Simplicial vertices in graphs with no induced four-edge path or four-edge antipath, and the H₆-conjecture

Let G be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos conjectured that every prime graph in 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 article, we give a counterexample to their conjecture, and prove a slightly weaker version. Additionally, applying a result of the first author and Seymour we give a short proof of Fouquet's result on the structure of the subclass of bull-free graphs contained in G.