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Weisfeiler-Leman on graphs of small twin-width

Irene Heinrich, Moritz Lichter, Klara Pakhomenko, Simon Raßmann

TL;DR

This work clarifies how the Weisfeiler-Leman hierarchy interacts with small twin-width, showing that fixed-dimension WL cannot handle twin-width 4 graphs due to CFI-based constructions, while 3-WL suffices for twin-width 1 graphs via canonical 1-contraction sequences and modular decomposition. It further resolves a conjecture by proving that stable graphs of twin-width 2 have bounded rank-width, yielding FP+C-definable canonization and fixed-dimension WL recognizability on this class. The combined approach uses subdivision tricks, canonical contraction sequences, modular decomposition, and well-linkedness to connect local contraction behavior to global width parameters. These results delineate tractable regimes for isomorphism testing and definability in relation to twin-width and suggest avenues for extending similar techniques to related width measures.

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, Gajarský, Guspiel, Hlinený, Pokrývka, 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.

Weisfeiler-Leman on graphs of small twin-width

TL;DR

This work clarifies how the Weisfeiler-Leman hierarchy interacts with small twin-width, showing that fixed-dimension WL cannot handle twin-width 4 graphs due to CFI-based constructions, while 3-WL suffices for twin-width 1 graphs via canonical 1-contraction sequences and modular decomposition. It further resolves a conjecture by proving that stable graphs of twin-width 2 have bounded rank-width, yielding FP+C-definable canonization and fixed-dimension WL recognizability on this class. The combined approach uses subdivision tricks, canonical contraction sequences, modular decomposition, and well-linkedness to connect local contraction behavior to global width parameters. These results delineate tractable regimes for isomorphism testing and definability in relation to twin-width and suggest avenues for extending similar techniques to related width measures.

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 , the -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 such that the -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 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.
Paper Structure (15 sections, 38 theorems, 25 equations, 1 figure)

This paper contains 15 sections, 38 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twin-width 4
  4. Twin-width 1
  5. Prime graphs of twin-width 1
  6. From prime graphs to all graphs
  7. Stable classes of twin-width 2
  8. Cuts induced by red edges in graphs of twin-width 1
  9. Stable graphs of twin-width 2 have bounded rank-width
  10. Conclusion
  11. Proof of Lemma 3.3
  12. Proof of Theorem 3.4
  13. Proof of Lemma 4.4
  14. Definable Canonization for graphs of twin-width 1
  15. Sparse graphs of twin-width 2 have bounded tree-width

Key Result

Lemma 2.1

If $G$ is a graph and $A$, $B$, and $C$ are pairwise disjoint vertex subsets of $G$, then

Figures (1)

  • Figure 5: $D$ has to map middle path vertices to middle path vertices. The bijection chosen by $D$ is indicated by the arrows.

Theorems & Definitions (73)

  • Lemma 2.1: Monotonicity and subadditivity of the rank-connectivity function
  • proof
  • Theorem 2.2: rw_linked_sets
  • Theorem 2.3: well-linked_sets
  • Theorem 2.4: Theorem 5.2 of cfi
  • Lemma 2.5: cfi
  • Lemma 3.1: cfi
  • Lemma 3.2: twwleq4NPharda
  • Lemma 3.3
  • Theorem 3.4
  • ...and 63 more