## Abstract

The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G)=∩{S: S is a maximum stable set in G}, and ξ(G)=|core(G)|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ(G)1, then G is quasi-regularizable; (ii) if the order of G is n, and α(G)(n+k-min{1, |N(core(G))|})/2, for some k1, then ξ(G)k+1; moreover, if n+k-min{1,|N(core(G))|} is even, then ξ(G)k+2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ(G)1 is true whenever α(G)n/2. In the case of König- Egerváry graphs, i.e., for graphs enjoying the equality α(G)+μ(G)=n, where μ(G) is the maximum size of a matching of G, we prove that |core(G)||N(core(G))| is a necessary and sufficient condition for α(G)n/2. Furthermore, for bipartite graphs without isolated vertices, ξ(G)2 is equivalent to α(G)n/2. We also show that Hall's Marriage Theorem is true for König-Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core(G).

Original language | English |
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Pages (from-to) | 149-161 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 117 |

Issue number | 1-3 |

DOIs | |

State | Published - 15 Mar 2002 |

Externally published | Yes |

## Keywords

- Bipartite graph
- Hall's Marriage Theorem
- König-Egerváry graph
- Maximum Matching
- Maximum stable set
- Quasi-regularizable graph
- α-stable graph