Kleinberg, Jon. Algorithm Design $($p$.110,$ q$.9).$ There's a natural intuition that two nodes that are far apart in a communication network $-$ separated by many hops $-$ have a more tenuous connection than two nodes that are close together. There are a number of algorithmic results that are based to some extent on different ways of making this notion precise. Here's one that involves the susceptibility of paths to the deletion of nodes.

Suppose that an $n-$node $($connected$)$ undirected graph $G=(V,E)$ contains two nodes $s$ and $t$ such that the distance between $s$ and $t$ is strictly greater than $\frac{n}{2}.$

$($a$)$ Show that there must exist some node $v,$ not equal to either $s$ or $t,$ such that deleting $v$ from $G$ destroys all $s-t$ paths. $($In other words, the graph obtained from $G$ by deleting $v$ contains no path from $s$ to $t.)$