On heroes in digraphs with forbidden induced forests

TL;DR

This paper characterizes heroes in hereditary classes of digraphs with forbidden induced subgraphs, revealing that transitive tournaments are often the only heroes, with some exceptions involving joins and specific forbidden structures.

Contribution

It provides a complete characterization of heroes in classes of digraphs with forbidden induced stars and certain other structures, extending previous work.

Findings

Transitive tournaments are the only heroes for digraphs with forbidden oriented stars of degree at least five.

For degree four, heroes include transitive tournaments and certain joins of them.

Certain forbidden orientations of brooms lead to all transitive tournaments being heroes.

Abstract

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.