Safe set problem on graphs

Abstract

A non-empty subset $S$ of the vertices of a connected graph $G=(V\left(G\right),E\left(G\right))$ is a safe set if, for every connected component $C$ of $G\left[S\right]$ and every connected component $D$ of $G-S$, we have $\left|C\right|\ge \left|D\right|$ whenever there exists an edge of $G$ between $C$ and $D$. If $G\left[S\right]$ is connected, then $S$ is called a connected safe set. We discuss the minimum sizes of safe sets and connected safe sets in connected graphs.