  For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION
(resp. ${\cal F}$-TM-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 (resp. topological minor). We are interested in the parameterized
complexity of both problems when the parameter is the treewidth of $G$, denoted
by $tw$, and specifically in the cases where ${\cal F}$ contains a single
connected planar graph $H$. We present algorithms running in time $2^{O(tw)}
\cdot n^{O(1)}$, called single-exponential, when $H$ is either $P_3$, $P_4$,
$C_4$, the paw, the chair, and the banner for both $\{H\}$-M-DELETION and
$\{H\}$-TM-DELETION, and when $H=K_{1,i}$, with $i \geq 1$, for
$\{H\}$-TM-DELETION. Some of these algorithms use the rank-based approach
introduced by Bodlaender et al. [Inform Comput, 2015]. This is the second 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 $H$.
