Vertex-Removal in $K$-Domination

Nader Jafari Rad, Lutz Volkmann

Let $G=(V,E)$ be any graph without isolated vertices. For some $\alpha$ with $0<\alpha\leq 1$ and a dominating set $S$ of $G$, we say that $S$ is an $\alpha$-dominating set if for any $v\in V-S,|N(v)\cap S|\geq \alpha |N(v)|$. The cardinality of a smallest $\alpha$-dominating set of $G$ is called the $\alpha$-\textit{domination number} of $G$ and is denoted by $\gamma_{\alpha}(G)$. We study the effect of vertex removal on $\alpha$-domination.