  The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a
planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool
for designing heuristics that minimize edge crossings in drawings of general
graphs. Unfortunately, some graphs do not have a geometric embedding $\Gamma$
such that $\Gamma+e$ has the same number of crossings as the embedding $G+e$.
This motivates the study of the computational complexity of the following
problem: Given a combinatorially embedded graph $G$, compute a geometric
embedding $\Gamma$ that has the same combinatorial embedding as $G$ and that
minimizes the crossings of $\Gamma+e$. We give polynomial-time algorithms for
special cases and prove that the general problem is fixed-parameter tractable
in the number of crossings. Moreover, we show how to approximate the number of
crossings by a factor $(\Delta-2)$, where $\Delta$ is the maximum vertex degree
of $G$.
