Stability of the Strong Domination Number of Graphs
Saeid Alikhani, Mazharuddin Mehraban, Hossein Shojaaldini Ardakani
TL;DR
This paper introduces and analyzes the stability of the strong domination number, $st_{\gamma_{st}}(G)$, defined as the smallest number of vertices whose removal changes $\gamma_{st}(G)$. It derives exact stability values for a wide range of graph classes (including paths, cycles, wheels, bipartite and multipartite graphs, and standard families like friendship and book graphs), and establishes general upper bounds and Nordhaus–Gaddum type inequalities. The work further studies how stability behaves under graph operations such as join, corona, and Cartesian product, and provides structural insights and open problems for graphs with a given stability level. Collectively, the results illuminate the robustness of strong domination against vertex deletions and lay groundwork for future characterizations and algorithmic questions in domination stability.
Abstract
This paper introduces and studies the stability of the strong domination number of a graph, denoted $\operatorname{st}_{γ_{st}}(G)$, defined as the minimum number of vertices whose removal changes the strong domination number $γ_{st}(G)$. 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 $\operatorname{st}_{γ_{st}}(G)$ 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.