# Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

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

Pick a uniformly random bipartite planar graph $$G$$ with $$n$$ vertices of each color and choose new additional edge on condition that the new bipartite graph $$H$$ is planar. Denote $$f(G)$$ and $$f(H)$$ to number of perfect matchings of each color respectively. The probability distribution of $$G$$ is different from $$H$$ since $$H$$ is no longer picked from uniform distribution.

What is the probability distribution or at least mean and variance of

1. number of perfect matchings $$f(G)$$

2. number of perfect matchings $$f(H)$$ (note the probability distribution of $$H$$ is different from $$G$$)

3. number of additional perfect matchings $$f(H) - f(G)$$?

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 of $$H$$ in general and this too depends on starting graph $$G$$ and is not a Markov process.