On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties

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 $k=2$.