Defensive k-alliances in graphs

Abstract

Let $\Gamma =(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, ${\delta }_X(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in $\Gamma =(V,E)$ if ${\delta }_S(v)\ge {\delta }_{\bar{S}}(v)+k,$ $\forall v\in S.$ The defensive $k$-alliance number of $\Gamma$, denoted by $a_k(\Gamma )$, is defined as the minimum cardinality of a defensive $k$-alliance in $\Gamma$. We study the mathematical properties of $a_k(\Gamma )$.