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arXiv · 2510.19601

A characterization of graphs of diameter two with fewer lines than vertices

Martín Matamala
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Abstract

In 2008 Chen and Chvátal 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.

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Martín Matamala. 2025-10-22. A characterization of graphs of diameter two with fewer lines than vertices. https://arxiv.org/abs/2510.19601

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