  For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION
problem consists in, given a graph $G$ and an integer $k$, deciding whether
there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$
does not contain any of the graphs in ${\cal F}$ as a minor. We are interested
in the parameterized complexity of ${\cal F}$-M-DELETION when the parameter is
the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a
fixed ${\cal F}$, the smallest function $f_{{\cal F}}$ such that {${\cal
F}$-M-DELETION can be solved in time $f_{{\cal F}}(tw) \cdot n^{O(1)}$ on
$n$-vertex graphs. We prove that $f_{{\cal F}}(tw) = 2^{2^{O(tw \cdot\log
tw)}}$ for every collection ${\cal F}$, that $f_{{\cal F}}(tw) = 2^{O(tw
\cdot\log tw)}$ if ${\cal F}$ contains a planar graph, and that $f_{{\cal
F}}(tw) = 2^{O(tw)}$ if in addition the input graph $G$ is planar or embedded
in a surface. We also consider the version of the problem where the graphs in
${\cal F}$ are forbidden as topological minors, called ${\cal F}$-TM-DELETION.
We prove similar results for this problem, except that in the last two
algorithms, instead of requiring ${\cal F}$ to contain a planar graph, we need
it to contain a subcubic planar graph. This is the first of a series of
articles on this topic.
