article published yes Alvaro Carbonero author Hidde Koerts author Benjamin Moore author 6dc1a1be-bf1c-11ed-8d2b-d044840f49d6 Sophie Spirkl author MaKw department Randomness and structure in combinatorics project 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. https://research-explorer.ista.ac.at/download/18753/19577/2025_EuropJournCombinatorics_Carbonero.pdf application/pdfno Elsevier2025 eng 0195-6698 2306.04710 00140011370000110.1016/j.ejc.2024.104104 125 Carbonero, A., Koerts, H., Moore, B., &#38; Spirkl, S. (2025). On heroes in digraphs with forbidden induced forests. <i>European Journal of Combinatorics</i>. Elsevier. <a href="https://doi.org/10.1016/j.ejc.2024.104104">https://doi.org/10.1016/j.ejc.2024.104104</a> A. Carbonero, H. Koerts, B. Moore, S. Spirkl, European Journal of Combinatorics 125 (2025). Carbonero A, Koerts H, Moore B, Spirkl S. 2025. On heroes in digraphs with forbidden induced forests. European Journal of Combinatorics. 125, 104104. Carbonero, Alvaro, Hidde Koerts, Benjamin Moore, and Sophie Spirkl. “On Heroes in Digraphs with Forbidden Induced Forests.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.ejc.2024.104104">https://doi.org/10.1016/j.ejc.2024.104104</a>. A. Carbonero, H. Koerts, B. Moore, and S. Spirkl, “On heroes in digraphs with forbidden induced forests,” <i>European Journal of Combinatorics</i>, vol. 125. Elsevier, 2025. Carbonero A, Koerts H, Moore B, Spirkl S. On heroes in digraphs with forbidden induced forests. <i>European Journal of Combinatorics</i>. 2025;125. doi:<a href="https://doi.org/10.1016/j.ejc.2024.104104">10.1016/j.ejc.2024.104104</a> Carbonero, Alvaro, et al. “On Heroes in Digraphs with Forbidden Induced Forests.” <i>European Journal of Combinatorics</i>, vol. 125, 104104, Elsevier, 2025, doi:<a href="https://doi.org/10.1016/j.ejc.2024.104104">10.1016/j.ejc.2024.104104</a>. 187532025-01-05T23:01:55Z2025-05-19T14:06:00Z