Safe set problem on graphs

A non-empty subset S of the vertices of a connected graph G=(V(G),E(G)) is a safe set if, for every connected component C of G[S] and every connected component D of G−S, we have |C|≥|D| whenever there exists an edge of G between C and D. If G[S] 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.