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 $G$ *permissive*1 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.](http://www.akademiai.com/index/u257n5n54n98071u.pdf) 

[2] [Extending matchings in graphs: A survey. Plummer, 1994](http://www.sciencedirect.com/science/article/pii/0012365X9200485A)

[3] [An $O(VE)$ algorithm for ear decompositions of matching-covered graphs. de Carvalho and Cheriyan, 2005](http://dl.acm.org/citation.cfm?id=1103969)

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1 My terminology, not standard.