  A graph is said to be equimatchable if all its maximal matchings are of the
same size. In this work we introduce two extensions of the property of
equimatchability by defining two new graph parameters that measure how far a
graph is from being equimatchable. The first one, called the matching gap,
measures the difference between the sizes of a maximum matching and a minimum
maximal matching. The second extension is obtained by introducing the concept
of equimatchable sets; a set of vertices in a graph $G$ is said to be
equimatchable if all maximal matchings of $G$ saturating the set are of the
same size. Noting that $G$ is equimatchable if and only if the empty set is
equimatchable, we study the equimatchability defect of the graph, defined as
the minimum size of an equimatchable set in it. We develop several
inapproximability and parameterized complexity results and algorithms regarding
the computation of these two parameters, a characterization of graphs of unit
matching gap, exact values of the equimatchability defect of cycles, and sharp
bounds for both parameters.
