TITLE: Is it true that all paths between two vertices share a common point?
QUESTION [5 upvotes]: I'm stuck on following graph theory problem:
suppose for two vertices $u$ and $v$ in an undirected graph $G$, any pair of paths between $u$ and $v$ share a common vertex (different from $u$ and $v$), then there exists vertex $w$ ($w\neq u,v$), which lies on all paths between $u$ and $v$.
I don't know if this statement is true, but can't come up with any counterexamples. how can I approach this problem?

REPLY [3 votes]: This is a special case of Menger's theorem. 
If any two $u,v$-paths share a common vertex, then the maximum number of vertex-disjoint $u,v$-paths we can find is at $1$; therefore by Menger's theorem, there is a $u,v$-cut of size $1$, which is the vertex $w$.
This is assuming we're not in one of several trivial cases:

If there are no $u,v$-paths, then it's vacuously true that any two share a vertex. Let $w$ be any vertex other than $u$ and $v$, and the conclusion still holds.
But if there are no vertices other than $u$ and $v$, we're in trouble.
If $u$ and $v$ are adjacent, and there are no other $u,v$-paths, it's still vacuously true that any two $u,v$-paths share a vertex. But there is no vertex $w$ other than $u$ and $v$ which lies on the path consisting only of the edge $uv$, so in this case the conclusion does not hold.