Path (graph theory)

In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. The first vertex is called the "start vertex" and the last vertex is called the "end vertex". Both of them are called "end or terminal vertices" of the path. The other vertices in the path are "internal vertices". A cycle is a path such that the start vertex and end vertex are the same. Notice however that unlike with paths, any vertex of a cycle can be chosen as the start, so the start is often not specified.

Paths and cycles 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.

Different types of path

The same concepts apply both to undirected graphs and directed graphs, with the edges being directed from each vertex to the following one. Often the terms "directed path" and "directed cycle" are used in the directed case.

A path with no repeated vertices is called a simple path, and cycle with no repeated vertices aside from the start/end vertex is a simple cycle. In modern graph theory, most often "simple" is implied; i.e., "cycle" means "simple cycle" and "path" means "simple path", but this convention is not always observed, especially in applied graph theory. Some authors (e.g. Bondy and Murty 1976) use the term "walk" for a path in which vertices or edges may be repeated, and reserve the term "path" for what is here called a simple path.

A path such that no graph edges connect two nonconsecutive path vertices is called an induced path.

A simple cycle that includes every vertex of the graph is known as a Hamiltonian cycle.

Two paths are "independent" (alternatively, "internally vertex-disjoint") if they do not have any internal vertex in common.

The "length" of a path is the number of edges that the path uses, counting multiple edges multiple times. The length can be zero for the case of a single vertex.

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.

See also

*Glossary of graph theory
*Shortest path problem
*Traveling salesman problem
*Average path problem
*Cycle space

References

*cite book
author = Bondy, J. A.; Murty, U. S. R.
title = Graph Theory with Applications
year = 1976
publisher = North Holland
id = ISBN 0-444-19451-7
pages = 12–21

*cite book
author = Diestel, Reinhard
title = Graph Theory
edition = 3rd ed.
url = http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
publisher = Graduate Texts in Mathematics, vol. 173, Springer-Verlag
year = 2005
id = ISBN 3-540-26182-6
pages = 6–9

*cite book
author = Gibbons, A.
title = Algorithmic Graph Theory
year = 1985
publisher = Cambridge University Press
pages = 5–6
id = ISBN 0-521-28881-9

*cite book
author = Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (Eds.)
title = Paths, Flows, and VLSI-Layout
publisher = Algorithms and Combinatorics 9, Springer-Verlag
year = 1990
id = ISBN 0-387-52685-4