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Let $G$ be a graph, and let $w$ be a positive real-valued weight function on$V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ Anon-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, forevery component $C$ of the subgraph induced by $S$ and every component $D$ of$G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$.If the subgraph of $G$ induced by a weighted safe set $S$ is connected, thenthe set $S$ is called a connected weighted safe set of $(G,w)$. The weightedsafe number $\mathrm{s}(G,w)$ and connected weighted safe number$\mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weightedsafe sets and all connected weighted safe sets of $(G,w)$, respectively. Notethat for every pair $(G,w)$, $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ by theirdefinitions. Recently, it was asked which pair $(G,w)$ satisfies the equalityand shown that every weighted cycle satisfies the equality. In this paper, wegive a complete list of connected bipartite graphs $G$ such that$\mathrm{s}(G,w)=\mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.

Stable structure on safe set problems in vertex-weighted graphs