Domination by Union of Complete Graphs

Abstract

Let $G$ be a graph with domination number $\gamma (G)$. A dominating set $S\subseteq V(G)$ has property $\mathcal{U}\mathcal{K}$ if all components of the subgraph it induces in $G$ are complete. The union of complete graphs domination number of a graph $G$, denoted ${\gamma }_{uk}(G)$, is the minimum possible size of a dominating set of $G$, which has property $\mathcal{U}\mathcal{K}$. Results on changing and unchanging of ${\gamma }_{uk}(G)$ after vertex removal are presented. Also forbidden subgraph conditions sufficient to imply $\gamma (G)={\gamma }_{uk}(G)$ are given.