Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and locally the out-neighbourhood in-neighbourhood of vertex induce a digraph. In this paper we study various subclasses digraphs for which give structural decomposition theorems. As consequence obtain several applications, among an answer to conjecture Naserasr first third authors (Aboulker et al., 2021): oriented graph such that every induces transitive tournament, then can partition its set into acyclic digraphs.