Given an undirected graph $\displaystyle G=(V,E) $ and two vertices $\displaystyle u $ and $\displaystyle v $, if the minimum number of edges connecting $\displaystyle u $ and $\displaystyle v $ is greater than $\displaystyle \frac n2 $ (where $\displaystyle n=|V| $), show there exists a node $\displaystyle w\neq u,v $ such that all $\displaystyle u $-$\displaystyle v $ paths contain $\displaystyle w $.

I'm not sure how to get started but something tells me to use breadth first search and apply the pigeon hole principle.

Any help is appreciated! Thanks!