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Concept
Path (graph theory)
Summary
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. , , or . cover more advanced algorithmic topics concerning paths in graphs.
A walk is a finite or infinite sequence of edges which joins a sequence of vertices.
Let G = (V, E, φ) be a graph. A finite walk is a sequence of edges (e1, e2, ..., en − 1) for which there is a sequence of vertices (v1, v2, ..., vn) such that φ(ei) = {vi, vi + 1} for i = 1, 2, ..., n − 1. (v1, v2, ..., vn) is the vertex sequence of the walk. The walk is closed if v1 = vn, and it is open otherwise. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.
A trail is a walk in which all edges are distinct.
A path is a trail in which all vertices (and therefore also all edges) are distinct.
If w = (e1, e2, ..., en − 1) is a finite walk with vertex sequence (v1, v2, ..., vn) then w is said to be a walk from v1 to vn. 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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