Graph theory- Deleting a vertex from a graph Given an undirected connected graph $G=(V,E)$ and two vertices $s,t \in V$ . We know that the length of the shortest path from s to t is bigger than $V / 2  $.

I would love your help with proving that there must be a vertex which if we would delete it, along with the edges going out of it, there won't be a path between $s$ and $t$.

sadly I didn't come with any smart conclusions.

Thanks!
Hint: Prove that $s$ and $t$ do not belong to the same biconnected component of $G$.
I do not have enough reputation to post something in a comment so i would post it here. Thank you for your cooperation .

So my confusion is: 

if we delete a vertex v:

v ∈ V - {s,t} 


from an undirected connected graph G = (V,E).

Can we prove that the deletion of v disconnects G?