Talk:Path in Tree is Unique

It says here that $T$ is a tree if and only if there is exactly one path between any two vertices.

Shouldn't this be changed to "between any two different vertices".

A path is a trail, and a trail is a walk with distinct edges in its edge set. But if the two vertices are the same, the edge set could not be distinct. The vertex set could be distinct, e.g. $v \to u \to v$ for some adjacent vertex $u$ would technically be a trail with distinct vertices (except perhaps the first and last ones), as per the definition of a trail, but the edge set would not be distinct since it traverses the same edge twice.

Off the top of my head, the only graphs with distinct edge and vertex sets for paths between all pairs of vertices, even $(v, v)$, have to contain loops so they aren't trees.

Yes indeed, good call. Thanks for the heads up. --prime mover (talk) 12:37, 1 March 2024 (UTC)