Distance between two vertices in a graph greater than n/2 If distance between two vertices in a graph is greater than n/2 then there exists a vertex whose removal would result in removal of all paths between those 2 vertices.
I've been trying to prove this for an hours without any progress

Let the vertices be $u$ and $v$. We are told any $uv$ path contains more than half of the vertices in $G$. If we take two $uv$ paths they have a vertex in common. This means the maximum number of vertex-independent paths is $1$. By Menger's Theorem this number is equal to the size of the minimum vertex cut for $v$ and $w$.
