## Abstract

A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König-Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = {pipe}V(G){pipe} -2μ(G) is the deficiency of G. The number def(G)={pipe}V(G){pipe}-2μ(G) is the deficiency of G. The number d(G)=max{{pipe}S{pipe}-{pipe}N(A){pipe}=d(G)}, where N(S) is the neighbourhood of S, and α _{c}(G) denotes the maximum size of a critical independent set. Lrson (Eur J Comb 32:294-300, 2011)demonstrated that G is a k̈nig-Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α _{c}(G)=α(G). In this paper we prove that: (i) d(G)={pipe}(G){pipe}-{pipe}N(core(G)){pipe}=α(G)=def(G) holds for every König-Egerváry graph G; (ii) G is König-Egerváry graph if and only if each maximum independent set of G is critical.

Original language | English |
---|---|

Pages (from-to) | 243-250 |

Number of pages | 8 |

Journal | Graphs and Combinatorics |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2012 |

Externally published | Yes |

## Keywords

- Core
- Critical difference
- Critical independent set
- Deficiency
- Maximum independent set
- Maximum matching