There is a fairly rich classification on graphs with respect to the existence of Hamiltonian cycles either in unmodified graphs or after certain small modifications. 


Do there also exist such classifications with respect to perfect matchings?
Specifically, I would like to know, whether there exist


* "Hypo-Matching" graphs that do not contain a perfect matching, but removing an arbitrary pair of vertices generates a graph with perfect matching.
* "Hyper-Matching" graphs with a perfect matching, for which removing an arbitrary pair of vertices generates a graph with perfect matching.



"Hypo" and "Hyper" for Perfect Matching
A hypo-matching graph cannot exist. Assume $G$ does not contain a perfect matching, but $G - \{u,v\}$ does. This means $uv$ is not an edge is $G$ or else $G$ has a perfect matching. This cannot hold for an arbitrary pair of vertices since that would force $G$ to be edgeless.


A hyper-matching graph definitely exists. Just consider the complete graph on $2n$ vertices.


