AI 中文总结
本文研究隐式线图WL（ILG-$k$-WL）的表达能力，证明ILG-$3$-WL严格强于$3$-WL，并给出反向包含关系，实验验证其区分能力。
AI 中文摘要
Whitney定理允许将连通简单图的同构测试（除$K_3$和$K_{1,3}$外）转化为区分它们的线图。然而，固定维度的Weisfeiler--Leman（WL）在线图上的表达能力与其根图上的表达能力之间的关系仍未解决。我们通过隐式线图WL（ILG-$k$-WL）研究这一关系，该算法恰好是$L(G)$上的$k$-WL，在$G$的边上执行，利用由端点关联导出的线图关系，且不显式构造$L(G)$。在Whitney一般类上，根域与线图WL之间的关系取决于$k$。对于$k=1,2$，ILG-$k$-WL不增加超越根域$1$-WL的区分能力，并遗漏了$1$-WL能区分的某些对。对于$k=3$，我们证明了反向包含关系$L(G)\equiv_{3\text{-WL}}L(H)\Rightarrow G\equiv_{3\text{-WL}}H$。强正则见证对，包括Shrikhande/rook对，表明ILG-$3$-WL严格强于$3$-WL。反向包含关系也扩展到无孤立顶点的非连通图，当每个连通分量都是Whitney一般时。确定性ILG-$3$-WL能区分所有三个子结构计数见证对、SR25中的全部105对以及BREC的400对中的359对。未训练的密集ILG-$3$-GNN在这些评估上给出相同的成对判定。
英文摘要
Whitney's theorem allows isomorphism testing for connected simple graphs, apart from $K_3$ and $K_{1,3}$, to be formulated as distinguishing their line graphs. However, the relation between fixed-dimensional Weisfeiler--Leman (WL) expressivity on line graphs and on their roots remains unresolved. We study this relation through Implicit Line-Graph WL (ILG-$k$-WL), which is exactly $k$-WL on $L(G)$, executed over the edges of $G$ with line-graph relations derived from endpoint incidence and without explicitly constructing $L(G)$. On the Whitney-general class, the relation between root-domain and line-graph WL depends on $k$. For $k=1,2$, ILG-$k$-WL adds no distinguishing power beyond root-domain $1$-WL and misses some pairs that $1$-WL separates. For $k=3$, we prove the backward containment $L(G)\equiv_{3\text{-WL}}L(H)\Rightarrow G\equiv_{3\text{-WL}}H$. Strongly regular witness pairs, including the Shrikhande/rook pair, show that ILG-$3$-WL is strictly more expressive than $3$-WL. The backward containment also extends to disconnected graphs with no isolated vertices when every connected component is Whitney-general. Deterministic ILG-$3$-WL separates all three substructure-counting witness pairs, all $105$ pairs in SR25, and $359$ of $400$ BREC pairs. An untrained dense ILG-$3$-GNN gives the same pairwise verdicts on these evaluations.
Comments9 pages of main text. 40 pages in total
