Trees with unique minimum glolal offensive alliance sets

TL;DR

This paper characterizes trees that have a unique minimum global offensive alliance set, providing insights into the structure and properties of such trees within graph theory.

Contribution

It offers a constructive characterization of trees with a unique minimum global offensive alliance, a novel contribution to the study of offensive alliances in graphs.

Findings

Identifies structural properties of trees with unique minimum global offensive alliances

Provides a constructive method to characterize such trees

Enhances understanding of offensive alliance sets in tree graphs

Abstract

Let $G=$ $\left( V,E\right) $ be a simple graph.\ A non-empty set $S \subseteq V$ is called a global offensive alliance if $S$ is a dominating set and for every vertex $v$ in $V-S$, at least half of the vertices from the closed neighborhood of $v$ are in $S$. The global offensive alliance number is the minimum cardinality of a global offensive alliance in $G$. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.