  We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every
oriented graph in which the out-neighborhood of every vertex induces a
transitive tournament can be partitioned into two acyclic induced subdigraphs.
We prove multiple extensions of this result to larger classes of digraphs
defined by a finite list of forbidden induced subdigraphs. We thereby resolve
several special cases of an extension of the famous Gy\'{a}rf\'{a}s-Sumner
conjecture to directed graphs by Aboulker et al.
