Twin-width of Planar Graphs is at most 8, and some Related Bounds

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS 2020], and has interesting applications in the areas of logic on graphs and in parameterized algorithmics. Very briefly, the essence of twin-width is in a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. While for many natural graph classes it is known that their twin-width is bounded, published upper bounds on the twin-width in non-trivial cases are very often "astronomically large". We focus on planar graphs, which are known to have bounded twin-width already since the introduction of it, but it took some time for the first explicit "non-astronomical" upper bounds to come. Namely, in the order of preprint appearance, it was the bound of at most 183 by Jacob and Pilipczuk [arXiv, January 2022], and 583 by Bonnet, Kwon and Wood [arXiv, February 2022]. Subsequent arXiv manuscripts in 2022 improved the bound down to 37 (Bekos et al.), 11 and 9 (both by Hlin\v{e}n\'y). We further elaborate on the approach used in the latter manuscripts, proving that the twin-width of every planar graph is at most 8, and construct a witnessing contraction sequence in linear time. Note that the currently best lower-bound planar example is of twin-width 7, by Kr\'al and Lamaison [arXiv, September 2022]. We also prove small explicit upper bounds on the twin-width of bipartite planar and 1-planar graphs (6 and 16), and of map graphs (38). The common denominator of all these results is the use of a novel specially crafted recursive decomposition of planar graphs, which may be found useful also in other areas.