Deniz, Z.Ekim, T.2024-08-232024-08-2320241034-4942https://hdl.handle.net/20.500.12684/14785A 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 creates a graph that is not equimatchable, 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 [Eur. J. Comb. 20 (2019), 261–272] on the characterization of well-covered graphs with no shedding vertex. We also introduce equimatchable graphs such that the removal of any vertex creates a graph that is not equimatchable, called vertex-critical equimatchable graphs (VCEgraphs). To conclude, we clarify 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. © The author(s).eninfo:eu-repo/semantics/closedAccessCritical equimatchable graphsArticle881711932-s2.0-85184456988Q2