On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.
Comment: 15 pages, 1 figure