On the Expressibility of the Reconstructional Color Refinement

TL;DR

This paper demonstrates that the connectedness of a graph can be identified using color refinement equivalence, extending classical results and showing that modern GNNs can recognize connectedness based on this property.

Contribution

It proves that connectedness is detectable from color refinement equivalence classes of vertex-deleted subgraphs, strengthening the classical reconstruction result.

Findings

Connectedness is recognizable via color refinement equivalence.

Reconstruction Graph Neural Networks can determine graph connectedness.

Strengthens the link between graph reconstruction and GNN capabilities.

Abstract

One of the most basic facts related to the famous Ulam reconstruction conjecture is that the connectedness of a graph can be determined by the deck of its vertex-deleted subgraphs, which are considered up to isomorphism. We strengthen this result by proving that connectedness can still be determined when the subgraphs in the deck are given up to equivalence under the color refinement isomorphism test. Consequently, this implies that connectedness is recognizable by Reconstruction Graph Neural Networks, a recently introduced GNN architecture inspired by the reconstruction conjecture (Cotta, Morris, Ribeiro 2021).