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

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

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 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

References

• Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. North Holland. ISBN 0-444-19451-7.
• Diestel, Reinhard (2005). Graph Theory. 3rd ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag. ISBN 3-540-26182-6.
• Gibbons, A. (1985). Algorithmic Graph Theory. Cambridge University Press. ISBN 0-521-28881-9.
• Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (Eds.) (1990). Paths, Flows, and VLSI-Layout. Algorithms and Combinatorics 9, Springer-Verlag. ISBN 0-387-52685-4.