The structure of strong arc-locally in-semicomplete digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices $x,y$, every in-neighbor of $x$ and every in-neighbor of $y$ either are adjacent or are the same vertex. In this paper, we study the structure of strong arc-locally in-semicomplete digraphs and prove that a strong arc-locally in-semicomplete digraph is either arc-locally semicomplete or in a special class of digraphs. Using this structural characterization, we show that a 2-strong arc-locally in-semicomplete digraph is arc-locally semicomplete and a conjecture of Bang-Jensen is true.