## Abstract

The stability number of a graph G, denoted by α(G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called α ^{+}-stable. G is a König-Egerváry graph if its order equals α(G)+μ(G), where μ(G) is the size of a maximum matching in G. In this paper, we characterize α ^{+}-stable König-Egerváry graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egerváry graph G=(V,E) of order at least two is α ^{+}-stable if and only if G has a perfect matching and |∩{V-S: S∈Ω(G)}|1 (where Ω(G) denotes the family of all maximum stable sets of G). We also show that the equality |∩{V-S: S∈Ω(G)}|=|∩{S: S∈Ω(G)}| is a necessary and sufficient condition for a König-Egerváry graph G to have a perfect matching. Finally, we describe the two following types of α ^{+}-stable König-Egerváry graphs: those with |∩{S: S∈Ω(G)}|=0 and |∩{S: S∈Ω(G)}|=1, respectively.

Original language | English |
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Pages (from-to) | 179-190 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 263 |

Issue number | 1-3 |

DOIs | |

State | Published - 28 Feb 2003 |

Externally published | Yes |

## Keywords

- Blossom
- König-Egerváry graph
- Maximum matching
- Maximum stable set
- Perfect matching
- α -stable graph

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