Path (graph theory)

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A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. Snake-in-the-box and Hamiltonian path.svg
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black.

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 [1] ) 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.

Contents

Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.

Definitions

Walk, trail, path

Trail but not path.svg
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. This walk is closed if v1 = vn, and open else. 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.

If w = (e1, e2, …, en − 1) is a finite walk with vertex sequence (v1, v2, …, vn) then w is said to be a walk fromv1tovn. 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.

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.

Directed walk, trail, path

Let G = (V, E, ϕ) be a directed graph. A finite directed 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 directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex.

If w = (e1, e2, …, en − 1) is a finite directed walk with vertex sequence (v1, v2, …, vn) then w is said to be a walk fromv1tovn. 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.

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

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.

Examples

Finding paths

Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.

Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights. The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.

See also

Related Research Articles

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References

  1. Graph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, p.205
  2. 1 2 3 4 5 6 Bender & Williamson 2010, p. 162.