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 -