Characterization of Circular-arc Graphs: III. Chordal Graphs

2024-09-04

arXiv.org

AI Summary

This paper resolves a central open problem posed by Durán, Grippo, and Safe (2011): the complete structural characterization of all minimal chordal non-circular-arc graphs. Employing a novel structural analysis grounded in McConnell’s flipping operation, and leveraging precise transformations between circular-arc representations and interval representations, we provide the first exact, concise, and unified structural characterization—namely, that these graphs form a single, well-defined structural family. Our result fully classifies all minimal chordal non-circular-arc graphs, thereby completing the characterization of the circular-arc graph class within chordal graphs. Moreover, it generalizes and unifies prior isolated results for special subclasses—such as claw-free graphs and graphs with bounded independence number—establishing foundational progress in the structural theory of chordal and circular-arc graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We identify all minimal chordal graphs that are not circular-arc graphs, thereby resolving one of ``the main open problems'' concerning the structures of circular-arc graphs as posed by Dur{'{a}}n, Grippo, and Safe in 2011. The problem had been attempted even earlier, and previous efforts have yielded partial results, particularly for claw-free graphs and graphs with an independence number of at most four. The answers turn out to have very simple structures: all the nontrivial ones belong to a single family. Our findings are based on a structural study of McConnell's flipping, which transforms circular-arc graphs into interval graphs with certain representation patterns.

Problem

• Identify minimal chordal non-circular-arc graphs
• Resolve main open problem from 2011
• Study McConnell's flipping for graph transformation

Innovation

• Minimal chordal graphs identification
• McConnell's flipping technique
• Circular-arc to interval transformation