On the expressibility of the reconstructional color refinement

In this note we explore the color refinement procedure — also known as the 1-dimensional Weisfeiler-Leman procedure, well-studied in connection with the Graph Isomorphism problem — in the context of the Graph Reconstruction conjecture of Ulam. A basic fact about the Ulam reconstruction conjecture is that the connectedness of a graph is determined by the deck of its vertex-deleted subgraphs, which are considered up to isomorphism. We strengthen this result by proving that connectedness of a graph can even be determined from the deck of its vertex-deleted subgraphs given only by their stable colorings (i.e., up to equivalence under color refinement). It follows as a consequence that connectedness is recognizable by Reconstruction Graph Neural Networks, which is a recently introduced GNN architecture inspired by the reconstruction conjecture (Cotta, Morris, Ribeiro 2021).