  We study the perfect matching reconfiguration problem: Given two perfect
matchings of a graph, is there a sequence of flip operations that transforms
one into the other? Here, a flip operation exchanges the edges in an
alternating cycle of length four. We are interested in the complexity of this
decision problem from the viewpoint of graph classes. We first prove that the
problem is PSPACE-complete even for split graphs and for bipartite graphs of
bounded bandwidth with maximum degree five. We then investigate polynomial-time
solvable cases. Specifically, we prove that the problem is solvable in
polynomial time for strongly orderable graphs (that include interval graphs and
strongly chordal graphs), for outerplanar graphs, and for cographs (also known
as $P_4$-free graphs). Furthermore, for each yes-instance from these graph
classes, we show that a linear number of flip operations is sufficient and we
can exhibit a corresponding sequence of flip operations in polynomial time.
