Completeness of 3-WL for distinguishing all 3D graphs

Establish whether the 3-dimensional Weisfeiler–Lehman (3-WL) test distinguishes all non-isomorphic 3D graphs by either proving completeness or providing a counterexample of two non-isomorphic 3D graphs that receive identical labels under 3-WL.

Background

While the authors argue that 3-FWL can generate a unique 3D graph and hence distinguish all 3D graphs, they do not have a comparable proof for 3-WL. The paper proposes an “edge equality analysis” framework to paper 3-WL but reports no definitive resolution.

The explicit lack of a proof or counterexample leaves open whether 3-WL is complete for 3D graph isomorphism, a central question for the practical and theoretical utility of WL-based methods in geometric settings.