TY - JOUR

T1 - Local maximum stable set greedoids stemming from very well-covered graphs

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2012/8

Y1 - 2012/8

N2 - A maximum stable setin a graph G is a stable set of maximum cardinality. The set 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. (1975) [28] proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (2002) [16] we showed that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, and well-covered while Ψ(G) is a greedoid were analyzed in Levit and Mandrescu (2004) [18], Levit and Mandrescu (2007) [20], and Levit and Mandrescu (2008) [23], 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.

AB - A maximum stable setin a graph G is a stable set of maximum cardinality. The set 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. (1975) [28] proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (2002) [16] we showed that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, and well-covered while Ψ(G) is a greedoid were analyzed in Levit and Mandrescu (2004) [18], Levit and Mandrescu (2007) [20], and Levit and Mandrescu (2008) [23], 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.

KW - Greedoid

KW - König-Egerváry graph

KW - Local maximum stable set

KW - Perfect matching

KW - Very well-covered graph

UR - http://www.scopus.com/inward/record.url?scp=84861186532&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2012.03.017

DO - 10.1016/j.dam.2012.03.017

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AN - SCOPUS:84861186532

SN - 0166-218X

VL - 160

SP - 1864

EP - 1871

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 12

ER -