The structure of strong arc-locally semicomplete digraphs

Abstract

Arc-locally semicomplete digraphs were introduced in (Preprint, No. 10, 1993, Department of Mathematics and Computer Science, University of Southern Denmark) as a common generalization of semicomplete digraphs and semicomplete bipartite digraphs. In this note, we prove that the structure of these digraphs is very closely related to that of semicomplete and semicomplete bipartite digraphs. In fact, we show that if a strong arc-locally semicomplete digraph is neither semicomplete nor semicomplete bipartite, then it is obtained from a directed cycle by substituting independent sets of vertices for each vertex of the cycle. We also identify a new class of digraphs for which the hamiltonian cycle problem seems tractable and non-trivial. As it is the case for arc-locally semicomplete digraphs, this new class of digraphs contains all semicomplete digraphs and all semicomplete bipartite digraphs.