A classification of arc-locally semicomplete digraphs

Tournaments are without doubt the best studied class of directed graphs [3, 6]. The generalizations of tournaments arise in order to extend the well-known results on tournaments to more general classes of directed graphs. Moreover, the knowledge about generalizations of tournaments has allowed to deepen our understanding of tournaments themselves. The semicomplete digraphs, the semicomplete multipartite digraphs, the locally semicomplete digraphs, the quasi-transitive digraphs, the path-mergeable digraphs are some generalizations of tournaments and they have been studied in a large number of papers. Particularly, Jørgen Bang-Jensen introduced the arc-locally semicomplete digraphs as a common generalization of both semicomplete and bipartite semicomplete digraphs and he proved some of their properties in [1]. Later in [2], Bang-Jensen claimed that the only strong arc-locally semicomplete digraphs are the extensions of cycles, the semicomplete digraphs and the bipartite semicomplete digraphs. But one family of strong arc-locally semicomplete digraphs is missing. Let C∗ 3 be the digraph with vertex set {v1, v2, v3} and arc set {v1v2, v2v3, v3v1, v1v3}. It is easy to check that C∗ 3 [E1, En, E1] is a family of strong arc-locally semicomplete digraphs, with the composition of digraphs as defined in [4] and where Ei denotes the independent set of i ≥ 1 vertices. In this paper we look at the structure of arc-locally semicomplete digraphs in order to extend the classification of Bang-Jensen to all arc-locally semicomplete digraphs. We give some characterizations of extensions of paths and cycles. Additionally, we found three special digraphs which play a important rôle in the characterization of arc-locally semicomplete digraphs.