# Distribution of global defensive $k$-alliances over some graph products

## TL;DR

This paper investigates the properties of global defensive $k$-alliances in various graph products, providing bounds and analyzing their sharpness across hierarchical, lexicographic, corona, and edge corona products.

## Contribution

It introduces bounds for the global defensive $k$-alliance number in multiple graph products, expanding understanding of alliance distribution in complex graph structures.

## Key findings

- Upper bounds for alliance numbers in different graph products
- Bounds are expressed with invariants of the factor graphs
- Discussion on the sharpness of these bounds

## Abstract

If $G=(V_G, E_G)$ is a graph, then $S\subseteq V_G$ is a global defensive $k$-alliance in $G$ if (i) each vertex not in $S$ has a neighbor in $S$ and (ii) each vertex of $S$ has at least $k$ more neighbors inside $S$ than outside of it. The global defensive $k$-alliance number of $G$ is the minimum cardinality among all global defensive $k$-alliance in $G$. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds are also discussed.