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Source. Section 8, printed pp. 129–130 (published original).

Scope. These are announced results. The introduction explicitly assigns the degree-constrained extension to another paper. Section 8 states them but does not prove them. No full-proof credit is attached to this page.

Let each vertex have an integer capacity $d_v$. In the usual nonempty case $d_v\ge0$: a negative capacity contradicts nonnegativity of its incident sum. A zero capacity forces all incident coordinates to zero. Use $E(S)$ for internal edges and $\delta(S)$ for edges with exactly one endpoint in $S$.

Polyhedron I has the inequalities

$$x_e\ge0,\qquad \sum_{e\ni v}x_e\le d_v,\qquad \sum_{e\in E(S)}x_e\le r \quad\left(\sum_{v\in S}d_v=2r+1,\ r\ge1\right).$$

The announcement says all its extreme points have integer coordinates. They are edge multiplicities; no bound $x_e\le1$ is imposed in this first polyhedron.

Polyhedron II has

$$0\le x_e\le1,\qquad \sum_{e\ni v}x_e\le d_v,$$

and, for every $F\subseteq\delta(S)$ with $t=|F|$ satisfying

$$|\{e\in F:e\ni v\}|<d_v\quad(v\in S),\qquad t+\sum_{v\in S}d_v=2r+1,\quad r\ge1,$$

the inequality

$$\sum_{e\in E(S)\cup F}x_e\le r.$$

The announcement again says that all extreme points are integral; here they are zero-one. The local strict restriction on selected boundary-edge counts and the source's positive integer $r$ are retained. In particular, the displayed restriction admits no such $S$ containing a zero-capacity vertex.

When every $d_v=1$, Polyhedron I becomes the matching polytope, and in Polyhedron II the boundary restriction forces $F$ empty. The vertex inequalities already imply the individual upper bounds. Thus both specialize to Theorem P.

The last paragraph also announces an efficient algorithm for maximum-weight degree-constrained subgraphs. Neither its algorithm nor either general integrality proof is supplied by this six-page article. The present compilation does not replace those missing proofs by an appeal to the matching special case.