I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing exponentially. My request is different from the one [here](https://en.wikipedia.org/wiki/FKT_algorithm) because right now, I have no interest in knowing what these perfect matchings are.


The [FKT](https://en.wikipedia.org/wiki/FKT_algorithm) algorithm gives a polynomial time algorithm for a planar graph and I am wondering if there is anything similar for non planar graphs. Does anyone have any ideas?



Algorithm to count the number of perfect matchings in non planar graph
Counting the number of perfect matchings in arbitrary graphs (i.e. non planar, non bipartite...) seems to be quite more difficult than the restricted cases that FKT-type algorithms can handle.


In particular, Valiant proved that the problem is in $\mathrm{\text{#}P}$.


With that in mind, you can find information on approximation algorithms, see for example:


* Steve Chien, [A determinant-based algorithm for counting matchings in a general graph](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.125.5577&rep=rep1&type=pdf)
* Martin Fürer, Shiva Prasad Kasiviswanathan, [Approximately Counting Perfect Matchings in General Graphs](https://siam.org/meetings/analco05/papers/10mfurer.pdf)


