A Note on Global Domination in Graphs

Abstract

Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is called a dominating set of $G$ if every vertex in $V–S$ is adjacent to at least one vertex in $S$. A global dominating set is a subset $S$ of $V$ which is a dominating set of both $G$ as well as its complement $\overline{G}$. The domination number (global domination number) $\gamma ({\gamma }_g)$ of $G$ is the minimum cardinality of a dominating set (global dominating set) of $G$. In this paper, we obtain a characterization of bipartite graphs with ${\gamma }_g=\gamma +1$. We also characterize unicyclic graphs and bipartite graphs with ${\gamma }_g={\alpha }_0(G)+1$, where ${\alpha }_0(G)$ is the vertex covering number of $G$.