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We continue a line of research which studies which hereditary families ofdigraphs have bounded dichromatic number. For a class of digraphs$\mathcal{C}$, a hero in $\mathcal{C}$ is any digraph $H$ such that $H$-freedigraphs in $\mathcal{C}$ 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 degreeexactly four, we show the only heroes in $F$-free digraphs are transitivetournaments, or possibly special joins of transitive tournaments. Aboulker etal. characterized the set of heroes of $\{H, K_{1} + \vec{P_{2}}\}$-freedigraphs almost completely, and we show the same characterization for the classof $\{H, rK_{1} + \vec{P_{3}}\}$-free digraphs. Lastly, we show that if weforbid two "valid" orientations of brooms, then every transitive tournament isa hero for this class of digraphs.

On heroes in digraphs with forbidden induced forests