On heroes in digraphs with forbidden induced forests Carbonero, Alvaro Koerts, Hidde Moore, Benjamin Spirkl, Sophie ddc:510 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. Elsevier 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/18753 https://research-explorer.ista.ac.at/download/18753/19577 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.ejc.2024.104104 info:eu-repo/semantics/altIdentifier/issn/0195-6698 info:eu-repo/semantics/altIdentifier/wos/001400113700001 info:eu-repo/semantics/altIdentifier/arxiv/2306.04710 info:eu-repo/semantics/openAccess