  Catch digraphs was introduced by Hiroshi Maehara in 1984 as an analog of
intersection graphs where a family of pointed sets represents a digraph. After
that Prisner continued his research particularly on interval catch digraphs by
characterizing them diasteroidal triple free. It has numerous applications in
the field of real world problems like network technology and telecommunication
operations. Recently, we characterized three important subclasses of interval
catch digraphs. In this article we introduce a new class of catch digraphs,
namely circular-arc catch digraphs. The definition is same as interval catch
digraph, only the intervals are replaced by circular-arcs here. We present the
characterization of proper circular-arc catch digraphs, which is a natural
subclass of circular-arc catch digraphs where no circular-arc is contained in
other properly. For this we introduce a concept, namely monotone circular
ordering for the vertices of the augmented adjacency matrix of it. Next we find
that underlying graph of a proper oriented circular-arc catch digraph is a
proper circular-arc graph. Also we characterize proper oriented circular-arc
catch digraphs by defining a certain kind of circular vertex ordering of its
vertices. Another interesting result is to characterize oriented circular-arc
catch digraphs which are tournaments in terms of forbidden subdigraphs. Further
we study some properties of an oriented circular-arc catch digraph. In
conclusion we discuss the relations between these subclasses of circular-arc
catch digraphs.
