Hitting minors, subdivisions, and immersions in tournaments

The Erdős–Pósa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph H (resp. strongly-connected directed graph H), the class of directed graphs that contain H as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erdős–Pósa property in the class of tournaments. We also prove that if H is a strongly-connected directed graph, the class of directed graphs containing H as an immersion has the edge-Erdős–Pósa property in the class of tournaments. Our results are orthogonal to the recent results of Amiri et al. [arXiv:1603.02504, March 2016] in the sense that we restrict the class of “host graphs”, whereas they restrict the class of “guest graphs”.