The Weisfeiler-Lehman Method and Graph Isomorphism Testing

TL;DR

This paper analyzes the properties of certain graph families related to the Weisfeiler-Lehman method and introduces an extension that effectively characterizes counterexample graphs, under verified assumptions.

Contribution

It presents an extension to the recursive k-dimensional Weisfeiler-Lehman method that efficiently characterizes all known counterexample graphs.

Findings

The extension accurately characterizes all known counterexample graphs.

Assumptions for the method hold in all known cases.

Provides insights into graph isomorphism testing techniques.

Abstract

Properties of the `$k$-equivalent' graph families constructed in Cai, F\"{u}rer and Immerman, and Evdokimov and Ponomarenko are analysed relative the the recursive $k$-dim WL method. An extension to the recursive $k$-dim WL method is presented that is shown to efficiently characterise all such types of `counterexample' graphs, under certain assumptions. These assumptions are shown to hold in all known cases.