Determining Factorial Speed Fast

2026-02-27

AI Summary

This work addresses the long-standing challenge of classifying the growth rates (or "speeds") of numerous graph classes listed on graphclasses.org that previously lacked known asymptotic bounds. By introducing a novel analytical framework based on finite binary language representations and integrating tools from formal language theory with the combinatorial structure of graph classes, the authors establish a general criterion guaranteeing that such classes have at most factorial speed, i.e., $2^{\Theta(n \log n)}$. Applying this framework, they resolve several open cases by proving for the first time that multiple previously unclassified graph classes indeed exhibit factorial growth. Furthermore, they rigorously demonstrate that k-letter graphs have exponential speed ($2^{\Theta(n)}$) and clarify strict containment relations among several graph classes, thereby significantly advancing the theory of graph class speeds.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{\Theta(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{\Theta(n)}$.

Problem

• graph class
• speed
• factorial speed
• exponential speed
• graph enumeration