Characterization of Circular-arc Graphs: III. Chordal Graphs

TL;DR

This paper fully characterizes minimal chordal graphs that are not circular-arc graphs, solving a long-standing open problem by identifying a simple family of such graphs through structural analysis.

Contribution

It provides a complete characterization of minimal non-circular-arc chordal graphs, resolving a major open problem in the structure of circular-arc graphs.

Findings

All nontrivial minimal chordal graphs not circular-arc belong to a single simple family.

Structural analysis of McConnell's flipping reveals key transformation patterns.

The results clarify the boundary between chordal and circular-arc graph classes.

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.