DRESS and the WL Hierarchy: Climbing One Deletion at a Time

TL;DR

This paper provides a theoretical foundation for DRESS and its extension, demonstrating their ability to distinguish complex graph structures and relate to the Weisfeiler-Lehman hierarchy.

Contribution

It offers the first unconditional proof that $ riangle^k$-DRESS distinguishes all CFI pairs for all k, and a conditional proof linking $ riangle^k$-DRESS to the WL hierarchy.

Findings

$ riangle^k$-DRESS distinguishes all CFI$(K_{k+3})$ pairs for all $k \\geq 0$

$ riangle^k$-DRESS is at least as powerful as $(k+2)$-WL under a conjecture

Theoretical justification for the expressive power of DRESS extensions

Abstract

DRESS is a deterministic, parameter-free framework that iteratively refines the structural similarity of edges in a graph to produce a canonical fingerprint: a real-valued edge vector, obtained by converging a non-linear dynamical system to its unique fixed point. $\Delta^k$-DRESS extends the framework by running DRESS on every $k$-vertex-deleted subgraph of $G$; it was introduced and empirically evaluated in the companion paper, where the CFI staircase showed that $\Delta^k$-DRESS matches $(k{+}2)$-WL for $k = 0, 1, 2, 3$. This paper provides the theoretical justification. The main contributions are: (i) an unconditional proof that $\Delta^k$-DRESS distinguishes every CFI$(K_{k+3})$ pair for all $k \geq 0$ (CFI Staircase Theorem), established via a new CFI Deck Separation theorem and the Virtual Pebble Lemma; and (ii) a conditional proof that $\Delta^k$-DRESS $\geq$ $(k{+}2)$-WL for all graphs and all $k \geq 0$, assuming a single structural conjecture about the WL hierarchy (WL-Deck Separation).