Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler--Leman

TL;DR

This paper demonstrates that canonical labeling and isomorphism testing for graphs with bounded rank-width can be efficiently computed in parallel complexity classes, extending previous results for treewidth.

Contribution

It introduces a parallel algorithm placing graph isomorphism for bounded rank-width in extsf{TC}^{2}, utilizing Weisfeiler--Leman and rank decompositions, advancing the understanding of graph isomorphism complexity.

Findings

Canonical labelings are in extsf{TC}^{2}.

Graphs of bounded rank-width are identified by extsf{FO} + extsf{C} formulas.

Isomorphism testing for these graphs is now in extsf{NC}.

Abstract

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 $\textsf{NC}$.