Hitting minors on bounded treewidth graphs. III. Lower bounds

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide 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 provide lower bounds under the ETH on $f{{\cal F}}$ for several collections ${\cal F}$. We first prove that for any ${\cal F}$ containing connected graphs of size at least two, $f{{\cal F}}(tw)= 2^{{\Omega(tw)}$,} even if the input graph $G$ is planar. Our main contribution consists of superexponential lower bounds for a number of collections ${\cal F}$, inspired by a reduction of Bonnet et al.~[IPEC, 2017]. In particular, we prove that when ${\cal F}$ contains a single connected graph $H$ that is either $P5$ or is not a minor of the banner (that is, the graph consisting of a $C4$ plus a pendent edge), then $f_{{\cal F}}(tw)= 2^{\Omega(tw \cdot \log tw)}$. This is the third of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of ${H}$-M-DELETION, in terms of $H$, when $H$ is connected.