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 $\mathcal{ G}$.
Categories: math.CO cs.DM