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 $\{H,K_1+\overrightarrow{P_2}\}$-free digraphs almost completely, and we show the same characterization for the class of $\{H,r{K}_1+\overrightarrow{P_3}\}$-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.