# Generalizations of $k$-Weisfeiler-Leman stabilization

## TL;DR

This paper explores various generalizations and approximations of graph isomorphism using Weisfeiler-Leman refinement, logic, and polynomial schemes, establishing relationships among these methods.

## Contribution

It introduces a unified framework for analyzing different isomorphism approximations based on refinement operators and SPAS, connecting multiple approaches.

## Key findings

- Established relationships between various Weisfeiler-Leman based approximations.
- Unified framework for refinement operators and isomorphism tests.
- Analyzed parameters of refinement operators in the context of graph isomorphism.

## Abstract

The family of Weisfeiler-Leman equivalences on graphs is a widely studied approximation of graph isomorphism with many different characterizations. We study these, and other approximations of isomorphism defined in terms of refinement operators and Schurian Polynomial Approximation Schemes (SPAS). The general framework of SPAS allows us to study a number of parameters of the refinement operators based on Weisfeiler-Leman refinement, logic with counting, lifts of Weisfeiler-Leman as defined by Evdokimov and Ponomarenko, and the invertible map test introduced by Dawar and Holm, and variations of these, and establish relationships between them.