We describe a framework for counting and enumerating various types of
crossing-free geometric graphs on a planar point set. The framework generalizes
ideas of Alvarez and Seidel, who used them to count triangulations in time
O(2nn2) where n is the number of points. The main idea is to reduce the
problem of counting geometric graphs to counting source-sink paths in a
directed acyclic graph.
The following new results will emerge. The number of all crossing-free
geometric graphs can be computed in time O(cnn4) for some c<2.83929.
The number of crossing-free convex partitions can be computed in time
O(2nn4). The number of crossing-free perfect matchings can be computed in
time O(2nn4). The number of convex subdivisions can be computed in time
O(2nn4). The number of crossing-free spanning trees can be computed in time
O(cnn4) for some c<7.04313. The number of crossing-free spanning cycles
can be computed in time O(cnn4) for some c<5.61804.
With the same bounds on the running time we can construct data structures
which allow fast enumeration of the respective classes. For example, after
O(2nn4) time of preprocessing we can enumerate the set of all crossing-free
perfect matchings using polynomial time per enumerated object. For
crossing-free perfect matchings and convex partitions we further obtain
enumeration algorithms where the time delay for each (in particular, the first)
output is bounded by a polynomial in n.
All described algorithms are comparatively simple, both in terms of their
analysis and implementation