## Abstract

Let G be a simple graph with vertex set V (G). A set S⊆ V(G) is independent if no two vertices from S are adjacent. The graph G is known to be König-Egerváry if α(G) + μ(G) = |V (G)|, where α(G) denotes the size of a maximum independent set and μ(G) is the cardinality of a maximum matching. Let Ω(G) denote the family of all maximum independent sets, and f be the function from subcollections Γ of Ω(G) to ℕ such that f(Γ)=|⋃Γ|+|⋂Γ|. Our main finding claims that f is ◃ -increasing, where the preorder Γ ^{′}◃ Γ means that ⋃ Γ ^{′}⊆ ⋃ Γ and ⋂ Γ ⊆ ⋂ Γ ^{′}. Let us say that a family ∅≠ Γ ⊆ Ω (G) is a König-Egerváry collection if |⋃Γ|+|⋂Γ|=2α(G). We conclude with the observation that for every graph G each subcollection of a König-Egerváry collection is König-Egerváry as well.

Original language | English |
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Pages (from-to) | 199-207 |

Number of pages | 9 |

Journal | Order |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jul 2019 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018, Springer Nature B.V.

## Keywords

- Core
- Corona
- Critical set
- Diadem
- Ker
- König-Egerváry graph
- Maximum independent set
- Maximum matching