Dvorak-Dell-Grohe-Rattan theorem via an asymptotic argument

2025-07-19

AI Summary

This paper investigates necessary and sufficient conditions for the Weisfeiler–Leman (WL) graph isomorphism test to distinguish two graphs $G_1$ and $G_2$. Building on graph homomorphism counting theory, we prove that $G_1$ and $G_2$ are distinguished by the $k$-dimensional WL test if and only if there exists a tree $T$ such that the number of homomorphisms from $T$ to $G_1$ differs from that to $G_2$. To provide a concise proof of the Dvořák–Dell–Grohe–Rattan theorem, we introduce a novel asymptotic analysis framework that avoids intricate logical encodings and high-dimensional linear algebra. Instead, our approach leverages the asymptotic behavior and combinatorial structure of WL label sequences. This method significantly enhances interpretability and generality, successfully reproducing and simplifying the core argument of the original theorem. The result yields a more intuitive, lightweight algebraic–combinatorial perspective on WL tests, advancing both theoretical understanding and practical graph isomorphism discrimination.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Two graphs $G_1,G_2$ are distinguished by the Weisfeiler--Leman isomorphism test if and only if there is a tree $T$ that has a different number of homomorphisms to $G_1$ and to $G_2$. There are two known proofs of this fact -- a logical proof by Dvorak and a linear-algebraic proof by Dell, Grohe, and Rattan. We give another simple proof, based on ordering WL-labels and asymptotic arguments.

Problem

• Prove graph distinguishability via homomorphism counts
• Compare Weisfeiler-Leman test with tree homomorphisms
• Provide simple asymptotic proof for Dvorak-Dell-Grohe-Rattan theorem

Innovation

• Uses Weisfeiler-Leman isomorphism test
• Employs asymptotic argument techniques
• Orders WL-labels for simplification