On the super domination number of lexicographic product graphs

TL;DR

This paper investigates the super domination number in lexicographic product graphs, providing formulas and bounds, and proves that computing this number is NP-Hard.

Contribution

It offers new closed-form formulas and tight bounds for the super domination number in lexicographic product graphs, linking it to invariants of the factor graphs.

Findings

Derived formulas for super domination number

Established tight bounds for specific graph classes

Proved NP-Hardness of computing the super domination number

Abstract

The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$ we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.