A connected bipartite graph has s(G,w)=cs(G,w) for every positive weight function w exactly when it is an even cycle, a double star, a book graph, K3,3 minus an edge, or one of the explicitly listed D or D* graph families.
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Stable structure on safe set problems in vertex-weighted graphs
A connected bipartite graph has s(G,w)=cs(G,w) for every positive weight function w exactly when it is an even cycle, a double star, a book graph, K3,3 minus an edge, or one of the explicitly listed D or D* graph families.