Subclasses of Normal Helly Circular-Arc Graphs

TL;DR

This paper explores subclasses of Helly circular-arc graphs, providing characterizations, forbidden subgraph descriptions, and algorithms for recognition, while relating these classes to other graph types and digraphs.

Contribution

It introduces characterizations and recognition algorithms for subclasses of Helly circular-arc graphs, extending properties known from interval graphs and connecting to digraph classes.

Findings

Characterizations of Helly circular-arc subclasses

Forbidden induced subgraph descriptions

Efficient recognition algorithms

Abstract

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.