# On the super domination number of graphs

## TL;DR

This paper investigates the super domination number in graphs, providing formulas and bounds, and explores specific cases like corona and Cartesian product graphs to deepen understanding of this graph invariant.

## Contribution

It introduces closed formulas and tight bounds for the super domination number, including special cases for product graphs, advancing theoretical understanding of this graph parameter.

## Key findings

- Derived closed formulas for super domination number.
- Established tight bounds based on graph invariants.
- Analyzed super domination in corona and Cartesian product graphs.

## Abstract

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting 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 of $G$ 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 domination number of $G$ in terms of several invariants of $G$. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered.