Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

For a finite collection of graphs $\mathcal{F}$, the $\mathcal{F}$-TM-Deletion problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in $\mathcal{F}$ as a topological minor. We prove that for every such $\mathcal{F}$, $\mathcal{F}$-TM-Deletion is fixed parameter tractable on planar graphs. Our algorithm runs in a $2^{\mathcal{O}(k^2)}\cdot n^{2}$ time or, alternatively in $2^{\mathcal{O}(k)}\cdot n^{4}$ time. Our techniques can easily be extended to graphs that are embeddable to any fixed surface.