## 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 language | English (US) |
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Pages (from-to) | 17-25 |

Number of pages | 9 |

Journal | Discrete Applied Mathematics |

Volume | 124 |

Issue number | 1-3 |

DOIs | |

State | Published - Dec 15 2002 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics