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 , 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 My terminology, not standard.