#perfectMatchings is self-reducible How can one show that the counting problem:
Given a  graph, output the number of perfect matchings
Is self reducible?
I found a hint in Moore's Chapter on Counting, Sampling and Statistical Physics:

Let G be a graph and e one of its edges. Show that we can modify G to
produce two graphs, G0 and G1, such that the number of perfect matchings
of G that include and exclude e is the number of perfect matchings of
G1 and G0 respectively.

But I could not come up with a proof. The main question here is: How can we proof self-reducability of #perfectMatchings?
Aside: I don't know of a formal definition of self-reduciability. I just know the intuitive phrasing &quot;partially solving the problem results in a smaller problem of the same kind&quot;. But is there a formal definition?
To count all perfect matchings containing an edge $e = (x,y)$, simply remove both endpoints $x,y$. To count all perfect matching not containing the edge $e$, simply remove the edge.