## Abstract

It is a classic result that a set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal conjectured in 2008 that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. One way to study lines in an space is though its betweenness. Given a quasi-metric space (V,ρ), its induced quasi-metric be-tweenness is the set of triples (x, y, z) μ V^{3}such that ρ(x, z)=ρ(x, y) +ρ(y, z). In this work, we prove the existence of a quasi-metric space on four points a, b, c and d whose quasi-metric betweenness is = {(c, a, b), (a, b, c), (d, b, a), (b, a, d)}. This space has only three lines, none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we conclude that Chen and Chvatal's conjecture is valid for any metric space on four points.

Original language | English |
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Pages (from-to) | 308-315 |

Number of pages | 8 |

Journal | Procedia Computer Science |

Volume | 223 |

DOIs | |

Publication status | Published - 2023 |

Event | 12th Latin-American Algorithms, Graphs and Optimization Symposium, LAGOS 2023 - Huatulco, Mexico Duration: 18 Sept 2023 → 22 Sept 2023 |

## Keywords

- betweenness
- lines
- quasi-metric spaces

## ASJC Scopus subject areas

- General Computer Science