Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

A matchingM is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139–154]. G is a König–Egerváry graph provided (G)+ (G)= |V (G)| [R.W. Deming, Independence numbers of graphs—an extension of the König–Egerváry theorem, Discrete Math. 27 (1979) 23–33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228–229], where (G) is the size of a maximummatching and (G) is the cardinality of a maximum stable set. S is a local maximum stable set ofG, and we write S ∈ (G), if S is a maximum stable set of the subgraph spanned by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232–248], proved that any S ∈ (G) is a subset of a maximum stable set of G. In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163–174] we have proved that for a bipartite graphG, (G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G is a triangle-free graph, then (G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S ∈ (G), the subgraph spanned by S ∪N(S) is a König–Egerváry graph. © 2007 Elsevier B.V. All rights reserved.