Hitting Topological Minors is FPT

In the Topological Minor Deletion (TM-Deletion) problem input consists of an undirected graph $G$, a family of undirected graphs ${\cal F}$ and an integer $k$. The task is to determine whether $G$ contains a set of vertices $S$ of size at most $k$, such that the graph $G\setminus S$ obtained from $G$ by removing the vertices of $S$, contains no graph from ${\cal F}$ as a topological minor. We give an algorithm for TM-Deletionwith running time $f(h^{\star,k)\cdot} |V(G)|^{4}$. Here $h^\star$ is the maximum size of a graph in ${\cal F}$ and $f$ is a computable function of $h^\star$ and $k$. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on $k$ and $h^\star$.