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. A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... This article just presents the basic definitions. ... This article just presents the basic definitions. ... In the mathematical field of graph theory a cycle graph or circle graph is a graph that consists of a cycle. ...

A directed cycle. Without the arrows, it is just a cycle. This is not a simple cycle, since the blue vertices are used twice.

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. Image File history File links A directed cycle graph, created by Derrick Coetzee, who releases all rights to it and places it in the public domain, in Adobe Photoshop and Illustrator. ...

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. This article just presents the basic definitions. ... A labeled graph on 6 vertices and 7 edges. ...

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 with no edges connecting two nonconsecutive path vertices is called an induced path. A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... An induced path of length four in a cube. ...

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. In the mathematical field of graph theory, a Hamiltonian path is a path in a undirected graph which visits each vertex exactly once. ...

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. In the graph shown, (1, 2, 5, 1, 2, 3) is a path of length 5, and (5, 2, 1) is a simple path of length 2.

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. One major problem that has plagued graph theory since its inception is the consistent lack of consistency in terminology. ...

See also

• Glossary of graph theory
• Shortest path problem
• Traveling salesman problem

Graph theory is a growth area in mathematical research, and has a large specialized vocabulary. ... In graph theory, the single-source shortest path problem is the problem of finding a path between two vertices such that the sum of the weights of its constituent edges is minimized. ... The traveling salesman problem (TSP), is a problem in discrete or combinatorial optimization. ...

References

• Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. North Holland, 12–21. ISBN 0-444-19451-7.

• Diestel, Reinhard (2005). Graph Theory, 3rd ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, 6–9. ISBN 3-540-26182-6.

• Gibbons, A. (1985). Algorithmic Graph Theory. Cambridge University Press, 5–6. 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.