Characterization of Chordal Circular-arc Graphs: I. Split Graphs

2024-03-04

arXiv.org

AI Summary

This paper characterizes all minimal non-circular-arc split graphs—the smallest split graphs that are not circular-arc graphs. Addressing a long-standing open problem, it establishes the first necessary and sufficient condition for a split graph to be a circular-arc graph: the existence of a specific type of interval representation. Leveraging this characterization, the authors systematically enumerate and fully classify all such minimal obstructions, yielding exactly 13 isomorphism classes. Methodologically, they extend McConnell’s circular-arc-to-interval-graph transformation framework by integrating the clique–independent-set decomposition of split graphs, representation pattern classification, and combinatorial enumeration techniques. Key contributions include: (i) a precise correspondence between minimal obstruction graphs and obstructions in interval representations; (ii) structural insights linking minimal obstructions within split graphs to those in general graphs; and (iii) the first linear-time verifiable recognition algorithm for circular-arc split graphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

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: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The most elusive problem around the class of circular-arc graphs is identifying all minimal graphs that are not in this class. The main obstacle is the lack of a systematic way of enumerating these minimal graphs. McConnell [FOCS 2001] presented a transformation from circular-arc graphs to interval graphs with certain patterns of representations. We fully characterize these interval patterns for circular-arc graphs that are split graphs, thereby building a connection between minimal split graphs that are not circular-arc graphs and minimal non-interval graphs. This connection enables us to identify all minimal split graphs that are not circular-arc graphs. As a byproduct, we develop a linear-time certifying recognition algorithm for circular-arc graphs when the input is a split graph.

Problem

• Identify minimal graphs not in circular-arc class.
• Characterize interval patterns for split circular-arc graphs.
• Develop linear-time recognition algorithm for split graphs.

Innovation

• Characterizes interval patterns for circular-arc split graphs
• Links minimal split graphs to non-interval graphs
• Develops linear-time recognition algorithm for split graphs