The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given graph G=(V,E), a dominating set of G is a subset S⊆V such that every vertex in V∖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∈S¯=V∖S, there exists a v∈S such that N(v)∩S¯=N(v)∖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 been shown to be NP-hard. In this paper, we present tight bounds for the super domination number of the neighbourhood corona product, r-clique sum, and the Hajós 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,5n2−10n8} based on nmod4.

Let G be a graph. A dominating set D⊆V(G) is a super dominating set if for every vertex x∈V(G)∖D there exists y∈D such that NG(y)∩(V(G)∖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 it is 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.