  The most elusive problem around the class of circular-arc graphs is
identifying all minimal graphs that are not in this class. The main obstacle is
the lack of a systematic way of enumerating these minimal graphs. McConnell
[FOCS 2001] presented a transformation from circular-arc graphs to interval
graphs with certain patterns of representations. We fully characterize these
interval patterns for circular-arc graphs that are split graphs, thereby
building a connection between minimal split graphs that are not circular-arc
graphs and minimal non-interval graphs. This connection enables us to identify
all minimal split graphs that are not circular-arc graphs. As a byproduct, we
develop a linear-time certifying recognition algorithm for circular-arc graphs
when the input is a split graph.
