Some results on structure of arbitrary arc-locally (out) in-semicomplete digraphs

Arc-locally semicomplete digraphs 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-Sánchez and Goldfeder (2009) and Wang and Wang ( 2009) provided a characterization of strong arc-locally semicomplete digraphs.In 2009, Wang and Wang provided a characterization of strong arc-locally in-semicomplete digraphs.In 2012, Galeana-Sánchez and Goldfeder provided a characterization of arbitrary arc-locally semicomplete digraphs which generalizes some results by Bang-Jensen.In this paper, we characterize the structure of arbitrary arc-locally (out) in-semicomplete digraphs and arbitrary arc-locally semicomplete digraphs.