A characterization of graphs of diameter two with fewer lines than vertices
Combinatorics
2025-10-23 v1 Metric Geometry
Abstract
In 2008 Chen and Chv\'atal conjectured that any metric space on n points has at least n lines, unless all the points belong to one line. Chv\atal proved in 2014 that this is indeed the case for metric spaces with distances 0, 1 and 2. In this work, we prove that there exists a family of ten graphs such that a metric space defined by a graph of diameter two has fewer lines than points if and only if the associated graph belongs to that family.
Cite
@article{arxiv.2510.19601,
title = {A characterization of graphs of diameter two with fewer lines than vertices},
author = {Martín Matamala},
journal= {arXiv preprint arXiv:2510.19601},
year = {2025}
}