Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs   The Weisfeiler-Leman procedure is a widely-used technique for graph
isomorphism testing that works by iteratively computing an
isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool
in structural graph theory, which is often exploited in approaches to tackle
the graph isomorphism problem, is the decomposition into 2- and 3-connected
components.
  We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
computes the decomposition of a graph into its 3-connected components. This
implies that the dimension of the algorithm needed to distinguish two given
non-isomorphic graphs is at most the dimension required to distinguish
non-isomorphic 3-connected components of the graphs (assuming dimension at
least 2).
  To obtain our decomposition result, we show that, for k >= 2, the
k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices
that separate the graph, from other vertex k-tuples. As a byproduct, we also
obtain insights about the connectivity of constituent graphs of association
schemes.
  In an application of the results, we show the new upper bound of k on the
Weisfeiler-Leman dimension of the class of graphs of treewidth at most k. Using
a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound
that is asymptotically tight up to a factor of 2.

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The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph