  A graph G is equimatchable if every maximal matching of G has the same
cardinality. In this paper, we investigate equimatchable graphs such that the
removal of any edge harms the equimatchability, called edge-critical
equimatchable graphs (ECE-graphs). We show that apart from two simple cases,
namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected
factor-critical. Accordingly, we give a characterization of factor-critical
ECE-graphs with connectivity 2. Our result provides a partial answer to an open
question posed by Levit and Mandrescu on the characterization of well-covered
graphs with no shedding vertex. We also introduce equimatchable graphs such
that the removal of any vertex harms the equimatchability, called
vertex-critical equimatchable graphs (VCE-graphs). To conclude, we enlighten
the relationship between various subclasses of equimatchable graphs (including
ECE-graphs and VCE-graphs) and discuss the properties of factor-critical
ECE-graphs with connectivity at least 3.
