Two remarks on circular arc graphs

In our first remark we observe a property of circular arcs which is similar to the Helly property and is helpful in describing all maximal cliques in circular are graphs (as well as allowing us to generalize a result of Tucker). Our main result is a new simple characterization of circular are graphs of clique covering number two. These graphs play a crucial role in recognition algorithms for circular are graphs, and have been characterized by several authors. Specifically, we show that a graph with clique covering number two is a circular are graph if and only if its edges can be coloured by two colours so that no induced four-cycle contains two opposite edges of the same colour. Our proof of the characterization depends on the 'lexicographic method' we have recently introduced. Both remarks could be useful in designing efficient algorithms for (maximum cliques in, respectively recognition of) circular are graphs.