Stability of $2$-domination number of a graph

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.