Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition
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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 πΌπ-ππ―-complete, the problem of computing the well-covered vector space of a given graph is πΌπ-ππ―-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.
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