A characterization of graphs of diameter two with fewer lines than vertices
Martín Matamala
TL;DR
The paper addresses the question of when a diameter-two graph can have fewer lines than vertices in its associated metric line system. It proves that there exists a finite family of ten graphs such that a diameter-two graph has $|\mathcal{L}(G)|<|V|$ if and only if it is isomorphic to one of these graphs, using a detailed structural analysis based on $N(x)$, $N^2(x)$ and the line-sets $\mathcal{L}^x_1$ and $\mathcal{L}^x_2$. The work also situates these results within the broader Chen-Chvátal framework, cites known diameter-two examples like $K_{2,3}$, $K_{1,2,2}$, and $M_{2p}$, and discusses the implications for bridgeless graphs and the diameter-three case, including nine diameter-two bridgeless examples and three diameter-three exemplars $M_6'$, $M_8'$, and $\hat{M}_8$. It outlines that while path gluing yields arbitrarily large graphs with $|\mathcal{L}(G)|<|V|$, finiteness results emerge in the bridgeless setting, guiding future complete characterizations for higher diameters. Overall, the paper advances a precise, case-based classification of diameter-two graphs with few lines, contributing to a deeper understanding of line-based metric properties in graphs.
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.