Definition:Path (Graph Theory)/Warning

Path: Warning

It seems at first glance that a path could also be defined as a walk in which all vertices (except perhaps the first and last ones) are distinct.

By this definition it would appear that a path is automatically a trail, because if an edge were to be retraced in any walk, then the vertices at either end of it would necessarily be visited more than once.

However, under this looser definition, the walk $u \to v \to u$ for two adjacent vertices $u$ and $v$, for example, would fit the definition of a path, and therefore be a cycle.

But such a walk is not a trail, as the edge $u v$ would be traversed twice.

Hence the insistence that a path is a type of trail.