The clique operator on circular-arc graphs

A circular-arc graph  GG is the intersection graph of a collection of arcs on the circle and such a collection is called a model   of GG. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graph  K(G)K(G) of a graph GG is the intersection graph of its cliques. The iterated clique graph  Ki(G)Ki(G) of GG is defined by K0(G)=GK0(G)=G and Ki+1(G)=K(Ki(G))Ki+1(G)=K(Ki(G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph KK-converges, if it is KK-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.