The intersection between some subclasses of circular-arc and circle graphs

The intersection graph of a family of arcs on a circle is called a circular-arc graph. This class of graphs admits some interesting subclasses: proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and clique-Helly circular-arc graphs. The intersection graph of a family of chords in a circle is called a circle graph. Analogously, this class of graphs admits some subclasses too: proper circular-arc graphs, unit circle graphs, Helly circle graphs and cliqueHelly circle graphs. In this paper, all possible intersections of these subclasses are studied. After eliminating trivially empty regions, twenty six regions remain. Two of them are empty as a consequence of a theorem by Durán and Lin. Twenty three regions are nonempty and we construct a minimal graph in each of them. Our main result is that the twenty-sixth region is empty, namely we prove that if a graph is Helly circle and unit circle, then it is also a Helly circular-arc graph. Finally, we show that all the trees are included in three of these regions and present an efficient algorithm to classify them.