  A (possibly denerate) drawing of a graph $G$ in the plane is approximable by
an embedding if it can be turned into an embedding by an arbitrarily small
perturbation. We show that testing, whether a straight-line drawing of a planar
graph $G$ in the plane is approximable by an embedding, can be carried out in
polynomial time, if a desired embedding of $G$ belongs to a fixed isotopy
class. In other words, we show that c-planarity with embedded pipes is
tractable for graphs with fixed embeddings.
  To the best of our knowledge an analogous result was previously known
essentially only when $G$ is a cycle.
