On the polynomial time computability of the circular-chromatic number for some superclasses of perfect graphs

A main result in combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grِtschel, Lovلsz and Schrijver 1981). The circular-clique and circular-chromatic number are well-studied refinements of these graph parameters, and circular-perfect graphs form the corresponding superclass of perfect graphs. So far, it is unknown whether the (weighted) circular-clique and circular-chromatic number of a circular-perfect graph are computable in polynomial time. In this paper, we show the polynomial time computability of these two graph parameters for some super-classes of perfect graphs with the help of polyhedral arguments.