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Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and the H6-Conjecture


  • Contract grant sponsor: NSF: contract grant numbers: IIS-1117631 and DMS-1001091.


Let math formula 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 math formula 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 math formula.