Given bipartite planar graph $G$ with $f(G)$ perfect matchings if we add one vertex of each color and choose new additional edge on condition that the new bipartite graph is planar.

>What is the distribution or at least mean and variance of number of additional perfect matchings?

>Planar graph permanent can be reduced to determinants and so statistics should be amenable.

Note number of additional perfect matchings is not a Markov process (it depends on $f(G)$).

Also since number of edges of planar bipartite graphs are bound we do not have much room to draw edges in general and this too depends on starting graph $G$ and is not a Markov process.