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A Plünnecke graph is a finite directed layered graph $V_0,\dots ,V_n$, with edges only from $V_i$ to $V_{i+1}$, satisfying two commutation conditions. For each edge $u\to v$, the successors of $v$ can be matched injectively into the successors of $u$, so that the matched predecessor has an edge to the original successor. Dually, the predecessors of $u$ can be matched injectively into the predecessors of $v$, so that each original predecessor has an edge to its match. These are upward and downward commutation. For nonempty bottom subsets define the graph magnification ratioAddition graphs have layer $A+iB$ and edges $x\to x+b$: exchanging the order of two additions gives exactly these matchings.

We prove a slightly stronger weighted cut lemma for commutative graphs, which also explains part (ii). Give vertices in $V_i$ weight $C^{-i}$, for any $C>0$. A separating set meets every bottom-to-top path. There is a minimum-weight separating set contained in $V_0\cup V_n$. We first establish its local three-layer step.

Consider a three-layer channel $U_0,U_1,U_2$, every vertex of which lies on a full path, and suppose the middle layer is a minimum-weight cut. Replacing a subset $Z\subseteq U_1$ by all its successors or all its predecessors still gives a cut. Minimality impliesLet $X_i\subseteq U_1$ be the vertices of incoming degree $i$. Partition $U_2$ into $T_i$ according to the largest incoming degree of one of their predecessors. ThenDownward commutation makes incoming degree nondecreasing along an edge. Thus every vertex of $T_i$ has at least $i$ predecessors, and summing the last inequalities over $j$ givesReverse all edges and interchange the endpoint layers. The graph still satisfies both commutation conditions, and the expansion factor becomes $C^{-1}$. The same argument now gives the opposite inequality $|E(U_0,U_1)|\ge C^{-1}|E(U_1,U_2)|$. Equality must therefore hold at every step. In particular the suffix inequality at $j=1$ is an equality, and the reversed argument givesConsequently the bottom layer has exactly the middle layer's weight. This proves the local replacement lemma.

For a general minimum cut $S$, write $S_i=S\cap V_i$, and take its highest nonempty intermediate portion $S_j$, $0<j<n$. Form the three-layer channel whose bottom vertices are reachable from $V_0$ while avoiding the earlier cut portions, and whose top vertices can reach $V_n\setminus S_n$. Its middle layer is exactly $S_j$: any other middle vertex would produce an uncut full path, while every member of $S_j$ is essential in a positive-weight minimum cut and hence lies on such a path when that vertex alone is allowed. Channels inherit the commutation matchings: the replacement intermediate vertices still join the same prefix and suffix paths.

The middle layer is a minimum cut within this channel, since a cheaper local separator would replace $S_j$ and make the global cut cheaper. Apply the local lemma, with weights scaled by the common factor $C^{-(j-1)}$, to replace $S_j$ by the channel's bottom layer. This does not increase weight and removes the highest intermediate cut layer. Repetition leaves a minimum cut in the endpoint layers, proving the weighted cut lemma. Vertices lying on no full path can first be deleted; when $D_n>0$ all bottom vertices survive, and deleting such branches can only decrease intermediate images, so proving the lower bounds in the channel suffices.

Now take $C=1$ and assume $D_n\ge 1$. If a minimum endpoint cut is $S_0\cup S_n$, then ${\mathrm{Im}}^{(n)}(V_0\setminus S_0)\subseteq S_n$, soThe bottom layer itself is a cut, so the minimum cut size is $|V_0|$. For every nonempty $Z\subseteq V_0$, the set $(V_0\setminus Z)\cup {\mathrm{Im}}^{(k)}Z$ is also a cut. Its size must be at least $|V_0|$, provingOne may also apply the permitted Menger theorem to obtain $|V_0|$ vertex-disjoint full paths, giving the same conclusion. The proof above establishes the needed cut statement directly.