Counting Triangulations of Planar Point Sets

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30ⁿ. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43ⁿ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs (207.84ⁿ), spanning cycles (O(68.67ⁿ)), spanning trees (O(146.69ⁿ)), and cycle-free graphs (O(164.17ⁿ)).