发表机构
College of Engineering and Technology; American University of the Middle East（工程学院与技术学院; 中东美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过$\text{F}_2$上的秩分解关联图结构与WL算法维数，得出有界秩宽图等的WL区分维数，且上界经CFI构造验证在常数因子内是紧的。
AI 中文摘要
Weisfeiler-Leman（WL）算法通过为顶点的$d$元组分配颜色，并根据邻域配置迭代细化着色直至稳定，来测试图同构。增大$d$可使该测试检测到图之间更细微的结构差异，但计算成本随$d$呈指数增长。核心问题是，哪一维数能够区分非同构图。我们通过在$\boldsymbol{\text{F}_2}$上的秩分解，将该充分维数与图$G$的结构关联起来。决定该界的两个参数是最大割秩$w$和叉负载$\boldsymbol{\textit{l}}$，叉负载是分解树中一个叉处父节点与两个子节点割秩的最大和。我们证明，维数为$\boldsymbol{\text{max}\{\textit{l}+1,2w+2\}}$的WL算法，可将每个具有此类分解的有限非空顶点着色图$G$与每个非同构于$G$的图区分开来。因此，对于$k\boldsymbol{\text{≥}1}$，秩宽至多为$k$的图可在维数$3k+1$下与所有非同构图区分，线性秩宽至多为$k$的图可在维数$2k+2$下区分，无边图可在维数1下区分。未着色的Cai-Fürer-Immerman（CFI）构造为两个宽度参数提供了线性下界，因此上述上界在常数因子范围内是紧的。
英文摘要
The Weisfeiler-Leman (WL) algorithm tests graph isomorphism by assigning colours to $d$-tuples of vertices and iteratively refining the colouring according to neighbourhood configurations until it stabilizes. Increasing $d$ allows the test to detect finer structural differences between graphs, but the computational cost grows exponentially with $d$. The central question is which dimension allows us to distinguish nonisomorphic graphs. We relate a sufficient such dimension to the structure of a graph $G$ through a rank decomposition over $\mathbb F_2$. The two parameters governing the bound are the maximum cut rank $w$ and the fork load $\ell$, which is the largest sum of the parent and two child cut ranks at a fork in the decomposition tree. We prove that WL in dimension $\max\{\ell+1,2w+2\}$ distinguishes every finite nonempty vertex-coloured graph $G$ with such a decomposition from every graph not isomorphic to $G$. Consequently, for $k\geq1$, graphs of rank width at most $k$ can be distinguished from every nonisomorphic graph in dimension $3k+1$, graphs of linear rank width at most $k$ in dimension $2k+2$, and edgeless graphs in dimension one. Uncoloured Cai-Fürer-Immerman (CFI) constructions give linear lower bounds for both width parameters, so the upper bounds are tight up to constant factors.
Comments44 pages. Includes ancillary verification code
