2000/11/21 by Vadim E. Levit, Levit, Vadim E., Eugen Mandrescu +1
Mathematics · #05C69 #05C70 #51D10 (Primary) 90C10 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C69 #msc:05C70 #msc:51D10 #msc:90C10
paper · pdf · doi:10.48550/arxiv.math/0011156
arxiv created 2000/11/21 · arxiv updated 2009/11/30
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.