  Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating
set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that
$N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super
dominating set of $G$ is the super domination number of $G$. An exact formula
for the super domination number of a tree $T$ is obtained and demonstrated that
a smallest super dominating set of $T$ can be computed in linear time. It is
proved that it is NP-complete to decide whether the super domination number of
a graph $G$ is at most a given integer if $G$ is a bipartite graph of girth at
least $8$. The super domination number is determined for all $k$-subdivisions
of graphs. Interestingly, in half of the cases the exact value can be
efficiently computed from the obtained formulas, while in the other cases the
computation is hard. While obtaining these formulas, II-matching numbers are
introduced and proved that they are computationally hard to determine.
