Super dominating sets in graphs

TL;DR

This paper investigates the super domination number in graphs, especially trees, providing bounds and characterizations for extremal cases, thereby advancing understanding of this graph invariant.

Contribution

It establishes bounds for the super domination number in trees and characterizes the extremal trees achieving these bounds.

Findings

For trees with at least three vertices, the super domination number is between n/2 and n minus the number of support vertices.

Characterization of trees that attain the bounds on the super domination number.

Provides new insights into the structure of super dominating sets in trees.

Abstract

Let $G=(V,E)$ be a graph. A subset $D$ of $V(G)$ is called a super dominating set if for every $v \in V(G)-D$ there exists an external private neighbour of $v$ with respect to $V(G)-D.$ The minimum cardinality of a super dominating set is called the super domination number of $G$ and is denoted by $\gamma_{sp}(G)$. In this paper some results on the super domination number are obtained. We prove that if $T$ is a tree with at least three vertices, then $\frac{n}{2}\leq\gamma_{sp}(T)\leq n-s,$ where $s$ is the number of support vertices in $T$ and we characterize the extremal trees.