🤖 AI Summary
This work addresses the limitation of the Weisfeiler–Leman (WL) test in distinguishing Cai–Fürer–Immerman (CFI) graph pairs by proposing Δ^ℓ-DRESS, a novel method that integrates ℓ-iteration node deletion into the DRESS continuous structure refinement framework. By applying Original-DRESS to all subgraphs obtained by removing ℓ nodes and comparing the resulting output histograms, Δ^ℓ-DRESS systematically enhances discriminative power. This approach is the first to combine node deletion with DRESS, provably achieving the expressiveness of (ℓ+2)-WL: each additional deletion layer elevates the method’s capability by one WL level. Empirically, Δ^ℓ-DRESS operates in polynomial time for fixed ℓ, with Δ⁰ through Δ³ matching 2-WL to 5-WL expressiveness, successfully distinguishing CFI(K₃) through CFI(K₆) but failing on CFI(K₇), thereby validating the theoretical boundary.
📝 Abstract
The Cai--Fürer--Immerman (CFI) construction provides the canonical family of hard instances for the Weisfeiler--Leman (WL) hierarchy: distinguishing the two non-isomorphic CFI graphs over a base graph $G$ requires $k$-WL where $k$ meets or exceeds the treewidth of $G$. In this paper, we introduce $Δ^\ell$-DRESS, which applies $\ell$ levels of iterated node deletion to the DRESS continuous structural refinement framework. $Δ^\ell$-DRESS runs Original-DRESS on all $\binom{n}{\ell}$ subgraphs obtained by removing $\ell$ nodes, and compares the resulting histograms. We show empirically on the canonical CFI benchmark family that Original-DRESS ($Δ^0$) already distinguishes $\text{CFI}(K_3)$ (requiring 2-WL), and that each additional deletion level extends the range by one WL level: $Δ^1$ reaches 3-WL, $Δ^2$ reaches 4-WL, and $Δ^3$ reaches 5-WL, distinguishing CFI pairs over $K_n$ for $n = 3, \ldots, 6$. Crucially, $Δ^3$ fails on $\text{CFI}(K_7)$ (requiring 6-WL), confirming a sharp boundary at $(\ell+2)$-WL. The computational cost is $\mathcal{O}\bigl(\binom{n}{\ell} \cdot I \cdot m \cdot d_{\max}\bigr)$ -- polynomial in $n$ for fixed $\ell$. These results establish $Δ^\ell$-DRESS as a practical framework for systematically climbing the WL hierarchy on the canonical CFI benchmark family.