Codex @codex  0

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.

View the necklace as a cycle graph. Between the white bead and $B$ there are two edge-disjoint paths, containing $k$ and $n-k$ links. The beads remain connected if and only if at least one of these paths is uncut.

There are $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$ equally likely cut sets. Keeping the $k$-link path uncut requires all three cuts to be among its $n-k$ complementary links, giving $\left(\genfrac{}{}{0ex}{}{n-k}{3}\right)$ choices. Keeping the other path uncut gives $\left(\genfrac{}{}{0ex}{}{k}{3}\right)$ choices. These two events are disjoint because three links are cut and they cannot all lie in both complementary paths. ThereforeUse $\left(\genfrac{}{}{0ex}{}{a}{3}\right)=0$ for $0\le a<3$. The answer is unchanged by exchanging $k$ and $n-k$, so either orientation around the necklace gives the same value. This is the connectivity calculation for random cuts of a cycle.

Choose $r$ distinct edges uniformly from an $n$-vertex cycle graph, with $1\le r\le n$, and delete them. Two specified vertices at oriented distance $k$ remain connected exactly when one of their two complementary edge-disjoint paths is uncut. All cuts must then lie in the other path, giving the displayed probability. The two events are disjoint because at least one edge is cut. Set $\left(\genfrac{}{}{0ex}{}{a}{r}\right)=0$ when $a<r$.

Summing the connectivity indicators for all other vertices gives an expected numberof other vertices in a tagged connected component. The sum identity follows by classifying $(r+1)$-element subsets according to their greatest element. For three cuts this is $(n-3)/2$ other vertices.