Hitting minors on bounded treewidth graphs. I. General upper bounds

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.