Twin-width of Planar Graphs is at most 9, and at most 6 when Bipartite Planar

TL;DR

This paper establishes that the twin-width of planar graphs is at most 9, and at most 6 for bipartite planar graphs, providing the first small explicit upper bounds for these classes.

Contribution

The paper improves the known upper bounds on twin-width for planar and bipartite planar graphs to 9 and 6 respectively, offering the first small explicit bounds.

Findings

Twin-width of planar graphs is at most 9.

Twin-width of bipartite planar graphs is at most 6.

Provides the first small explicit bounds for these classes.

Abstract

The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have seen first small explicit upper bounds of 183 by Jacob and Pilipczuk [arXiv, January 2022], 583 by Bonnet et al. [arXiv, February 2022], and of 37 by Bekos et al. [arXiv, April 2022]. We prove that the twin-width of planar graphs is at most 9. Furthermore, if a planar graph is also bipartite, then its twin-width is at most 6.