In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in $\textsf{TC}^{2}$. Our approach builds on the framework of K\"obler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. Nonetheless, these separators may not in general split the graph into connected components of sufficiently small size. This presents an obstacle for controlling the depth of our circuit. In order to overcome this obstacle, we leverage the fact that any graph of rank-width $k$ admits a rank decomposition of width $\leq 2k$ and height $O(\log n)$ (Courcelle & Kant\'e, WG 2007), which allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in $\textsf{TC}^{1}$. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the $(6k+3)$-dimensional Weisfeiler--Leman (WL) algorithm can identify graphs of rank-width $k$ using only $O(\log n)$ rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by $\textsf{FO} + \textsf{C}$ formulas with $6k+4$ variables and quantifier depth $O(\log n)$. Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in $\textsf{NC}$.

翻译：本文证明有界秩宽图的规范标号计算属于$\textsf{TC}^{2}$类。我们的方法建立在Köbler与Verbitsky（CSR 2008）的研究框架之上，该框架已建立了有界树宽图的类似结论。此处我们采用Grohe与Neuen（ACM Trans. Comput. Log., 2023）的框架，通过拆分对与翻转函数枚举分隔符。然而，这些分隔符通常无法将图分割成足够小规模的连通分量，这给电路深度的控制带来障碍。为克服此障碍，我们利用如下事实：任意秩宽为$k$的图均存在宽度$\leq 2k$且高度为$O(\log n)$的秩分解（Courcelle & Kanté, WG 2007），从而得以借鉴Wagner（CSR 2011）的思路追踪递归计算深度。此外，在将图分割为连通分量后，需要判断这些分量在$\textsf{TC}^{1}$中是否同构。为此，我们扩展了Grohe与Neuen（同上）的工作，证明$(6k+3)$维Weisfeiler–Leman（WL）算法仅需$O(\log n)$轮即可识别秩宽为$k$的图。由此可得，有界秩宽图可由包含$6k+4$个变量且量词深度为$O(\log n)$的$\textsf{FO} + \textsf{C}$公式唯一标识。在本研究之前，有界秩宽图同构测试是否属于$\textsf{NC}$类尚属未知。