Breaking Hard Isomorphism Benchmarks with DRESS

TL;DR

DRESS is a novel, deterministic graph fingerprinting framework that effectively distinguishes complex graphs, including strongly regular graphs, by iteratively refining edge similarities to produce unique, robust signatures.

Contribution

This paper introduces DRESS and its variant Δ-DRESS, a parameter-free, iterative graph refinement method that achieves high accuracy in distinguishing hard graph instances, surpassing existing theoretical limits.

Findings

Δ-DRESS produces unique fingerprints for 33 of 34 benchmark families.

It resolves all but one within-family collision among over 576 million pairs.

Δ-DRESS successfully separates the Rook L2(4)/Shrikhande pair, surpassing 3-WL boundary.

Abstract

DRESS is a deterministic, parameter-free framework for structural graph refinement 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 nonlinear dynamical system to its unique fixed point. $\Delta$-DRESS is a member of the DRESS family of graph fingerprints that applies a single level of vertex deletion. We test it on a benchmark of 51,813 distinct graphs across 34 hard families, including the complete Spence collection of strongly regular graphs (43,703 SRGs, 12 families), four additional SRG families (8,015 graphs), and 18 classical hard constructions (102 family entries corresponding to 99 distinct graphs). $\Delta$-DRESS produces unique fingerprints in 33 of 34 benchmark families at $k=1$, resolving all but one within-family collision among over 576 million non-isomorphic pairs. One genuine collision exists at deletion depth $k=1$, between two vertex-transitive SRGs in SRG(40,12,2,4), which is resolved by a single-step fallback to $\Delta^2$-DRESS. For every family with pairwise-comparable full sorted-multiset fingerprints, the minimum observed separation margin remains at least $137 \times \epsilon$, confirming that the reported separations are numerically robust and not artifacts of the convergence threshold. We also show that $\Delta$-DRESS separates the Rook $L_2(4)$/Shrikhande pair, proving it escapes the theoretical boundary of 3-WL. The method runs in $\mathcal{O}(n \cdot I \cdot m \cdot d_{\max})$ time per graph.