Alliance free sets in Cartesian product graphs

TL;DR

This paper investigates the properties and relationships of alliance free sets in Cartesian product graphs, focusing on defensive, offensive, and powerful alliances, and how they relate to the factor graphs.

Contribution

It introduces new insights into the structure of alliance free sets in Cartesian products and their connection to alliances in the factor graphs.

Findings

Characterization of alliance free sets in Cartesian product graphs

Relationships between alliances in product graphs and factor graphs

Conditions under which alliance free sets are preserved or transformed

Abstract

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the neighborhood of $S$ is called an \emph{offensive} $k$-\emph{alliance}. A subset of vertices $S\subseteq V$, is a \emph{powerful} $k$-\emph{alliance} if it is both a defensive $k$-alliance and an offensive $(k +2)$-alliance. Moreover, a subset $X\subset V$ is a defensive (an offensive or a powerful) $k$-alliance free set if $X$ does not contain any defensive (offensive or powerful, respectively) $k$-alliance. In this article we study the relationships between defensive (offensive, powerful) $k$-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) $k$-alliance free sets in the factor graphs.