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- W4297632386 abstract "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 $Ssubseteq V$ 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 $uin overline{S}=V setminus S$, there exists a $vin S$ 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 $nmod4$." @default.
- W4297632386 created "2022-09-30" @default.
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- W4297632386 date "2022-09-05" @default.
- W4297632386 modified "2023-09-27" @default.
- W4297632386 title "Super Domination: Graph Classes, Products and Enumeration" @default.
- W4297632386 doi "https://doi.org/10.48550/arxiv.2209.01795" @default.
- W4297632386 hasPublicationYear "2022" @default.
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