  We discuss the problem of embedding graphs in the plane with restrictions on
the vertex mapping. In particular, we introduce a technique for drawing planar
graphs with a fixed vertex mapping that bounds the number of times edges bend.
An immediate consequence of this technique is that any planar graph can be
drawn with a fixed vertex mapping so that edges map to piecewise linear curves
with at most $3n + O(1)$ bends each. By considering uniformly random planar
graphs, we show that $2n + O(1)$ bends per edge is sufficient on average.
  To further utilize our technique, we consider simultaneous embeddings of $k$
uniformly random planar graphs with vertices mapping to a fixed, common point
set. We explain how to achieve such a drawing so that edges map to piecewise
linear curves with $O(n^{1-1/k})$ bends each, which holds with overwhelming
probability. This result improves upon the previously best known result of O(n)
bends per edge for the case where $k \geq 2$. Moreover, we give a lower bound
on the number of bends that matches our upper bound, proving our results are
optimal.
