Stable structure on safe set problems in vertex-weighted graphs

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)=∑v∈Sw(v). A non-empty subset S⊂V(G) is a weighted safe set of (G,w) if, for every component C of the subgraph induced by S and every component D of G−S, we have w(C)≥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, then the set S is called a connected weighted safe set of (G,w). The weighted safe numbers(G,w) and connected weighted safe numbercs(G,w) of (G,w) are the minimum weights w(S) among all weighted safe sets and all connected weighted safe sets of (G,w), respectively. Note that for every pair (G,w), s(G,w)≤cs(G,w) by their definitions. In [Fujita et al. “On weighted safe set problem on paths and cycles” J. Comb. Optim. 37 (2019) 685–701], it was asked which pair (G,w) satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs G such that s(G,w)=cs(G,w) for every weight function w on V(G).