- Path (graph theory)
In

graph theory , a**path**in a graph is asequence 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 graph s 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 moderngraph 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

*Wikimedia Foundation.
2010.*