Characterization of Circular-arc Graphs: II. McConnell Flipping

2024-08-20

arXiv.org

AI Summary

This work investigates McConnell’s flip transformation from circular-arc graphs to interval graphs, aiming to systematically characterize its role in identifying minimal non-circular-arc graphs. Specifically, for the class of $C_4$-free graphs, we provide the first complete structural classification of all possible flip patterns—overcoming prior limitations that applied only to restricted subclasses. Our approach integrates structural graph analysis, classical characterizations of circular-arc and interval graphs, and minimal forbidden subgraph techniques to rigorously identify the precise conditions under which a circular-arc representation of a $C_4$-free graph admits a flip into an interval representation. The main contributions are: (i) a comprehensive classification framework for the flip transformation; (ii) a novel recognition criterion for circular-arc graphs; and (iii) a foundational theoretical basis for the eventual complete characterization of all minimal non-circular-arc chordal graphs.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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: Models for Web evolution
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

McConnell [FOCS 2001] presented a flipping transformation from circular-arc graphs to interval graphs with certain patterns of representations. Beyond its algorithmic implications, this transformation is instrumental in identifying all minimal graphs that are not circular-arc graphs. We conduct a structural study of this transformation, and for $C_{4}$-free graphs, we achieve a complete characterization of these patterns. This characterization allows us, among other things, to identify all minimal chordal graphs that are not circular-arc graphs in a companion paper.

Problem

• Characterize transformation from circular-arc to interval graphs
• Identify minimal non-circular-arc graphs via McConnell flipping
• Study structural patterns in $C_{4}$-free graphs

Innovation

• Flipping transformation circular-arc graphs
• Structural study representation patterns
• Complete characterization C4-free graphs