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arXiv · 1302.0404

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

Maria Chudnovsky·Peter Maceli
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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 $\mathcal{G}$.

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Maria Chudnovsky, Peter Maceli. 2013-02-02. Simplicial vertices in graphs with no induced four-edge path or four-edge antipath, and the $H_6$-conjecture. https://arxiv.org/abs/1302.0404

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