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

TL;DR

This paper characterizes specific graphs of diameter two that have fewer lines than points in their associated metric spaces, confirming a conjecture for a particular class of graphs.

Contribution

It identifies a unique family of ten graphs that precisely determine when a diameter-two graph's metric space has fewer lines than points.

Findings

Existence of a family of ten graphs with this property

Characterization of diameter-two graphs with fewer lines

Confirmation of the conjecture for these graphs

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.