Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and theH6-Conjecture
Maria Chudnovsky, Peter Maceli · Journal of Graph Theory · 2013
Let be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos 6 conjectured that every prime graph in 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 1 we give a short proof of Fouquet's result 3 on the structure of the subclass of bull-free graphs contained in .