  We study cross-graph charging schemes for graphs drawn in the plane. These
are charging schemes where charge is moved across vertices of different graphs.
Such methods have been recently applied to obtain various properties of
triangulations that are embedded over a fixed set of points in the plane. We
show how this method can be generalized to obtain results for various other
types of graphs that are embedded in the plane. Specifically, we obtain a new
bound of $O^*(187.53^N)$ (where the $O^*()$ notation hides polynomial factors)
for the maximum number of crossing-free straight-edge graphs that can be
embedded over any specific set of $N$ points in the plane (improving upon the
previous best upper bound $207.85^N$ in Hoffmann et al.). We also derive upper
bounds for numbers of several other types of plane graphs (such as connected
and bi-connected plane graphs), and obtain various bounds on expected
vertex-degrees in graphs that are uniformly chosen from the set of all
crossing-free straight-edge graphs that can be embedded over a specific point
set.
  We then show how to apply the cross-graph charging-scheme method for graphs
that allow certain types of crossings. Specifically, we consider graphs with no
set of $k$ pairwise-crossing edges (more commonly known as $k$-quasi-planar
graphs). For $k=3$ and $k=4$, we prove that, for any set $S$ of $N$ points in
the plane, the number of graphs that have a straight-edge $k$-quasi-planar
embedding over $S$ is only exponential in $N$.
