Core Concepts
The Weisfeiler-Leman dimension of a graph is bounded by 0.15n + o(n).
Abstract
This article analyzes the upper bound on the Weisfeiler-Leman dimension of graphs, focusing on structural complexity measurement. The proof involves various techniques to analyze coherent configurations' structure, including recursive proofs and interspace analysis. The Weisfeiler-Leman dimension determines the difficulty of testing isomorphism using combinatorial algorithms.
- Introduction
- Weisfeiler-Leman (WL) dimension as a measure for graph complexity.
- Core Message
- WL-dimension of a graph is at most 0.15n + o(n).
- Results
- Polynomial time isomorphism algorithm for certain graph classes.
- Techniques
- Reduction in vertices to simplify coherent configurations.
- Critical Configurations
- Criteria for identifying critical configurations.
- Small Fibers and Interspaces
- Classification and analysis of small fiber-induced coherent configurations.
- Large and Small Fiber Interspaces
- Examination of interspaces between large and small fibers.
- Inquiry and Critical Thinking Questions
Stats
証明は、WL次元が最大で3/20・n + o(n) = 0.15・n + o(n)であることを示しています。
グラフの頂点数に関する証拠を使用して、WL次元の上限を決定します。
多項式時間同型性アルゴリズムについて述べられています。