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：孪生宽度是最近由Bonnet、Kim、Thomassé和Watrigant引入的一种图宽度参数。给定两个图$G$和$H$以及一种图乘积$\star$，我们研究以下问题：$G\star H$的孪生宽度是否由$G$和$H$的孪生宽度及其最大度数的函数所界定？已知这类界对于强乘积成立（Bonnet、Geniet、Kim、Thomassé和Watrigant；SODA 2021）。我们证明相同形式的界对于笛卡尔积、张量/直积、冠积、根积、替换积和锯齿积同样成立。对于字典序乘积，已知两个图的乘积的孪生宽度恰好等于各图孪生宽度的最大值（Bonnet、Kim、Reinald、Thomassé和Watrigant；IPEC 2021）。相比之下，对于模乘积，我们证明不存在任何界。此外，我们提供例子表明许多界是紧的，并针对某些图类给出改进的界。