Global Defensive Alliances in the Lexicographic Product of Paths and Cycles

TL;DR

This paper determines the exact size of the smallest global defensive alliances in the lexicographic products of paths and cycles, contributing to graph theory and alliance analysis.

Contribution

It provides the first exact calculations of global defensive alliance numbers for lexicographic products of paths and cycles.

Findings

Exact values for global defensive alliance numbers in these graph products

Extension of alliance theory to complex graph structures

New formulas for specific graph classes

Abstract

A set $S$ of vertices of graph $G$ is a \textit{defensive alliance} of $G$ if for every $v \in S$, it holds $|N[v] \cap S| \geq |N[v]-S|$. An alliance $S$ is called $global$ if it is also a dominating set. In this paper, we determine the exact values of the global defensive alliance number of lexicographic products of path and cycles.