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

• Published in: Discrete Applied Mathematics Vol. 283; pp. 189 - 198
• Main Author: Tankus, David
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.09.2020; Elsevier BV
• Subjects: Algorithms; Extendable vertex; Generating subgraph; Graph theory; Graphs; Independent set; Polynomials; Relating edge; Vertex sets; Weighting functions; Well-covered graph; Relating edge; Independent set; Extendable vertex; Generating subgraph; Well-covered graph
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2020.01.012

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.