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 Kőnig-Egerváry 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 $α$-critical graphs.