If G is a graph such that any two vertices of G are connected by exactly one path, then G is a tree.

Question

I am thinking about something along these lines:

Assume that $G$ is connected and let $u$ and $v$ be two vertices that are disjoint. In order for $u$ and $v$ to be connected there exists another vertex $w$ such that $uw$ and $wv$ are connected. Now since $u$ and $v$ are both connected with $w$ then $G$ is a tree.

Does that work OR is it more complicated than that?

Answer 1

You would need more than just there be a connected graph on $uw$ and a connected graph on $wv.$ Here is a way I would go about it.

Since every vertex of $G$ is linked by a path, $G$ is connected. If $G$ had a cycle, and $u,w$ were on that cycle, then they would be connected by two distinct paths. This cannot happen, so $G$ has no cycles. The only connected acyclic graphs are trees, hence $G$ is a tree.