Partitioning a graph into defensive k-alliances

TL;DR

This paper investigates how to partition a graph's vertices into defensive k-alliances, providing bounds and relationships for such partitions based on various graph parameters and extending to product graphs.

Contribution

It introduces the concept of defensive k-alliance partition numbers and derives tight bounds, also exploring their behavior in product graphs.

Findings

Derived tight bounds for defensive k-alliance partition numbers.

Established relationships between partitions of product graphs and component graphs.

Analyzed the impact of graph parameters on alliance partitions.

Abstract

A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $\Gamma=(V,E)$, ($\psi_{k}^{gd}(\Gamma)$) $\psi_k^{d}(\Gamma)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $\psi_k^{d}(\Gamma)$ and $\psi_{k}^{gd}(\Gamma)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $\Gamma_1\times \Gamma_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $\Gamma_i$ into (global) defensive $k_i$-alliances, $i\in \{1,2\}$.