  A graph is near-planar if it can be obtained from a planar graph by adding an
edge. We show the surprising fact that it is NP-hard to compute the crossing
number of near-planar graphs. A graph is 1-planar if it has a drawing where
every edge is crossed by at most one other edge. We show that it is NP-hard to
decide whether a given near-planar graph is 1-planar. The main idea in both
reductions is to consider the problem of simultaneously drawing two planar
graphs inside a disk, with some of its vertices fixed at the boundary of the
disk. This leads to the concept of anchored embedding, which is of independent
interest. As an interesting consequence we obtain a new, geometric proof of
NP-completeness of the crossing number problem, even when restricted to cubic
graphs. This resolves a question of Hlin\v{e}n\'y.
