To see that this subdivision into subsets characterizes the edges that belong to perfect matchings, suppose that two vertices "x" and "y" in "G" belong to the same subset of the decomposition, but are not already matched by the initial perfect matching. Then there exists a strongly connected component in "H" containing edge "x,y". This edge must belong to a simple cycle in "H" (by the definition of strong connectivity) which necessarily corresponds to an alternating cycle in "G" (a cycle whose edges alternate between matched and unmatched edges). This alternating cycle may be used to modify the initial perfect matching to produce a new matching containing edge "x,y".