  Graph neural networks are prominent models for representation learning over
graphs, where the idea is to iteratively compute representations of nodes of an
input graph through a series of transformations in such a way that the learned
graph function is isomorphism invariant on graphs, which makes the learned
representations graph invariants. On the other hand, it is well-known that
graph invariants learned by these class of models are incomplete: there are
pairs of non-isomorphic graphs which cannot be distinguished by standard graph
neural networks. This is unsurprising given the computational difficulty of
graph isomorphism testing on general graphs, but the situation begs to differ
for special graph classes, for which efficient graph isomorphism testing
algorithms are known, such as planar graphs. The goal of this work is to design
architectures for efficiently learning complete invariants of planar graphs.
Inspired by the classical planar graph isomorphism algorithm of Hopcroft and
Tarjan, we propose PlanE as a framework for planar representation learning.
PlanE includes architectures which can learn complete invariants over planar
graphs while remaining practically scalable. We empirically validate the strong
performance of the resulting model architectures on well-known planar graph
benchmarks, achieving multiple state-of-the-art results.
