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

Endre Boros, Martin C. Golumbic, Vadim E. Levit

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)17-25
Number of pages9
JournalDiscrete Applied Mathematics
Volume124
Issue number1-3
DOIs
StatePublished - Dec 15 2002

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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