Simplicial vertices in graphs with no induced four-edge path or four-edge antipath, and the $H_6$-conjecture

TL;DR

This paper investigates the structure of graphs with no induced four-edge path or antipath, providing a counterexample to a conjecture about their prime graphs and offering a simplified proof for a related structural result.

Contribution

It presents a counterexample to Hayward and Nastos's conjecture and proves a weaker version, advancing understanding of these graph classes.

Findings

Counterexample to the original conjecture.

A proof of a weaker structural property.

Simplified proof of Fouquet's result on bull-free graphs.

Abstract

Let $\mathcal{G}$ be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos \cite{MS} conjectured that every prime graph in $\mathcal{G}$ not isomorphic to the cycle of length five is either a split graph or contains a certain useful arrangement of simplicial and antisimplicial vertices. In this paper we give a counterexample to their conjecture, and prove a slightly weaker version. Additionally, applying a result of the first author and Seymour \cite{grow} we give a short proof of Fouquet's result \cite{C5} on the structure of the subclass of bull-free graphs contained in $\mathcal{G}$.