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Directed walk, directed trail, and directed path

If w = (e, e, …, e) is a finite directed walk with vertex sequence (v, v, …, v) then w is said to be a walk from v to v. Similarly for a directed trail or a path. If there is a finite directed walk between two distinct vertices then there is also a finite directed trail and a finite directed path between them.

A "simple directed path" is a path where all vertices are distinct.

A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.