Stable structure on safe set problems in vertex-weighted graphs

Let G be a graph, and let w positive real-valued weight function on V(G). For every subset S of V(G), w(S)=?v?Sw(v). A non-empty S?V(G) is weighted safe set (G,w) if, for component C the subgraph induced by D G?S, we have w(C)?w(D) whenever there an edge between D. If connected, then called connected (G,w). The number s(G,w) cs(G,w) are minimum weights w(S) among all sets (G,w), respectively. Note that pair s(G,w)?cs(G,w) their definitions. In [Fujita et al. “On problem paths cycles” J. Comb. Optim. 37 (2019) 685–701], it was asked which satisfies equality shown cycle equality. this paper, give complete list bipartite graphs such s(G,w)=cs(G,w)