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

TL;DR

This paper investigates the invariance properties of the Weisfeiler-Leman algorithm for graph isomorphism, focusing on subgraph counts and patterns for dimensions 1 and 2, providing a complete characterization for 1-WL and extending results for 2-WL.

Contribution

It provides a complete description of WL-invariant subgraph patterns for 1-WL and significantly extends the understanding for 2-WL, advancing graph invariance theory.

Findings

Complete characterization of 1-WL invariant subgraph patterns.

Extended the understanding of 2-WL invariant patterns.

Implications for graph isomorphism and AI applications.

Abstract

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 $k=2$.