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.

翻译：Cai–Fürer–Immerman（CFI）构造为Weisfeiler–Leman（WL）层次结构提供了经典的困难实例族：区分基于图$G$的两个非同构CFI图需要$k$-WL，其中$k$达到或超过$G$的树宽。本文提出$Δ^\ell$-DRESS方法，该方法将$\ell$层迭代节点删除应用于DRESS连续结构细化框架。$Δ^\ell$-DRESS通过移除$\ell$个节点得到所有$\binom{n}{\ell}$个子图，并在每个子图上运行原始DRESS算法，最终比较生成的直方图。我们在经典CFI基准族上的实验表明，原始DRESS（$Δ^0$）已能区分需要2-WL的$\text{CFI}(K_3)$，且每增加一层删除操作可将识别能力提升一个WL层级：$Δ^1$达到3-WL，$Δ^2$达到4-WL，$Δ^3$达到5-WL，成功区分$n = 3, \ldots, 6$时基于$K_n$的CFI图对。关键的是，$Δ^3$无法区分需要6-WL的$\text{CFI}(K_7)$，这证实了其在$(\ell+2)$-WL处存在明确的能力边界。该方法的计算复杂度为$\mathcal{O}\bigl(\binom{n}{\ell} \cdot I \cdot m \cdot d_{\max}\bigr)$——当$\ell$固定时关于$n$为多项式级别。这些结果确立了$Δ^\ell$-DRESS作为一个可系统提升经典CFI基准族上WL层级识别能力的实用框架。