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 $a_k(\Gamma )$.

Cited by in corpus (6)

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• On global offensive k-alliances in graphs
• Proportionally dense subgraph of maximum size: complexity and approximation
• Alliances and related parameters in graphs
• Alliance free and alliance cover sets
• On global defensive k-alliances in zero-divisor graph of finite commutative rings