TY - JOUR

T1 - On local maximum stable set greedoids

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2012/2/6

Y1 - 2012/2/6

N2 - A maximum stable set in a graph G is a stable set of maximum cardinality. 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. 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) [29], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (2002) [19] 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, unicycle, while Ψ (G) is a greedoid, were analyzed in Levit and Mandrescu (2004, 2007, 2008, 2001, 2009) [2022,18,25], respectively. In this paper, we demonstrate that if the family Ψ(G) satisfies the accessibility property, then, first, Ψ(G) is a greedoid, and, second, this greedoid, which we called the local maximum stable set greedoid defined by G, is an interval greedoid. We also characterize those graphs whose families of local maximum stable sets are either antimatroids or matroids. For these cases, some polynomial recognition algorithms are suggested.

AB - A maximum stable set in a graph G is a stable set of maximum cardinality. 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. 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) [29], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (2002) [19] 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, unicycle, while Ψ (G) is a greedoid, were analyzed in Levit and Mandrescu (2004, 2007, 2008, 2001, 2009) [2022,18,25], respectively. In this paper, we demonstrate that if the family Ψ(G) satisfies the accessibility property, then, first, Ψ(G) is a greedoid, and, second, this greedoid, which we called the local maximum stable set greedoid defined by G, is an interval greedoid. We also characterize those graphs whose families of local maximum stable sets are either antimatroids or matroids. For these cases, some polynomial recognition algorithms are suggested.

KW - Antimatroid

KW - Bipartite graph

KW - Interval greedoid

KW - König-Egerváry graph

KW - Matroid

KW - Simplicial graph

KW - Tree

KW - Triangle-free graph

KW - Unicycle graph

KW - Well-covered graph

UR - http://www.scopus.com/inward/record.url?scp=81955160873&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.04.015

DO - 10.1016/j.disc.2011.04.015

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AN - SCOPUS:81955160873

SN - 0012-365X

VL - 312

SP - 588

EP - 596

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

ER -