Definition
Directed cycle
A positive-length closed directed path with no repetitions except its common initial and final vertex.
A directed cycle is a directed path , , such that and are distinct. A loop is a cycle of length one.
Closed paths
Every positive-length closed directed path contains a directed cycle: if vertices repeat before the final return, select a shortest positive closed subpath. Minimality excludes an internal repetition. Consequently, excluding directed cycles also excludes all positive-length paths from a vertex to itself.