Number of planar graphs, given an embedding

Question

I want to find an upper bound on the number of planar graphs with $n$ vertices, assuming that we are given some embedding for those vertices beforehand. In particular, Im interested in either showing for every embedding there is $2^{o(n\log(n))}$ planar graphs, or showing that there exists an embedding with $2^{\Omega(n\log(n))}$ different planar graphs.

I know that without fixing embedding there is $2^{\Theta(n\log(n))}$, but my question differs from it since we fix an embedding beforehand.

I believe that still there is an embedding with $2^{\Omega(n\log(n))}$, but I couldn't prove or disprove it. Here is an attempt I made to try and calculate the number of planar graphs for some embedding I believed would be "hard", but it didn't work out as expected.

Thanks in advance!

Answer 1

Pach and Wenger proved in their paper Embedding planar graphs at fixed vertex locations that if $p_1,\ldots,p_n$ are $n$ different points on the plane, then every planar graph on $n$ vertices $v_1,\ldots,v_n$ has a planar embedding in which $v_i$ is found at position $p_i$.