Safety of links with respect to the Myerson value for communication situations

Let $$\mu (N,v,L)$$ be the Myerson value for graph games (N, v, L). We call a link ij of a graph L safe if $$\mu _k(N,v,L)\ge \mu _k(N,v,L\setminus \{ij\})$$ for any $$k\in N$$ , which means that none of players benefits from breaking the link ij. A link $$ij\in L$$ is called a bridge if N splits into more components after ij is deleted. We show that if (N, v) is convex, then any bridge is safe. Furthermore, if (N, v) is strictly convex, then a link is safe if and only if it is a bridge.