  Arc-locally semicomplete and arc-locally in-semicomplete digraphs were
introduced by Bang-Jensen as a common generalization of both semicomplete and
semicomplete bipartite digraphs in 1993. Later, Bang-Jensen (2004),
Galeana-Sanchez and Goldfeder (2009) and Wang and Wang (2009) provided a
characterization of strong arc-locally semicomplete digraphs. In 2009, Wang and
Wang characterized strong arc-locally in-semicomplete digraphs. In 2012,
Galeana-Sanchez and Goldfeder provided a characterization of all arc-locally
semicomplete digraphs which generalizes some results by Bang-Jensen. In this
paper, we characterize the structure of arbitrary connected arc-locally (out)
in-semicomplete digraphs and arbitrary connected arc-locally semicomplete
digraphs.
