f such that every graph either contains k disjoint subgraphs each

isomorphic to a member in F or contains at most f(k) vertices

intersecting all such subgraphs. In this talk I will address the

Erdos-Posa property with respect to three closely related graph

containment relations: minor, topological minor, and immersion. We

denote the set of graphs containing H as a minor, topological minor and

immersion by M(H),T(H) and I(H), respectively.

Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa

property if and only if H is planar. And they left the question for

characterizing H in which T(H) has the Erdos-Posa property in the same

paper. This characterization is expected to be complicated as T(H) has

no Erdos-Posa property even for some tree H. In this talk, I will

present joint work with Postle and Wollan for providing such a

characterization. For immersions, it is more reasonable to consider an

edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs

and covering them by edges. I(H) has no this edge-variant of the

Erdos-Posa property even for some tree H. However, I will prove that

I(H) has the edge-variant of the Erdos-Posa property for every graph H

if the host graphs are restricted to be 4-edge-connected. The

4-edge-connectivity cannot be replaced by the 3-edge-connectivity.