We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with m+1 edges to the number of graphs with m edges. We show how a relatively small improvement of this bound would improve existing bounds concerning numbers of plane graphs with a prescribed number of edges. Furthermore, we use a cross-graph charging scheme to derive lower bounds for the number of such graphs when the point set has few extreme points.