It is shown that a graphG has all matchings of equal size if and only if for every matching setλ inG, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle. (V(λ) denotes the set of vertices incident with some edge ofλ.) Subsequently edge-coverings of graphs are discussed. A characterization is supplied for graphs all whose minimal covers have equal size.