On the super domination number of graphs

Abstract

The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets of $G$. In this paper, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. We also obtain results on the super domination number of corona product graphs and Cartesian product graphs.