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

  • Authors:

  • V. E. Levit;Eugen Mandrescu

  • Affiliations:

  • Department of Computer Science, Holon Institute of Technology, 52 Golomb Street, P.O. Box 305, Holon 58102, Israel and Department of Computer Science and Mathematics, The College of Judea and Sama ...;Department of Computer Science, Holon Institute of Technology, 52 Golomb Street, P.O. Box 305, Holon 58102, Israel

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

A matching M 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 Konig-Egervary graph provided @a(G)+@m(G)=|V(G)| [R.W. Deming, Independence numbers of graphs-an extension of the Konig-Egervary 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 @m(G) is the size of a maximum matching and @a(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, and we write S@?@J(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@?@J(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 graph G,@J(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 @J(G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S@?@J(G), the subgraph spanned by S@?N(S) is a Konig-Egervary graph.