Counting the total number of matchings in a general graph is #P-complete (i.e. hard).
Also, counting the total number of perfect matchings is also #P-complete.

We know that matchings can be translated into independence sets,
so every matching counting problem can be made into a counting independence sets in a related graph. (But the converse is not true, independence sets are more general).

**Question:** Given a graph $G$, can we find (in some efficient manner) a graph $G'$
such that the number of matchings of $G$ is the same as the number of perfect matchings in $G'$? 

**Question 2:** If the answer is no for general graphs, what about bipartite graphs?