several fundamental graph classes, including paths, cycles, wheels, complete bipartite graphs, friendship graphs, book graphs, and balanced complete multipartite graphs. General bounds on $\operatorname{st}_{\gamma_{st}}(G)$ are established, along with a Nordhaus Gaddum type inequality. The behavior of stability under graph operations such as join, corona, and Cartesian product is also investigated. Structural characterizations of graphs with given stability values are provided, and several open problems and directions for future research are outlined.