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Source. Sections 4.17 and 4.20, printed pp. 459–460; used in 6.6, p. 465 (published PDF).

Statement. Let $F$ be a Hungarian forest in a finite graph, with collective inner set $I$ and outer set $O$, and let $W=V(G)\setminus V(F)$. Then

$$\nu(G)=|I|+\nu(G[W]).$$

More generally, replace some outer vertices by pairwise disjoint odd blocks $P_o$ of sizes $2r_o+1$. For ordinary outer vertices use singleton blocks with $r_o=0$. Assume each block has an internal matching omitting any specified vertex, and every edge leaving a block joins it to $I$. Keep the quotient Hungarian forest and all its edge identities. Then the expanded graph satisfies

$$\nu(G)=\sum_{o\in O}r_o+|I|+\nu(G[W]). \tag{1}$$

Proof. Every edge of the expanded graph belongs to one of three classes: it is internal to a block, it meets $I$, or it is internal to $W$. This follows from the stipulated neighbor condition. Any matching has at most $r_o$ edges inside each odd block, at most $|I|$ edges meeting $I$, and at most $\nu(G[W])$ edges in the third class. These are disjoint edge classes, proving the upper bound in (1).

Choose a maximum matching of each alternating tree in the quotient forest. Together they match every inner vertex to a distinct outer vertex and have $|I|$ edges. For each matched outer vertex, use in its block a near-perfect matching omitting the actual attachment endpoint of that chosen quotient edge. In an unmatched outer block, omit any vertex. These internal matchings, the forest matching, and a maximum matching on $W$ are pairwise disjoint and attain (1).

For singleton blocks this is the ordinary forest formula. The proof uses the collective neighbor condition and does not assume that each constituent tree is Hungarian. $\square$

Nested blossom expansions satisfy the block hypothesis by Section 4.14. This explicit counting form is supplied by the compilation to expand the source's forest analogy. It will be used in Section 6.6 both after deletion of an inner vertex and after deletion of a vertex in $W$.