Characterization of Circular-arc Graphs: II. McConnell Flipping

TL;DR

This paper studies a transformation that converts circular-arc graphs into interval graphs, providing a structural understanding and complete characterization of certain patterns, which aids in identifying minimal non-circular-arc graphs.

Contribution

It offers a structural analysis of McConnell's flipping transformation and characterizes patterns for C4-free graphs, advancing understanding of minimal non-circular-arc graphs.

Findings

Complete characterization of patterns in C4-free graphs.

Identification of all minimal chordal graphs not circular-arc.

Enhanced understanding of the transformation's structural properties.

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.