Weighted Well-Covered Claw-Free Graphs

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A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w 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. Given an input claw-free graph G, we present an O(n⁶)algortihm, whose input is a claw-free graph G, and output is the vector space of weight functions w, for which G is w-well-covered. A graph G is equimatchable if all its maximal matchings are of the same cardinality. Assume that a weight function w is defined on the edges of G. Then G is w-equimatchable if all its maximal matchings are of the same weight. For every graph G, the set of weight functions w such that G is w-equimatchable is a vector space. We present an O(m*n⁴ + n⁵*log(n)) algorithm which receives an input graph G, and outputs the vector space of weight functions w such that G is w-equimatchable.