The Weisfeiler-Leman dimension of a graph , sometimes known as the WL dimension, is the smallest integer
such that the -dimensional Weisfeiler-Leman algorithm correctly identifies
(Grohe 2017).

The Weisfeiler-Leman dimension of a graph with vertex count satisfies

(Kiefer 2020, p. 37). The only graph with is the singleton
graph .

If ,
then
is the maximum over all connected components of (Schweitzer 2018).

A graph has WL dimension 1 if it is distinguished from every non-isomorphic graph by color refinement (i.e., application of the 1-dimensional WL algorithm). Such a
graph is said to be "identified" (Schweitzer 2008) or "amenable"
(Arvind et al. 2015).

The graphs of WL-dimension 1 have been completely (and independently) characterized by Arvind et al. (2015) and Kiefer et al. (2015). Unfortunately, a
complete description of the graphs of WL-dimension 2 seems intractable (Li et
al. 2023).

The Weisfeiler-Leman dimension of a tree or forest is 1 (Arvind et al. 2015), of a distance-hereditary
graph is at most 2 (Gavrilyuk et al. 2020), and of a planar
graph is at most 3 (Kiefer et al. 2019, Li et al. 2023).

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Algorithm." In Proc. 7th International Symposium on Theoretical Aspects of
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A. L.; Nedela, R.; and Ponomarenko, I. "The Weisfeiler-Leman Dimension
of Distance-Hereditary Graphs." 24 May 2020. https://arxiv.org/abs/2005.11766.Grohe,
M. Descriptive Complexity, Canonisation and Definable Graph Structure Theory.
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and Limits of the Weisfeiler-Leman Algorithm. M.S. thesis. Aachen, Germany:
RWTH Aachen University, 2020.Kiefer, S. and Neuen, D. "A Study
of Weisfeiler-Leman Colorings on Planar Graphs." 49th International Colloquium
on Automata, Languages, and Programming (ICALP 2022). pp. 81:1-81:20, 2022.Kiefer,
S.; Ponomarenko, I.; and Schweitzer, P. "The Weisfeiler-Leman Dimension of Planar
Graphs Is at Most 3." J. ACM66, 1-31, 2019.Kiefer,
S.; Schweitzer, P.; and Selman, E. "Graphs Identified by Logics With Counting."
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H.; Ponomarenko, I.; and Zeman, P. "On the Weisfeiler-Leman Dimension of Some
Polyhedral Graphs." 26 May 2023. https://arxiv.org/abs/2305.17302.Schweitzer,
P. "The Weisfeiler-Leman Dimension of Graphs and Isomorphism Testing."
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