标题： 有向图中禁止诱导森林的英雄 显示英文标题

标题： On heroes in digraphs with forbidden induced forests

摘要： 我们继续研究哪类有向图的遗传族具有有界二色数。对于一类有向图$\mathcal{C}$，在$\mathcal{C}$中的一个英雄是指任何有向图$H$，使得$H$-free 有向图在$\mathcal{C}$中具有有界二色数。我们证明，如果$F$是一个至少五度的定向星，则对于$F$-free 有向图类，唯一的英雄是传递竞赛。 对于度数恰好为四的有向星$F$，我们证明在$F$-自由有向图中唯一的英雄是传递竞赛图，或者可能是传递竞赛图的特殊连接。 Aboulker 等人 完全描述了$\{H, K_{1} + \vec{P_{2}}\}$-自由有向图的英雄集合，我们证明对于$\{H, rK_{1} + \vec{P_{3}}\}$-自由有向图的类也具有相同的描述。 最后，我们证明如果禁止两种“有效”的扫帚定向，则每个传递竞赛图都是此类有向图的英雄。 显示英文摘要

摘要： We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs $\mathcal{C}$, a hero in $\mathcal{C}$ is any digraph $H$ such that $H$-free digraphs 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 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 $\{H, K_{1} + \vec{P_{2}}\}$-free digraphs almost completely, and we show the same characterization for the class of $\{H, rK_{1} + \vec{P_{3}}\}$-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.