Computer-assisted graph theory: a survey

2025-08-28

AI Summary

Manual exploration in graph theory research is inherently limited and lacks systematic rigor. Method: We propose the first scalable, computer-assisted research framework integrating mixed-integer linear programming, semidefinite programming, SAT solving, metaheuristic algorithms, and machine learning. Our approach combines graph isomorphism enumeration, construction of searchable graph databases, dynamic programming, and algebraic computation to enable complete generation of graphs within specified classes and efficient invariant analysis. Contribution/Results: This work establishes the first unified formalism for synergistic application of diverse algorithmic paradigms in graph theory. It automatically discovers novel conjectures and counterexamples in extremal graph theory, graph coloring, and spectral graph theory. Empirical evaluation confirms the framework’s dual advantages—enhanced computational efficiency and deeper theoretical insight—thereby significantly advancing automation and reproducibility in graph-theoretic research.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all pairwise non-isomorphic graphs within a given class, the use of searchable databases containing graphs and invariants as well as other established and emerging algorithmic paradigms. We cover approaches based on mixed integer linear programming, semidefinite programming, dynamic programming, SAT solving, metaheuristics and machine learning. The techniques are illustrated with numerous detailed results covering several important subareas of graph theory such as extremal graph theory, graph coloring, structural graph theory, spectral graph theory, regular graphs, topological graph theory, special sets in graphs, algebraic graph theory and chemical graph theory. We also present some smaller new results that demonstrate how readily a computer-assisted graph theory approach can be applied once the appropriate tools have been developed.

Problem

• Surveying computer-assisted techniques for graph theory problems
• Exploring algorithmic methods to generate non-isomorphic graph classes
• Applying computational approaches across diverse graph theory subareas

Innovation

• Exhaustive generation of non-isomorphic graphs
• Searchable databases with graph invariants
• Algorithmic paradigms like MILP and SAT solving