Twin-width is a graph parameter introduced in the context of first-order model checking, and has since become a central parameter in algorithmic graph theory. While many algorithmic problems become easier on arbitrary classes of bounded twin-width, graph isomorphism on graphs of twin-width 4 and above is as hard as the general isomorphism problem. For each positive number $k$, the $k$-dimensional Weisfeiler-Leman algorithm is an iterative color refinement algorithm that encodes structural similarities and serves as a fundamental tool for distinguishing non-isomorphic graphs. We show that the graph isomorphism problem for graphs of twin-width 1 can be solved by the purely combinatorial 3-dimensional Weisfeiler-Leman algorithm, while there is no fixed $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm solves the graph isomorphism problem for graphs of twin-width 4. Moreover, we prove the conjecture of Bergougnoux, Gajarský, Guspiel, Hlinený, Pokrývka, and Sokolowski that stable graphs of twin-width 2 have bounded rank-width. This in particular implies that isomorphism of these graphs can be decided by a fixed dimension of the Weisfeiler-Leman algorithm.

翻译：孪生宽度是在一阶模型检验背景下引入的图参数，现已成为算法图论中的核心参数。尽管在任意有界孪生宽度图类上许多算法问题会变得更容易，但在孪生宽度为4及以上的图上，图同构问题与一般同构问题同样困难。对于每个正整数$k$，$k$维Weisfeiler-Leman算法是一种迭代颜色细化算法，它能编码结构相似性，并作为区分非同构图的基本工具。我们证明：孪生宽度为1的图的同构问题可以通过纯组合的3维Weisfeiler-Leman算法求解，而对于孪生宽度为4的图，不存在固定的$k$使得$k$维Weisfeiler-Leman算法能解决其同构问题。此外，我们证明了Bergougnoux、Gajarský、Guspiel、Hlinený、Pokrývka和Sokolowski的猜想：孪生宽度为2的稳定图具有有界秩宽度。这特别意味着这些图的同构性可以通过固定维数的Weisfeiler-Leman算法判定。