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, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). 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：Bonnet、Kim、Thomas\'{e} & Waterrigant最近引入了一个图形宽度参数。 根据两张G$和$H$的图形和一张图表产品 $star$,我们讨论的问题是:G$star H$的双曲线是否受双曲线G$和$H$的函数及其最大度的约束?已知这种类型的捆绑为强型产品(Bonnet、Geniet、Kim、Thomass\'e} & Waterrigant; SOD 2021) 。我们展示了对Cartesian、高压/直接、corona、扎根、替换和zig-zag产品的同一表格的界限。对于这两个图表产品,已知两个图表的双曲线正好是单个图表(Bonet、Kim、Reinald、Tomas\'e} & Waterrigant; SODO; IPEC 2021) 的双边框框框框框框框框框框框框框框框框框框框框框框框框框的模型显示我们一定的模型中的某些模型。