ON CHORDAL PROPER CIRCULAR-ARC GRAPHS

Wegner proved that a chordal graph is a proper interval graph if and only if it is claw-free, net-free and tent-free. We observe that in fact the characterization can be refined to say that a chordal graph is a proper interval graph if and only if it is claw-free, net-free and not a multiple of the tent. This observation implies that a chordal graph is a proper circular arc graph if and only if it is claw-free and net-free. A further implication of this result is that a chordal graph can be oriented as a local tournament - an oriented graph in which the predecessors as well as the successors of any vertex induce a tournament - if and only if it is claw-free and net-free.