Non-factorizable graphs in which every edge can be extended to a maximum matching All graphs in this question are finite, simple, undirected, and unweighted. A graph is said to be factorizable if has a perfect matching, and non-factorizable otherwise. An edge in a graph is said to be allowed if it belongs to some maximum matching of the graph. Call a graph $G$ permissive1 if every edge of $G$ is allowed. 
I am interested in the properties of non-factorizable permissive graphs. What do we know about these? Where can I read up more on such graphs?
I could find quite a bit of existing work, both structural [1, 2] and algorithmic [3], on factorizable permissive graphs, which are called matching-covered graphs in the literature. However, I could not find anything at all about the case when every edge of a graph belongs to some maximum matching of the graph, and the graph itself does not have a perfect matching. What properties do such graphs have? Do we know something about their structure?
[1] On the structure of factorizable graphs. Lovasz, 1972. 
[2] Extending matchings in graphs: A survey. Plummer, 1994
[3] An $O(VE)$ algorithm for ear decompositions of matching-covered graphs. de Carvalho and Cheriyan, 2005

1 My terminology, not standard.
 A: The the Edmonds-Gallai decomposition could be what you're looking for.  It decomposes a graph into sets $A$, $B$ and $C$, where $A$ is all the vertices missed by at least one maximum matching, $B = N(A)$, and $C$ is all other vertices.  Basically, every maximum matching matches vertices in $B$ to odd components of $G - B$ in $A$.   In $G - B$, $C$ consists of even components with perfect matchings.  
If you follow what this means for a permissive graph, $B$ must be an independent set, since such edges must not be used in a maximum matching.  Furthermore, there are no edges from $C$ to the rest of the graph (so if we assume $G$ is connected, $C$ is empty).   That's quite a bit of structure right there.  I believe the components of $A$ are automatically permissive, and that any of the edges from $A$ to $B$ can automatically be used in maximum matchings, but I'm not sure.

Edit:  I noticed the Lovasz article you cite basically makes the same points I was making above, so you may have been aware of this information.  But regardless quite a bit is known about such graphs. 
A: I think your problem can be reduced to just looking at matching-covered graphs. Let $G$ be a graph and let $\mu$ be its matching number. Let $S$ be the set of all vertices covered by all maximum matchings. Let $G'$ be the larger graph obtained by adding a new independent set of size $|V(G)| - 2\mu$, adjacent to the vertices of $V(G)-S$.
If $G'$ is well-covered, then $G$ is permissive: given any $e \in E(G)$, we can extend to a perfect matching in $G'$, whose restriction to $G$ must be a matching of size $\mu$. On the other hand, if $G$ is permissive then $G'$ is well-covered: given any $e \in E(G')$, either $e \in E(G)$ in which case we can use permissiveness to extend to a maximum $G$-matching and then fix the leftover vertices with the independent set, or else $e$ joins some $v \in V(G) - S$ to the independent set, in which case we can take some maximum $G$-matching that misses $v$ and fix the leftovers (necessarily including $v$) with the independent set, using the edge $e$.
Using this trick, you can probably "transfer" results about matching-covered graphs to your context.
