  We study the computational complexity of graph planarization via edge
contraction. The problem CONTRACT asks whether there exists a set $S$ of at
most $k$ edges that when contracted produces a planar graph. We work with a
more general problem called $P$-RESTRICTEDCONTRACT in which $S$, in addition,
is required to satisfy a fixed MSOL formula $P(S,G)$. We give an FPT algorithm
in time $O(n^2 f(k))$ which solves $P$-RESTRICTEDCONTRACT, where $P(S,G)$ is
(i) inclusion-closed and (ii) inert contraction-closed (where inert edges are
the edges non-incident to any inclusion minimal solution $S$).
  As a specific example, we can solve the $\ell$-subgraph contractibility
problem in which the edges of a set $S$ are required to form disjoint connected
subgraphs of size at most $\ell$. This problem can be solved in time $O(n^2
f'(k,\ell))$ using the general algorithm. We also show that for $\ell \ge 2$
the problem is NP-complete.
