Local WL Invariance and Hidden Shades of Regularity

The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a powerful tool for testing isomorphism of two given graphs. We aim at investigating the ability of $k$-WL to capture properties of vertices (or small sets of vertices) in a single input graph $G$. In general, $k$-WL computes a canonical coloring of $k$-tuples of vertices of $G$, which determines a canonical coloring of $s$-tuples for each $s$ between 1 and $k$. We say that a property (or a numerical parameter) of $s$-tuples is $k$-invariant if it is determined by the tuple color. Our main result establishes $k$-invariance of the parameters counting the number of extensions of an $s$-tuple of vertices to a given subgraph pattern $F$. We state a sufficient condition for $k$-invariance in terms of the treewidth of $F$ and its homomorphic images, using suitable variants of these concepts for graphs with $s$ designated roots. As an application, we observe some non-obvious regularity properties of strongly regular graphs: For example, if $G$ is strongly regular, then the number of paths of length 6 between vertices $x$ and $y$ in $G$ depends only on whether or not $x$ and $y$ are adjacent (and the length 6 is here optimal). Despite the fact that $k$-WL indistinguishability of vertex tuples implies high degree of regularity, we prove, on the negative side, that no fixed dimension $k$ suffices for $k$-WL to recognize global symmetry of a graph. Specifically, for every $k$, there is a graph $G$ whose vertex set is colored by $k$-WL uniformly while $G$ is not vertex-transitive.