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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.
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. 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.
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.
★ Glossary of graph theory
★ Shortest path problem
★ Traveling salesman problem
★ Graph Theory with Applications, Bondy, J. A.; Murty, U. S. R., , , North Holland, 1976, ISBN 0-444-19451-7
★ Graph Theory, Diestel, Reinhard, , , Graduate Texts in Mathematics, vol. 173, Springer-Verlag, 2005, ISBN 3-540-26182-6
★ Algorithmic Graph Theory, Gibbons, A., , , Cambridge University Press, 1985, ISBN 0-521-28881-9
★ Paths, Flows, and VLSI-Layout, Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (Eds.), , , Algorithms and Combinatorics 9, Springer-Verlag, 1990, ISBN 0-387-52685-4
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.
| Contents |
| Different types of path |
| See also |
| References |
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. 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.
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
References
★ Graph Theory with Applications, Bondy, J. A.; Murty, U. S. R., , , North Holland, 1976, ISBN 0-444-19451-7
★ Graph Theory, Diestel, Reinhard, , , Graduate Texts in Mathematics, vol. 173, Springer-Verlag, 2005, ISBN 3-540-26182-6
★ Algorithmic Graph Theory, Gibbons, A., , , Cambridge University Press, 1985, ISBN 0-521-28881-9
★ Paths, Flows, and VLSI-Layout, Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (Eds.), , , Algorithms and Combinatorics 9, Springer-Verlag, 1990, ISBN 0-387-52685-4
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