Global defensive k-alliances in graphs

Abstract

Let $Γ=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $δ_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in $Γ=(V,E)$ if $δ_S(v)\ge δ_{\bar{S}}(v)+k,$ $\forall v\in S.$ A defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global defensive $k$-alliance number} of $Γ$, denoted by $γ_{k}^{a}(Γ)$, is the minimum cardinality of a defensive $k$-alliance in $Γ$. We study the mathematical properties of $γ_{k}^{a}(Γ)$.