Erd\"os-P\'osa property of minor-models with prescribed vertex sets

A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\mathrm{\ell }$ and a non-empty planar graph $H$ with at least $\mathrm{\ell }-1$ connected components, there exists a function $f_{H,\mathrm{\ell }}:\mathbb{N}\to \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1,\dots ,Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\mathrm{\ell }$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H,\mathrm{\ell }}(k)$ that meets all such minor-models of $H$. This function $f_{H,\mathrm{\ell }}$ is independent with the number $m$ of given sets, and thus, our result generalizes Mader’s $\mathcal{S}$-path Theorem, by applying $\mathrm{\ell }=2$ and $H$ to be the one-vertex graph. We prove that such a function $f_{H,\mathrm{\ell }}$ does not exist if $H$ consists of at most $\mathrm{\ell }-2$ connected components.