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

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).