Characterize all minimally $2$-matchable graphs

Question

A finite graph is minimally $k$-matchable if it has at least $k$ distinct perfect matchings, but deleting any edge results in a graph which does not. We want to characterize all minimally $2$-matchable graphs. Any hints?

Answer 1

Your graph has two perfect matchings $M_1$ and $M_2$, and any edge of the graph should be contained in $M_1 \cup M_2$, since otherwise you could delete it and be left with a 2-matchable graph.

In particular, your graph has maximum degree 2, since every vertex is incident to only one edge of $M_1$ and only one edge of $M_2$ (and these might be the same edge).

It's reasonably easy to characterize all graphs with maximum degree 2, and see which ones

1. Have any perfect matchings to begin with.
2. Have at least two distinct perfect matchings.
3. Lose this property if any edge is deleted.