  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.
