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arXiv · 1908.03788

Avoidable paths in graphs

Marthe Bonamy·Oscar Defrain·Meike Hatzel·Jocelyn Thiebaut
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Abstract

We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced P_k also contains an avoidable P_k. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for k in {1,2} (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chv\'atal et al. 2002, which assumed cycle restrictions. We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.

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Marthe Bonamy, Oscar Defrain, Meike Hatzel, Jocelyn Thiebaut. 2019-08-10. Avoidable paths in graphs. https://arxiv.org/abs/1908.03788

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