Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm

TL;DR

This paper provides the first forbidden structure characterization of circular-arc graphs, enabling the development of the first polynomial-time certifying recognition algorithm for this class.

Contribution

It introduces a novel forbidden structure characterization of circular-arc graphs, extending interval graph obstructions, and presents the first certifying recognition algorithm.

Findings

First forbidden structure characterization of circular-arc graphs

Development of a polynomial-time certifying recognition algorithm

Extension of interval graph obstructions to circular-arc graphs

Abstract

A circular-arc graph is the intersection graph of arcs of a circle. It is a well-studied graph model with numerous natural applications. A certifying algorithm is an algorithm that outputs a certificate, along with its answer (be it positive or negative), where the certificate can be used to easily justify the given answer. While the recognition of circular-arc graphs has been known to be polynomial since the 1980s, no polynomial-time certifying recognition algorithm is known to date, despite such algorithms being found for many subclasses of circular-arc graphs. This is largely due to the fact that a forbidden structure characterization of circular-arc graphs is not known, even though the problem has been intensely studied since the seminal work of Klee in the 1960s. In this contribution, we settle this problem. We present the first forbidden structure characterization of circular-arc graphs. Our obstruction has the form of mutually avoiding walks in the graph. It naturally extends a similar obstruction that characterizes interval graphs. As a consequence, we give the first polynomial-time certifying algorithm for the recognition of circular-arc graphs.