  In extension problems of partial graph drawings one is given an incomplete
drawing of an input graph $G$ and is asked to complete the drawing while
maintaining certain properties. A prominent area where such problems arise is
that of crossing minimization. For plane drawings and various relaxations of
these, there is a number of tractability as well as lower-bound results
exploring the computational complexity of crossing-sensitive drawing extension
problems. In contrast, comparatively few results are known on extension
problems for the fundamental and broad class of simple drawings, that is,
drawings in which each pair of edges intersects in at most one point. In fact,
only recently it has been shown that the extension problem of simple drawings
is NP-hard even when the task is to insert a single edge.
  In this paper we present tractability results for the crossing-sensitive
extension problem of simple drawings. In particular, we show that the problem
of inserting edges into a simple drawing is fixed-parameter tractable when
parameterized by the number of edges to insert and an upper bound on newly
created crossings. Using the same proof techniques, we are also able to answer
several closely related variants of this problem, among others the extension
problem for $k$-plane drawings. Moreover, using a different approach, we
provide a single-exponential fixed-parameter algorithm for the case in which we
are only trying to insert a single edge into the drawing.
