TY - JOUR

T1 - VERY WELL-COVERED GRAPHS of GIRTH at LEAST FOUR and LOCAL MAXIMUM STABLE SET GREEDOIDS

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

N1 - Publisher Copyright:
© 2011 World Scientific Publishing Company.

PY - 2011/6/1

Y1 - 2011/6/1

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 ϵψ(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. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232-248], proved that any S ϵ ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91-101] 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 [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163-174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414-2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89-94], 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.

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 ϵψ(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. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232-248], proved that any S ϵ ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91-101] 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 [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163-174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414-2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89-94], 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.

KW - König-Egerváry graph

KW - Very well-covered graph

KW - greedoid

KW - local maximum stable set

KW - triangle-free graph

UR - http://www.scopus.com/inward/record.url?scp=84861201946&partnerID=8YFLogxK

U2 - 10.1142/S1793830911001115

DO - 10.1142/S1793830911001115

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84861201946

SN - 1793-8309

VL - 3

SP - 245

EP - 252

JO - Discrete Mathematics, Algorithms and Applications

JF - Discrete Mathematics, Algorithms and Applications

IS - 2

ER -