  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. In order to control the depth of our circuit, 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). This 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 $\textsf{NC}$.
