  We leverage an algorithm of Deming [R.W. Deming, Independence numbers of
graphs -- an extension of the Koenig-Egervary theorem, Discrete Math.,
27(1979), no. 1, 23--33; MR534950] to decompose a matchable graph into
subgraphs with a precise structure: they are either spanning even subdivisions
of blossom pairs, spanning even subdivisions of the complete graph $K_4$, or a
K\H{o}nig-Egerv\'{a}ry graph. In each case, the subgraphs have perfect
matchings; in the first two cases, their independence numbers are one less than
their matching numbers, while the independence number of the KE subgraph equals
its matching number. This decomposition refines previous results about the
independence structure of an arbitrary graph and leads to new results about
$\alpha$-critical graphs.
