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. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics, a sequence is an ordered list of objects (or events For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out In Mathematics and Computer science, a graph is the basic object of study in Graph theory. 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. In Graph theory, a cycle graph is a graph that consists of a single cycle, or in other words some number of vertices connected in a closed chain

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. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation

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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. 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. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects 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. In the mathematical area of Graph theory, an induced path in an undirected graph G is a path that is an Induced subgraph of G

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 an 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. 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. Graph theory is a growing area in mathematical research and has a large specialized vocabulary 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. Graph theory is a growing area in mathematical research and has a large specialized vocabulary In Graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes such that the sum of the weights The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization. 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. László Lovász (born March 9, 1948 in Budapest Hungary) (ˈlaːsloː ˈlovaːs is a mathematician best known for his work in Combinatorics ) (1990). Paths, Flows, and VLSI-Layout. Algorithms and Combinatorics 9, Springer-Verlag. ISBN 0-387-52685-4.