Counting Plane Graphs: Cross-Graph Charging Schemes
  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 $N$. 
Micha Sharir and Adam Sheffer
cs.CG cs.DM math.CO