# Canonisation and Definability for Graphs of Bounded Rank Width

## TL;DR

This paper proves that a specific Weisfeiler-Leman algorithm can completely determine graph isomorphism for graphs with bounded rank width, leading to efficient algorithms for testing isomorphism and canonisation, and shows fixed-point logic with counting captures polynomial time on these classes.

## Contribution

It establishes a complete isomorphism test for graphs of bounded rank width using a fixed dimension of the Weisfeiler-Leman algorithm, and provides the first polynomial-time canonisation method for such graphs.

## Key findings

- Weisfeiler-Leman of dimension (3k+4) is complete for graphs of rank width at most k.
- The isomorphism test runs in time n^{O(k)} for graphs of rank width k.
- Fixed-point logic with counting captures polynomial time on all graphs of bounded rank width.

## Abstract

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.