Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, rooted, replacement, and zig-zag products. For the lexicographical product we prove that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs. In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

翻译：双曲线是Bonnet、Kim、Thomas\'{{e} & Waterrigant最近引入的图形宽度参数。 根据两张G$和$H$的图形和一个图形产品$Star$,我们处理的问题是:G$star H$的双曲线是否受双曲线(G$和$H$及其最大度的函数)的约束?已知这种类型的捆绑为强型产品(Bonnet、Geniet、Kim、Thomas\'{e} & Waterrigant;SODA 2021)所持有。我们提供的例子显示,我们许多条框绑定的卡泰斯、高压/直接、扎根、替换和zig-zag产品。对于词汇产品,我们证明,两张图的双曲线的双曲线正好是单张双曲线的最大界限。相比之下,我们所展示的模块产品没有约束。此外,我们提供的例子显示,我们许多条框定级的界限是紧的,并提供了改进的图表。