Strict monotonicity and completeness of k-WL on 3D graphs

Determine whether the isomorphism discrimination power of the k-dimensional Weisfeiler–Lehman (k-WL) test is strictly increasing as k increases for 3D graphs, and ascertain whether there exists a finite k such that k-WL distinguishes all non-isomorphic 3D graphs.

Background

The Weisfeiler–Lehman (WL) test is a standard method for graph isomorphism testing, with its k-dimensional variant (k-WL) known to strictly increase in power on general graphs as k grows. This paper focuses on 3D graphs—graphs embedded in three-dimensional space—and studies whether analogous monotonicity and completeness properties hold in this geometric setting.

The authors state that for 3D graphs, it has not been established whether k-WL’s discriminative ability strictly improves with k or whether some k suffices to distinguish all 3D graphs. These questions are central to understanding the limits of WL-type methods for geometric graph isomorphism.