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 Chvátal 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.
Highlights
A graph G is chordal if every induced cycle is of length three
Not all graphs exhibit the nice structure of chordal graphs, and the statement does not extend to general graphs
A vertex v in a graph G is avoidable if every induced path on three vertices with middle vertex v is contained in an induced cycle in G
Summary
A graph G is chordal if every induced cycle is of length three. A classical result of Dirac [Dir61] states that every chordal graph has a simplicial vertex, that is, a vertex. Not all graphs exhibit the nice structure of chordal graphs, and the statement does not extend to general graphs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
