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Weisfeiler-Leman on Graphs of Small Twin-Width

Authors Irene Heinrich , Moritz Lichter , Klara Pakhomenko , Simon Raßmann

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • twin-width
  • Weisfeiler-Leman algorithm
  • canonization
  • half-graph
  • rank-width

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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 integer 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 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, Gajarský, Guspiel, Hlinený, Pokrývka, and Sokolowski (ISAAC 2023) that stable graphs of twin-width 2 have bounded rank-width. This implies that isomorphism of these graphs is solved by a fixed dimension of the Weisfeiler-Leman algorithm.

Cite As Get BibTex

Irene Heinrich, Moritz Lichter, Klara Pakhomenko, and Simon Raßmann. Weisfeiler-Leman on Graphs of Small Twin-Width. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.26

Author Details

Irene Heinrich
  • TU Darmstadt, Germany
Moritz Lichter
  • RWTH Aachen University, Germany
Klara Pakhomenko
  • Hasselt University, Diepenbeek, Belgium
Simon Raßmann
  • TU Darmstadt, Germany

Funding

  • Heinrich, Irene: The research of this author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148)
  • Lichter, Moritz: The research of this author has received funding by the European Union (ERC, SymSim, 101054974).
  • Pakhomenko, Klara: This author is supported by the Flanders AI Research Program.

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