TY - JOUR

T1 - On the number of vertices belonging to all maximum stable sets of a graph

AU - Boros, Endre

AU - C. Golumbic, Martin

AU - E. Levit, Vadim

PY - 2002/12/15

Y1 - 2002/12/15

N2 - Let us denote by α(G) the size of a maximum stable set, and by μ(G) the size of a maximum matching of a graph G, and let ξ(G) be the number of vertices which belong to all maximum stable sets. We shall show that ξ(G)1+α(G)-μ(G) holds for any connected graph, whenever α(G)>μ(G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Levit and Mandrescu [(prE-print math. CO/9912047 (1999) 13pp.)]. We also prove that on one hand, ξ(G)>0 can be recognized in polynomial time whenever μ(G)<|V(G)|/3, and on the other hand determining whether ξ(G)>k is, in general, NP-complete for any fixed k0.

AB - Let us denote by α(G) the size of a maximum stable set, and by μ(G) the size of a maximum matching of a graph G, and let ξ(G) be the number of vertices which belong to all maximum stable sets. We shall show that ξ(G)1+α(G)-μ(G) holds for any connected graph, whenever α(G)>μ(G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Levit and Mandrescu [(prE-print math. CO/9912047 (1999) 13pp.)]. We also prove that on one hand, ξ(G)>0 can be recognized in polynomial time whenever μ(G)<|V(G)|/3, and on the other hand determining whether ξ(G)>k is, in general, NP-complete for any fixed k0.

UR - http://www.scopus.com/inward/record.url?scp=84867936915&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(01)00327-4

DO - 10.1016/S0166-218X(01)00327-4

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AN - SCOPUS:84867936915

SN - 0166-218X

VL - 124

SP - 17

EP - 25

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1-3

ER -