Proper Helly Circular-Arc Graphs

Abstract

A circular-arc model ${\mathcal {M}} =(C,\mathcal{A})$ is a circle C together with a collection $\mathcal{A}$ of arcs of C. If no arc is contained in any other then $\mathcal{M}$ is a proper circular-arc model, if every arc has the same length then $\mathcal{M}$ is a unit circular-arc model and if $\mathcal{A}$ satisfies the Helly Property then $\mathcal{M}$ is a Helly circular-arc model. A (proper) (unit) (Helly) circular-arc graph is the intersection graph of the arcs of a (proper) (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear time recognition algorithms have been described both for the general class and for some of its subclasses. In this article we study the circular-arc graphs which admit a model which is simultaneously proper and Helly. We describe characterizations for this class, including one by forbidden induced subgraphs. These characterizations lead to linear time certifying algorithms for recognizing such graphs. Furthermore, we extend the results to graphs which admit a model which is simultaneously unit and Helly, also leading to characterizations and a linear time certifying algorithm.