Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS 2020]. Very briefly, its essence is 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. Having such a sequence at hand allows us to solve many otherwise hard problems efficiently. Graph classes of bounded twin-width, in which appropriate contraction sequences are efficiently constructible, are thus of interest in combinatorics and in computer science. However, we currently do not know in general how to obtain a witnessing contraction sequence of low width efficiently, and published upper bounds on the twin-width in non-trivial cases are often "astronomically large". We focus on planar graphs, which are known to have bounded twin-width (already since the introduction of twin-width), but the first explicit "non-astronomical" upper bounds on the twin-width of planar graphs appeared just a year ago; namely 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 that the twin-width of every bipartite planar graph is at most 6, and again construct a witnessing contraction sequence in linear time.

翻译：双宽度是由Bonnet、Kim、Thomassé和Watrigant [FOCS 2020] 引入的一种结构宽度参数。简而言之，其本质是通过逐步缩减（即收缩序列）将给定图减少至单一顶点，同时保持顶点邻域差异的有限性，可被视为多种传统结构参数的广泛推广。拥有这样的序列使我们能够高效解决许多原本困难的问题。在组合数学和计算机科学中，具有有界双宽度且能高效构造适当收缩序列的图类备受关注。然而，我们目前普遍缺乏高效构造低宽度可证收缩序列的方法，且已知非平凡情形下的双宽度上界常"大得惊人"。我们聚焦于已知具有有界双宽度的平面图（自双宽度概念提出以来即被认知），但首批显式"非天文数字"的平面图双宽度上界于一年前才出现：Jacob和Pilipczuk [arXiv, 2022年1月] 给出的上界至多为183，Bonnet、Kwon和Wood [arXiv, 2022年2月] 给出的上界至多为583。2022年后续的arXiv手稿将上界改进至37（Bekos等）、11和9（均为Hliněný）。我们进一步优化了这些手稿中的方法，证明每个平面图的双宽度至多为8，并在线性时间内构造出可证收缩序列。需注意，目前最优的下界平面图示例（Kráľ和Lamaison [arXiv, 2022年9月]）的双宽度为7。我们还证明每个二部平面图的双宽度至多为6，同样在线性时间内构造出可证收缩序列。