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

Abstract

Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. The subgraph B is generating if there exists an independent set S such that each of S∪BX and S∪BY 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 K1,1 (Brown et al., 2007). We present a polynomial algorithm for recognizing generating subgraphs for graphs without cycles of lengths 6 and 7.