Path (graph theory)
Walk, trail, and path

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

Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path where all vertices are distinct.

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