Clique Graphs of Helly Circular-Arc Graphs

Clique graphs of several classes of graphs have been already characterized. Trees, interval graphs, chordal graphs, block graphs, clique-Helly graphs are some of them. However, no characterization of clique graphs of circular-arc graphs and some of their subclasses is known. In this paper, we present a characterization theorem of clique graphs of Helly circular-arc graphs and prove that this subclass of circular-arc graphs is properly contained in the intersection between proper circular-arc graphs, clique-Helly circular-arc graphs and Helly circular-arc graphs. Furthermore, we prove properties about the \(2^{\text{nd}}\) iterated clique graph of this family of graphs.