  We call a finite undirected graph minimally k-matchable if it has at least k
distinct perfect matchings but deleting any edge results in a graph which has
not. An odd subdivision of some graph G is any graph obtained by replacing
every edge of G by a path of odd length connecting its end vertices such that
all these paths are internally disjoint. We prove that for every k>0 there
exists a finite set of graphs S(k) such that every minimally k-matchable graph
is isomorphic to a disjoint union of an odd subdivision of some graph from S(k)
and any number of copies of the complete graph on two vertices.
