A graph is called matching-covered if every edge is containd in a perfect matching. (Such graphs are also sometimes called "elementary", e.g. in Chapter 4 of "Matching Theory" by Lovasz & Plummer). It is well-known that for bipartite graphs this is equivalent to the existence of ear decompositions.
What I'd like to know is whether the problem of counting the perfect matchings - which is very difficult in general, even for bipartite graphs, being equivalent to the permanent etc. - becomes easier when restricted to bipartite matching-covered graphs.
For example, it seems to me that the number of matchings ought to be somehow readable off the ear-decomposition but I don't quite see how.