More on the super domination number of graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex in $ V \setminus S$ is adjacent to at least one vertex in $S$. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V \setminus S$, there exists a $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a minimum size super dominating set of $G$ is the super domination number of $G$. The problem of finding the super domination number of a graph is $\mathcal{NP}$-hard. In this paper, we continue the study of the super domination number of graphs and present tight bounds for neighbourhood corona product, $r$-gluing, and Haj\'{o}s sum of two graphs. We present infinite families of graphs attaining our bounds. Finally, we give the exact number of minimum size super dominating sets for some