## 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. Let Ind(G) denote the family of all independent sets. The graph G is said 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. A family Γ ⊆ Ind(G) is a König-Egerváry collection if |S Γ|+ |T Γ| = 2α(G). The number d (X) = |X| − |N(X)| is the difference of X ⊆ V (G), and a set A ∈ Ind(G) is critical if d(A) = max{d (I): I ∈ Ind(G)}. In this paper, we show that if the family of all maximum critical independent sets of a graph G is a König-Egerváry collection, then G is a König-Egerváry graph. This result generalizes one of our conjectures verified by Short in 2016.

Original language | English |
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Article number | 1027 |

Journal | Art of Discrete and Applied Mathematics |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2019 University of Primorska. All right reserved.

## Keywords

- Core
- Corona
- Critical set
- Diadem
- Kernel
- König-Egerváry graph
- Maximum independent set
- Nucleus