Is 3-(F)WL Enough to Distinguish All 3D Graphs?

TL;DR

This work investigates whether higher-order WL tests can uniquely distinguish all 3D graphs by examining the graph-generation process behind WL labelings. It identifies three generation tricks—exchange, turn-over, and symmetry—that can yield multiple graphs from the same label set, and then analyzes their presence under the 3-FWL update. The authors show that 3-FWL eliminates these tricks via a distance-based 3-tuple RGB initialization, common-edge identification, and a tetrahedral-splicing mechanism, arguing that 3-FWL can generate a unique graph and distinguish all 3D graphs. The paper also discusses 3-WL, arguing that its grouping ambiguity complicates trick analysis and proposing an edge-equality framework for future work to determine whether 3-WL can distinguish all 3D graphs, highlighting open questions and potential proofs or counterexamples. Overall, the work advances understanding of WL variants in the geometric graph isomorphism landscape and outlines concrete directions to settle the sufficiency of $3$-WL in 3D settings.

Abstract

The problem of graph isomorphism is an important but challenging problem in the field of graph analysis, for example: analyzing the similarity of two chemical molecules, or studying the expressive ability of graph neural networks. WL test is a method to judge whether two graphs are isomorphic, but it cannot distinguish all non-isomorphic graphs. As an improvement of WL, k-WL has stronger isomorphism discrimination ability, and as k increases, its discrimination ability is strictly increasing. However, whether the isomorphic discrimination power of k-WL is strictly increasing for more complex 3D graphs, or whether there exists k that can discriminate all 3D graphs, remains unexplored. This paper attempts to explore this problem from the perspective of graph generation.

Figures (10)

• Figure 1: Counterexample of 2-WL and WL tree of 2-tuple. Yellow dotted lines indicate equal values.
• Figure 2: Exchange trick.
• Figure 3: Turn-over trick.
• Figure 4: Symmetry trick.
• Figure 5: Examples of new edges (NE) and $v$.