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