On the twin-width of near-regular graphs

2025-04-03

AI Summary

This work investigates twin-width behavior in regular and near-regular graphs—particularly cubic graphs—to resolve the mystery of the scarcity of bounded-degree graphs with high twin-width. Using combinatorial graph theory, structural analysis, symmetry arguments, and spectral theory of circulant and strongly regular graphs, we establish that all known bounded-degree extremal graphs with high twin-width must be asymmetric, revealing the fundamental obstruction to their explicit construction. We determine the exact twin-width of Johnson graphs $J(n,2)$ and cyclic Latin square graphs. Tight upper bounds are provided for $d$-degenerate graphs and circulant graphs. Moreover, we fully characterize the twin-width of several classical near-regular families, including strongly regular graphs and Paley graphs. Our results show that while twin-width is unbounded on cubic graphs, all currently known explicit constructions have twin-width at most 4.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known. This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and d-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.

Problem

• Investigates twin-width behavior in near-regular graphs
• Explores asymmetry in high twin-width bounded-degree graphs
• Determines twin-width for specific graph families

Innovation

• Explores twin-width in near-regular graphs
• Shows high twin-width graphs are asymmetric
• Determines twin-width of Johnson graphs