Twin-width of Planar Graphs; a Short Proof

TL;DR

This paper provides a concise and self-contained proof that the twin-width of planar graphs is at most 11, contributing to the ongoing effort to determine the exact maximum twin-width for such graphs.

Contribution

The paper introduces a short, simple proof establishing an upper bound of 11 for the twin-width of planar graphs, simplifying previous complex proofs.

Findings

Twin-width of planar graphs is at most 11.

Simplified proof technique for twin-width bounds.

Potential insights into twin-width bounds for beyond-planar graphs.

Abstract

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Kr\'al' and Lamaison [arXiv, September 2022], and an upper bound of 8 by Hlin\v{e}n\'y and Jedelsk\'y [arXiv, October 2022]. The upper bound (currently best) of 8, however, is rather complicated and involved. In the paper we give a short and simple self-contained proof that the twin-width of planar graphs is at most 11. We believe that this short proof can also shed more light on the topic of upper bound(s) on the twin-width of planar and beyond-planar graphs in general.