Recognizing Generating Subgraphs in Graphs without Cycles of Lengths 6 and 7
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Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition B_X and B_Y. The subgraph B is generating if there exists an independent set S such that each of S ∪ B_X and S ∪ B_Y is a maximal independent set in the graph. If B is generating, it produces the restriction w(B_X)=w(B_Y). Let w:V(G) ⟶R be a weight function. We say that G is w-well-covered if all maximal independent sets are of the same weight. The graph G is w-well-covered if and only if w satisfies all restrictions produced by all generating subgraphs of G. Therefore, generating subgraphs play an important role in characterizing weighted well-covered graphs. It is an NP-complete problem to decide whether a subgraph is generating, even when the subgraph is isomorphic to K_1,1 bnz:related. We present a polynomial algorithm for recognizing generating subgraphs for graphs without cycles of lengths 6 and 7.
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