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Source. Section 4.14 and its extension, printed p. 458 (published PDF).

Statement. If an odd circuit $B$ is contracted, every matching $N$ of $G/B$ lifts to a matching of $G$ by adding $(|V(B)|-1)/2$ edges of $B$. If the quotient vertex is exposed, the lift may leave any specified vertex of $B$ exposed.

More generally, a complete pseudovertex expansion $P$ on $2r+1$ original vertices has a near-perfect matching omitting any specified vertex. Every quotient matching lifts across $P$ by adding exactly $r$ internal edges. Consequently any increase in quotient matching size lifts to the same increase.

The extension printed after 4.14 (p. 458) asserts only a matching of $P$ leaving exactly one exposed vertex and compatible with a given matching of the contracted graph. The choice of an arbitrary exposed vertex in a complete expansion is the form Section 6.5 (pp. 464–465) invokes when it cites 4.14; it is proved below.

Proof. A quotient matching has at most one edge incident to the contracted vertex. If there is such an edge, let $b$ be its original endpoint in $B$; otherwise choose any $b$. The odd-circuit matching omitting $b$ is compatible with all edges of $N$, giving the first claim.

For a nested expansion, induct on the number of remembered contractions. An ordinary vertex has the empty matching. At the top contraction, let the odd circuit have child blocks $P_1,\ldots,P_{2k+1}$, each already possessing the asserted property. To omit a specified original vertex $v$, choose the circuit matching omitting the child containing $v$. For every other child, its incident selected circuit edge has a particular endpoint in that child. Match internally all but that endpoint, using induction. In the omitted child, match internally all but $v$.

These matchings and circuit edges are disjoint. Every original vertex except $v$ is covered, so the internal edge count is $(|V(P)|-1)/2$. The remembered edge identities supply exactly the required attachment vertices, even when parallel edges are present.

For a quotient matching meeting $P$ in one crossing edge, choose $v$ to be that edge's original endpoint. If it has no such edge, choose $v$ freely. This proves compatibility and the fixed edge-count increment. Applying it to disjoint current blocks or successively reversing nested contractions proves the final assertion. $\square$

No optimality converse for an arbitrary quotient is asserted. That converse requires the hypotheses in Section 4.15.