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

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 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 maximum matching and (G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, 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 graph G, (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.

Computing unique maximum matchings in $$O(m)$$ O ( m ) time for König–Egerváry graphs and unicyclic graphs

“…The following lemma, firstly presented in [20], shows that every König-Egerváry graph with a unique perfect matching has at least one leaf (see, for example, the graph G 1 , depicted in Figure 1). We give a proof here for the sake of selfcontainment.…”

Local maximum stable set greedoids stemming from very well-covered graphs

“…Theorem 1.9 [19] If G is a triangle-free graph, then Ψ(G) is a greedoid if and only if all maximum matchings of G are uniquely restricted and the closed neighborhood of every local maximum stable set of G induces a König-Egerváry graph.…”

“…S is called a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V, F) is called a local maximum stable set greedoid if there exists a graph G = (V, E) such that F = Ψ(G).Nemhauser and Trotter Jr. [27], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [15] we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [17], [19], [21], respectively.In this paper we demonstrate that if G is a very well-covered graph, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.…”

“…Nemhauser and Trotter Jr. [27], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [15] we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [17], [19], [21], respectively.…”

Very Well-Covered Graphs of Girth at Least Four and Local Maximum Stable Set Greedoids

“…S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N (S), where N (S) is the neighborhood of S.Nemhauser and Trotter Jr. [20], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [12] we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [14], [15], [17], respectively.In this paper we demonstrate that if G is a very well-covered graph of girth ≥ 4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.…”