Bisplit graphs satisfy the Chen-Chv\'atal conjecture

TL;DR

This paper proves that bisplit graphs, a specific class of graphs, satisfy the Chen-Chvátal conjecture by demonstrating their metric space has a universal line or at least as many lines as vertices.

Contribution

It provides a proof that bisplit graphs meet the Chen-Chvátal conjecture, a previously unverified case for this class of graphs.

Findings

Bisplit graphs satisfy the Chen-Chvátal conjecture.

Their metric space has a universal line or many lines.

The proof confirms the conjecture for this graph class.

Abstract

In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the Chen-Chv\'atal conjecture: their metric space (in the usual sense) has a universal line (in an unusual sense) or at least as many lines as the number of vertices.