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Paths are edge-disjoint when no graph edge occurs in two of them. They may share internal vertices. If $r$ edge-disjoint client-to-server paths exist, fewer than $r$ failed links cannot destroy them all. For simultaneous link and node failures one instead needs internally vertex-disjoint paths, since one shared internal node could destroy several edge-disjoint paths at once.

Name the outer pentagon vertices, in order from the client clockwise, $c,T,R,B,L$. Name the upper inner vertex $U$, the left inner vertex $A$, and the lower-right inner vertex $D$. The two remaining inner vertices are the labelled servers $s_1,s_2$.

Four edge-disjoint server paths areEach uses different links, although some intermediary vertices are shared. Any three failed links therefore leave at least one path intact. The client has exactly four incident links; failing those four disconnects it. Thus the minimum link cut has size four, givingFor mixed failures use the three paths $c-A-s_1$, $c-U-s_2$ and $c-T-R-B-D-s_1$. Their internal vertices are disjoint, as are their links. One failed link or intermediary node can destroy at most one of these paths, so any two failures leave a path intact. Failing the three intermediary nodes $A,U,D$ disconnects both servers: $s_1$ has neighbors $A,D$, and $s_2$ has neighbors $U,D$. ThereforeThe two certificates distinguish edge-disjoint paths from internally vertex-disjoint paths, exactly the distinction needed between the two failure models.

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.

The vertex form of Menger theorem states that if $x$ and $y$ are distinct nonadjacent vertices of a finite graph, then the maximum number of pairwise internally vertex-disjoint paths from $x$ to $y$ equals the minimum cardinality of an $x$-$y$ vertex separator.