Recognizing generating subgraphs in graphs without cycles of lengths 6 and 7

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 boolean OR B-X and S boolean OR B-Y is a maximal independent set in the graph. If B is generating, it produces the restriction w(BX) = w(BY). 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,K-1 (Brown et al., 2007). We present a polynomial algorithm for recognizing generating subgraphs for graphs without cycles of lengths 6 and 7. (C) 2020 Elsevier B.V. All rights reserved.