Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Authors Frank Fuhlbrück, Johannes Köbler, Oleg Verbitsky

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Author Details

Frank Fuhlbrück
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Johannes Köbler
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Oleg Verbitsky
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany


We thank Ilia Ponomarenko and the anonymous referees for their numerous detailed comments and Daniel Neuen and Pascal Schweitzer for a useful discussion of multipede graphs. The third author is especially grateful to Ilia Ponomarenko for his patient and insightful guidance through the theory of coherent configurations.

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Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Design and analysis of algorithms
  • Graph Isomorphism
  • Weisfeiler-Leman Algorithm
  • Cai-Fürer-Immerman Graphs
  • coherent Configurations


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