# A tight Erdős-Pósa function for planar minors

Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f:N→R such that for all k ∈N and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G-X has no H-minor. We prove that this remains true with f(k) = c k k for some constant c=c(H). This bound is best possible, up to the value of c, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with f(k) = c k ^d k for some universal constant d. The proof is constructive and yields a polynomial-time O(OPT)-approximation algorithm for packing subgraphs containing an H-minor.

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