 Solution

For nonadjacent vertices $a,b$, the local connectivity $\kappa (a,b;G)$ is the minimum cardinality of an $a$-$b$ vertex separator: a set $S\subseteq V(G)\setminus \{a,b\}$ such that $a$ and $b$ lie in different components of $G-S$.

To prove the vertex form of the Menger theorem, replace every vertex $v\notin \{a,b\}$ by an arc $v_{\mathrm{in}}\to v_{\mathrm{out}}$ of capacity one. Replace every undirected edge $uv$ by directed arcs from $u_{\mathrm{out}}$ to $v_{\mathrm{in}}$ and conversely, each of capacity larger than $|V(G)|$; leave $a,b$ unsplit. An integral $a$-$b$ flow of value $k$ decomposes into $k$ paths, and the unit-capacity vertex arcs force the corresponding graph paths to be internally vertex-disjoint paths. Conversely, $k$ such paths give an integral flow of value $k$.

Because $a$ and $b$ are nonadjacent, any finite minimum cut consists entirely of unit-capacity vertex arcs. These arcs correspond exactly to an $a$-$b$ vertex separator of the same size. The max-flow min-cut theorem, whose augmenting-path proof also guarantees an integral maximum flow for integral capacities, therefore givesIn particular, $G$ contains the required set of $\kappa (a,b;G)$ paths.