  Circular perfect graphs are those undirected graphs such that the circular
clique number is equal to the circular chromatic number for each induced
subgraph. They form a strict superclass of the perfect graphs, whose index
coding broadcast rates are well known. We present the broadcast rate of index
coding for side-information graphs whose complements are circular perfect,
along with an optimal achievable scheme. We thus enlarge the known classes of
graphs for which the broadcast rate is exactly characterized. In an attempt to
understand the broadcast rate of a graph given that of its complement, we
obtain upper and lower bounds for the product and sum of the vector linear
broadcast rates of a graph and its complement. We show that these bounds are
satisfied with equality even for some perfect graphs. Curating prior results,
we show that there are circular perfect but imperfect graphs which satisfy the
lower bound on the product of the broadcast rate of the complementary graphs
with equality.
