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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 thecardinality 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 everyvertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for everyvertex $v$ 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 ora powerful) $k$-alliance free set if $X$ does not contain any defensive(offensive or powerful, respectively) $k$-alliance. In this article we studythe relationships between defensive (offensive, powerful) $k$-alliance freesets in Cartesian product graphs and defensive (offensive, powerful)$k$-alliance free sets in the factor graphs.

Alliance free sets in Cartesian product graphs