TY - JOUR

T1 - A new Greedoid

T2 - The family of local maximum stable sets of a forest

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2002/12/15

Y1 - 2002/12/15

N2 - A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable of the subgraph of G spanned by S∪N(S), where N(S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. (Math. Programming 8 (1975) 232-248), working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S 1 of T, such that S 1⊂S and |S 1|=|S|-1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.

AB - A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable of the subgraph of G spanned by S∪N(S), where N(S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. (Math. Programming 8 (1975) 232-248), working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S 1 of T, such that S 1⊂S and |S 1|=|S|-1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.

KW - 05C05

KW - 05C12

KW - 05C70

KW - 05C75

UR - http://www.scopus.com/inward/record.url?scp=84867962087&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(01)00332-8

DO - 10.1016/S0166-218X(01)00332-8

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

AN - SCOPUS:84867962087

SN - 0166-218X

VL - 124

SP - 91

EP - 101

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1-3

ER -