Is it true that all paths between two vertices share a common point?

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.