  The dominating set problem (DSP) is one of the most famous problems in
combinatorial optimization. It is defined as follows. For a given simple graph
$G=(V,E)$, 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 $S$.
Furthermore, the DSP is the problem of finding a minimum-size dominating set
and the corresponding minimum size, the domination number of $G$.
  In this, work we investigate a variant of the DSP, the super dominating set
problem (SDSP), which has attracted much attention during the last years. 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 $N(v)\cap
\overline{S}=\{u\}$. Analogously, the SDSP is to find a minimum-size super
dominating set, and the corresponding minimum size, the super domination number
of $G$. The decision variants of both the DSP and the SDSP have shown to be
$\mathcal{NP}$-hard.
  In this paper, we present tight bounds for the super domination number of the
neighbourhood corona product, $r$-gluing, and the Haj\'{o}s sum of two graphs.
Additionally, we present infinite families of graphs attaining our bounds.
Finally, we give the exact number of minimum size super dominating sets for
some graph classes. In particular, the number of super dominating sets for
cycles has quite surprising properties as it varies between values of the set
$\{4,n,2n,\frac{5n^2-10n}{8}\}$ based on $n\mod4$.
