Weisfeiler-Leman on graphs of small twin-width

TL;DR

This paper investigates the effectiveness of the Weisfeiler-Leman algorithm in solving graph isomorphism problems on graphs with small twin-width, establishing bounds and solving specific cases.

Contribution

It proves that the 3-dimensional Weisfeiler-Leman algorithm can decide isomorphism for graphs of twin-width 1 and confirms a conjecture relating twin-width 2 to bounded rank-width.

Findings

Weisfeiler-Leman solves isomorphism for twin-width 1 graphs.

No fixed k makes Weisfeiler-Leman effective for twin-width 4 graphs.

Stable graphs of twin-width 2 have bounded rank-width.

Abstract

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, Gajarsk\'y, Guspiel, Hlinen\'y, Pokr\'yvka, 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.