@article{18753,
  abstract     = {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.},
  author       = {Carbonero, Alvaro and Koerts, Hidde and Moore, Benjamin and Spirkl, Sophie},
  issn         = {0195-6698},
  journal      = {European Journal of Combinatorics},
  publisher    = {Elsevier},
  title        = {{On heroes in digraphs with forbidden induced forests}},
  doi          = {10.1016/j.ejc.2024.104104},
  volume       = {125},
  year         = {2025},
}