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Bipartite graphs with uniquely restricted maximum matchings and their corresponding greedoids

Vadim E. Levit·Eugen Mandrescu
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Abstract

A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by the union of S and N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. One theorem of Nemhauser and Trotter Jr., working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In one of our previous papers it is proven that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. In this paper we obtain a generalization of this assertion claiming that the family of all local maximum stable sets of a bipartite graph G is a greedoid if and only if all maximum matchings of G are uniquely restricted.

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Vadim E. Levit, Eugen Mandrescu. 2000-11-21. Bipartite graphs with uniquely restricted maximum matchings and their corresponding greedoids. https://arxiv.org/abs/math/0011156

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