Recognizing generating subgraphs revisited

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph G is well-covered is co-NP-complete. Therefore, finding WCW(G) is co-NP-hard. A generating subgraph of a graph G is an induced complete bipartite subgraph B of G on vertex sets of bipartition B_X and B_Y, such that each of S ∪ B_X and S ∪ B_Y is a maximal independent set of G, for some independent set S. If B is generating, then w(B_X)=w(B_Y) for every weight function w ∈ WCW(G). Therefore, generating subgraphs play an important role in finding WCW(G). The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article, we prove NP-completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other and, we supply polynomial algorithms for recognizing generating subgraphs and finding WCW(G), when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds WCW(G) when G does not contain cycles of lengths 3, 4, 5, and 7.