A new class of graphs that satisfies the Chen-Chv\'atal Conjecture

TL;DR

This paper proves a stronger version of the Chen-Chvátal conjecture for a class of graphs including chordal and distance-hereditary graphs, extending a geometric combinatorial theorem to these graph families.

Contribution

It introduces a new class of graphs satisfying the Chen-Chvátal conjecture, expanding the conjecture's applicability beyond previously known cases.

Findings

Proved the conjecture for chordal graphs

Extended the conjecture to distance-hereditary graphs

Established a stronger version of the conjecture

Abstract

A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chv\'atal conjecture for a family of graphs containing chordal graphs and distance-hereditary graphs.