Stability of $2$-domination number of a graph

TL;DR

This paper investigates the stability of the 2-domination number in graphs, defining a new stability measure and analyzing its behavior, bounds, and effects of graph operations.

Contribution

It introduces the concept of 2-domination stability, computes it for specific graphs, and explores its bounds and behavior under graph operations.

Findings

Computed stability for specific graphs

Established bounds for 2-domination stability

Analyzed stability behavior under graph operations

Abstract

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $\gamma_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is adjacent to at least two members of the subset. We define the $2$-domination stability, $st_{\gamma_2}(G)$, as the smallest number of vertices whose removal causes a change in $\gamma_2(G)$. Our primary contributions include computing this parameter for specific graphs, establishing various bounds for this stability and determining its behavior under certain graph operations combining two graphs.