Twin-width is a width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$.

翻译：双宽度是由Bonnet、Kim、Thomassé和Watrigant [FOCS'20, JACM'22]引入的一个宽度参数，具有许多结构和算法应用。我们证明：每个可嵌入欧拉亏格$g$曲面的图，其双宽度为$18\sqrt{47g}+O(1)$，该结果在渐近意义下达到最优，因为其与下界仅相差一个常数乘法因子。我们的证明还给出一个二次时间算法，用于寻找相应的收缩序列。为证明可嵌入曲面图的双宽度上界，我们给出了欧拉亏格$g$图的乘积结构定理的一个加强版本：每个这样的图都是某条路径与某个图的强乘积的子图，该图具有一个树分解，其中除一个例外袋子大小为$\max\{8,32g-27\}$外，其余所有袋子大小至多为8。