Generalizations of $k$-dimensional Weisfeiler–Leman stabilization

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, the invertible map test introduced by Dawar and Holm, and variations of these, as well as to establish relationships between them.