  The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a fruitful
approach to the Graph Isomorphism problem. 2-WL corresponds to the original
algorithm suggested by Weisfeiler and Leman over 50 years ago. 1-WL is the
classical color refinement routine. Indistinguishability by $k$-WL is an
equivalence relation on graphs that is of fundamental importance for
isomorphism testing, descriptive complexity theory, and graph similarity
testing which is also of some relevance in artificial intelligence. Focusing on
dimensions $k=1,2$, we investigate subgraph patterns whose counts are $k$-WL
invariant, and whose occurrence is $k$-WL invariant. We achieve a complete
description of all such patterns for dimension $k=1$ and considerably extend
the previous results known for $k=2$.
