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- W4311997218 abstract "A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph $G$, a real-valued vertex weight function $w$ is said to be a well-covered weighting of $G$ if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph $G$ forms a vector space over the field of real numbers, called the well-covered vector space of $G$. Since the problem of recognizing well-covered graphs is $mathsf{co}$-$mathsf{NP}$-complete, the problem of computing the well-covered vector space of a given graph is $mathsf{co}$-$mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs." @default.
- W4311997218 created "2023-01-03" @default.
- W4311997218 creator A5027970574 @default.
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- W4311997218 date "2022-12-16" @default.
- W4311997218 modified "2023-10-01" @default.
- W4311997218 title "Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition" @default.
- W4311997218 doi "https://doi.org/10.48550/arxiv.2212.08599" @default.
- W4311997218 hasPublicationYear "2022" @default.
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