  The {\sc Weighted} $\mathcal{F}$-\textsc{Vertex Deletion} for a class ${\cal
F}$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$
such that $G-S\in{\cal F}.$ The case when ${\cal F}$ is minor-closed and
excludes some graph as a minor has received particular attention but a
constant-factor approximation remained elusive for \textsc{Weighted}
$\mathcal{F}$-{\sc Vertex Deletion}. Only three cases of minor-closed ${\cal
F}$ are known to admit constant-factor approximations, namely \textsc{Vertex
Cover}, \textsc{Feedback Vertex Set} and \textsc{Diamond Hitting Set}. We study
the problem for the class ${\cal F}$ of $\theta_c$-minor-free graphs, under the
equivalent setting of the \textsc{Weighted $c$-Bond Cover} problem, and present
a constant-factor approximation algorithm using the primal-dual method. For
this, we leverage a structure theorem implicit in [Joret et al., SIDMA'14]
which states the following: any graph $G$ containing a $\theta_c$-minor-model
either contains a large two-terminal {\sl protrusion}, or contains a
constant-size $\theta_c$-minor-model, or a collection of pairwise disjoint {\sl
constant-sized} connected sets that can be contracted simultaneously to yield a
dense graph. In the first case, we tame the graph by replacing the protrusion
with a special-purpose weighted gadget. For the second and third case, we
provide a weighting scheme which guarantees a local approximation ratio.
Besides making an important step in the quest of (dis)proving a constant-factor
approximation for \textsc{Weighted} $\mathcal{F}$-\textsc{Vertex Deletion}, our
result may be useful as a template for algorithms for other minor-closed
families.
