On heroes in digraphs with forbidden induced forests

We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs C, a hero in C is any digraph H such that H -free digraphs in C have bounded dichromatic number. We show that if F is an oriented star of degree at least five, the only heroes for the class of F -free digraphs are transitive tournaments. For oriented stars F of degree exactly four, we show the only heroes in F -free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of {H,K1+P2→} -free digraphs almost completely, and we show the same characterization for the class of {H,rK1+P3→} -free digraphs. Lastly, we show that if we forbid two "valid" orientations of brooms, then every transitive tournament is a hero for this class of digraphs.