Weighted well-covered graphs without cycles of lengths 5, 6 and 7

• Published in: Information processing letters Vol. 174; p. 106189
• Main Author: Tankus, David
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2022
• Subjects: Generating subgraph; Graph algorithms; Maximal independent set; Vector space; Weighted well-covered graph; Generating subgraph; Maximal independent set; Graph algorithms; Vector space; Weighted well-covered graph
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2021.106189

•There exists a polynomial algorithm for finding WCW(G) for a graph G without cycles of lengths 5, 6 and 7.•There exists a polynomial algorithm which recognizes well-covered graphs without cycles of lengths 5, 6 and 7.•The complexity of finding WCW(G) for graphs without cycles of lengths 5, 6 and 7 is O(|V|̂3(|V|+|E|)). A graph G is well-covered if all maximal independent sets are of the same cardinality. Assume that a weight function w:V(G)⟶R 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[2] denoted WCW(G)[1]. Deciding whether an input graph G is well-covered is co-NP-complete [4,17]. Therefore, finding WCW(G) is co-NP-hard. This paper presents an algorithm whose input is a graph G without cycles of lengths 5, 6, and 7. The algorithm finds polynomially WCW(G).