An Efficient Algorithm to Generate all Labeled Triangle-free Graphs with a given Graphical Degree Sequence

2026-01-22

AI Summary

This work addresses the problem of efficiently generating all labeled triangle-free graphs that realize a given degree sequence. To overcome the inefficiency of conventional generate-and-filter approaches, the authors propose a backtracking-based direct generation algorithm that incorporates tailored pruning strategies to dynamically eliminate partial constructions containing triangles during the graph-building process. The algorithm guarantees that the output graphs are produced in lexicographic order, significantly improves computational efficiency, and naturally supports parallelization. Furthermore, the method extends seamlessly to the generation of bipartite graphs, demonstrating both theoretical rigor and practical scalability.

Technology Category

Machine Learning: Graph-based Machine LearningConstraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologiesWeb Mining and Content Analysis: Web data generation and simulation

Abstract

We extend our previous algorithm that generates all labeled graphs with a given graphical degree sequence to generate all labeled triangle-free graphs with a given graphical degree sequence. The algorithm uses various pruning techniques to avoid having to first generate all labeled realizations of the input sequence and then testing whether each labeled realization is triangle-free. It can be further extended to generate all labeled bipartite graphs with a given graphical degree sequence by adding a simple test whether each generated triangle-free realization is a bipartite graph. All output graphs are generated in the lexicographical ordering as in the original algorithm. The algorithms can also be easily parallelized.

Problem

• triangle-free graphs
• graphical degree sequence
• labeled graphs
• graph generation

Innovation

• triangle-free graphs
• graphical degree sequence
• pruning techniques
• lexicographical ordering
• parallelization