  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 $h^\star$.
