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，其核心是通过逐步收缩序列将给定图缩减至单个顶点，同时保持顶点邻域差异的有限性，可视为对多种传统结构参数的广泛推广。拥有此类序列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，具有有界双宽的图类（且其对应的收缩序列可高效构造）具有重要意义。然而，目前我们通常无法高效地构造低宽度的见证收缩序列，且非平凡情形下已发表的双宽上界往往"天文数字般巨大"。我们聚焦于已知具有有界双宽的平面图（自双宽概念提出之初便已知），但首个显式"非天文级"平面图双宽上界在一年前才出现：即Jacob与Pilipczuk [arXiv, 2022年1月] 给出的183，以及Bonnet、Kwon与Wood [arXiv, 2022年2月] 给出的583。随后2022年的arXiv手稿将上界逐步改进至37（Bekos等）、11和9（均为Hliněný）。我们进一步深化了后者的方法，证明所有平面图的双宽至多为8，并在线性时间内构造出见证收缩序列。值得注意的是，当前最佳下界的平面图实例双宽为7，由Král'与Lamaison [arXiv, 2022年9月] 给出。我们还证明所有二分平面图的双宽至多为6，同样可在线性时间内构造出见证收缩序列。