Number of planar graphs within $K_n$

Question

Let $\mathcal{N}$ be collection $n$ points in the plane. The question is rather simple:

How many planar graphs $G$ are there such that the vertices of $G$ are all in $\mathcal{N}$? (all edges must be straight and no loops are allowed)

I know this is ambiguous, but what I am looking for are useful references for anything like upper/lower bounds, related results, special cases, characterisations of such graphs etc.

I would greately appreciate any help! Thank you.

Answer 1

Here is a reference (Sharir and Sheffer) which deals with the maximal number of triangualations (as a corollary, maximum number of planar graphs) in a configuration of n points in the plane: http://emis.impa.br/EMIS/journals/EJC/Volume_18/PDF/v18i1p70.pdf Many relevant references within.