Computing global offensive alliances in Cartesian product graphs

TL;DR

This paper derives formulas for the global offensive alliance number in Cartesian product graphs, establishes bounds involving domination numbers, and proposes a Vizing-like conjecture, advancing understanding of alliance properties in graph theory.

Contribution

It provides closed-form formulas for specific graph families, proves new bounds relating alliance and domination numbers, and introduces a conjecture analogous to Vizing's conjecture for global offensive alliances.

Findings

Established lower bounds for $ abla_o(G imes H)$ involving $ abla(G)$ and $ abla_o(H)$.

Proved the conjecture for certain graph families.

Derived closed formulas for the global offensive alliance number in multiple graph classes.

Abstract

A global offensive alliance in a graph $G$ is a set $S$ of vertices with the property that every vertex not belonging to $S$ has at least one more neighbor in $S$ than it has outside of $S$. The global offensive alliance number of $G$, $\gamma_o(G)$, is the minimum cardinality of a global offensive alliance in $G$. A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex not belonging to $S$ has at least one neighbor in $S$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. In this work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we also prove that $\gamma_o(G\square H)\ge \frac{\gamma(G)\gamma_o(H)}{2}$ for any graphs $G$ and $H$ and we show that if $G$ has an efficient dominating set, then $\gamma_o(G\square H)\ge \gamma(G)\gamma_o(H).$ Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.