## Abstract

Let α(G) denote the cardinality of a maximum independent set, while µ(G) be the size of a maximum matching in the graph G = (V, E). If α(G) + µ(G) = |V |, then G is a König-Egerváry graph. If d_{1} ≤ d_{2} ≤ · · · ≤ d_{n} is the degree sequence of G, then the annihilation number a (G) of G is the largest integer k such that ^{Pk}_{i}_{=1} d_{i} ≤ |E|. A set A ⊆ V satisfying ^{P}_{v}∈_{A} deg(v) ≤ |E| is an annihilation set; if, in addition, deg (x) + ^{P}_{v}∈_{A} deg(v) > |E|, for every vertex x ∈ V (G) − A, then A is a maximal annihilation set in G. In 2011, Larson and Pepper conjectured that the following assertions are equivalent: (i) α (G) = a (G); (ii) G is a König-Egerváry graph and every maximum independent set is a maximal annihilating set. It turns out that the implication “(i) =≻ (ii)” is correct. In this paper, we show that the opposite direction is not valid, by providing a series of generic counterexamples.

Original language | English |
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Pages (from-to) | 359-369 |

Number of pages | 11 |

Journal | Ars Mathematica Contemporanea |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:cb This work is licensed under https://creativecommons.org/licenses/by/4.0/

## Keywords

- Annihilation number
- Annihilation set
- König-Egerváry graph
- Maximum independent set
- Maximum matching

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