Global 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 ${\delta }_S(v)\ge {\delta }_{\bar{S}}(v)+k,$ $\forall v\in S.$ A defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global defensive $k$-alliance number} of $\Gamma$, denoted by ${\gamma }_k^a(\Gamma )$, is the minimum cardinality of a defensive $k$-alliance in $\Gamma$. We study the mathematical properties of ${\gamma }_k^a(\Gamma )$.

References in corpus (1)

• Defensive k-alliances in graphs

Cited by in corpus (4)

• On global offensive k-alliances in graphs
• Alliance free and alliance cover sets
• On global defensive k-alliances in zero-divisor graph of finite commutative rings
• Distribution of global defensive $k$-alliances over some graph products