Let $G$ be a finite graph with an eigenvalue $\mu$ of multiplicity $m$. A set $X$ of $m$ vertices in $G$ is called a {\em star set} for $\mu$ in $G$ if $\mu$ is not an eigenvalue of the {\em star complement} $G-X$. Various dominating properties of the vertices in $G-X$ are established and discussed in the context of memoryless communication networks.