Path (graph theory)

For the family of graphs known as paths, see Path graph.
A 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 connect a sequence of vertices which, by most definitions, are all distinct from one another. In a directed graph, a directed path (sometimes called dipath[1]) is again a sequence of edges (or arcs) which connect a sequence of vertices, but with the added restriction that the edges all be 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. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.


A path is a trail in which all vertices (except possibly the first and last) are distinct. A trail is a walk in which all edges are distinct. A walk of length in a graph is an alternating sequence of vertices and edges, , which begins and ends with vertices. If the graph is undirected, then the endpoints of are and . If the graph is directed, then is an arc from to . An infinite path is an alternating sequence of the same type described here, but with no first or last vertex, and a semi-infinite path (also ray) has a first vertex, , but no last vertex. Most authors require that all of the edges and vertices be distinct from one another. However, some authors do not make this requirement, and instead use the term simple path to refer to a path which contains no repeated vertices.

A weighted graph associates a value (weight) with every edge in the graph. The weight of a 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.


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, unless P=NP.

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


  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
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