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Source. Sections 6.0–6.6, printed pp. 463–465, with the lifting result 4.14 (published PDF).

Statement. Put

$$D=\{v:v\text{ is exposed by some maximum matching of }G\}, \quad A=N_G(D)\setminus D,\quad C=V(G)\setminus(D\cup A).$$

Every component $D_i$ of $G[D]$ has odd size $2r_i+1$ and an internal matching omitting any specified vertex. Every maximum matching has exactly $r_i$ internal edges in each $D_i$, matches each vertex of $A$ to a distinct component of $D$, and restricts to a perfect matching of $C$. If $G$ is bipartite, every $D_i$ is a singleton.

Proof. Start the source construction with the particular maximum matching under consideration. By Theorem 6.2, its complete outer blocks are exactly the intrinsic components $D_i$, and its ordinary inner vertices are exactly $A$. The nested lifting property gives odd order and internal matchings omitting any chosen vertex. The starting matching uses $r_i$ internal edges in each block.

Each inner vertex is matched to an outer quotient vertex in its planted tree, so each vertex of $A$ is matched to $D$. A block with $r_i$ internal edges has only one remaining vertex; hence two such matching edges cannot meet the same component. The remaining ordinary vertices form $C$ and are matched perfectly by the completed algorithm. Since the starting maximum matching was arbitrary, these assertions hold for every maximum matching.

In a bipartite graph there is no odd circuit. The algorithm therefore performs no blossom contraction. Every outer block, and hence every $D_i$, is a singleton. $\square$

In the notation of Edmonds–Fulkerson's matching input, $J=V(G)\setminus D$ and $Q=A$. The existence form used there follows in particular. Its separate local all-maximum and arbitrary-base lifting proof remains a distinct argument. This page extracts the consequences of the present original's arbitrary-starting-maximum construction; it does not claim the later paper's starred display appears verbatim here.