On the Expressibility of the Reconstructional Color Refinement

TL;DR

The paper studies how color refinement information from vertex-deleted subgraphs (the CR-deck) preserves graph structure. It proves that connectedness is determined by the deck-level CR invariants, not by CR of the graph itself alone. The result connects color refinement with graph reconstruction and implies that Reconstruction GNNs operating on decks can infer connectivity, as illustrated by the C6/P5 example. It clarifies relationships among CR, WL, and deck-based invariants and strengthens the case for deck-based refinement in graph neural networks.

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).

Theorems & Definitions (10)

• Theorem 1: Harary Harary64
• Theorem 2
• Corollary 3
• Lemma 4: Nash-Williams Nash-Williams78
• Definition 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• proof : Proof of Lemma \ref{['lem:subgraphs']}