Super Domination: Graph Classes, Products and Enumeration

TL;DR

This paper studies the super dominating set problem, a variant of the dominating set problem, providing bounds, exact counts for specific graph classes, and exploring properties of super dominating sets in cycles.

Contribution

It introduces tight bounds for super domination numbers in various graph operations and characterizes the number of minimum super dominating sets in certain graph classes.

Findings

Bounds for super domination number in graph products and sums

Exact counts of super dominating sets in specific graph classes

Surprising variation in the number of super dominating sets in cycles

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 $S\subseteq V$ 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 $n\mod4$.