Turboencabulator

2013-12-24T14:53:41Z

76.65.229.237: /* History */

{{infobox graph
| name = Path graph
| image = [[Image:Path-graph.svg|250px]]
| image_caption = A path graph on 6 vertices
| vertices = ''n''
| edges = ''n - 1''
| automorphisms = 2
| diameter = ''n - 1''
| radius = ''⌊n/2⌋''
| chromatic_number = 2
| chromatic_index = 2
| spectrum = {2 cos(''k'' π / (''n'' + 1))<sup>1</sup>; ''k''=1,...,''n''}
| properties = [[Unit distance graph|Unit distance]]<br/>[[Bipartite graph]]<br/>[[tree (graph theory)|Tree]]
|notation = <math>P_n</math>
}}
{{confused|line graph}}
In the [[Mathematics|mathematical]] field of [[graph theory]], a '''path graph''' or '''linear graph''' is a particularly simple example of a [[tree (graph theory)|tree]], namely a tree with two or more [[vertex (graph theory)|vertices]] that is not branched at all, that is, contains only vertices of [[degree (graph theory)|degree]] 2 and 1. In particular, it has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2.

A '''path''' in a [[graph (mathematics)|graph]] is a [[sequence]] of [[vertex (graph theory)|vertices]] such that from each of its vertices there is an [[edge (graph theory)|edge]] to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its ''start vertex'', and a last vertex, called its ''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. Note that the choice of the start vertex in a cycle is arbitrary.

[[Image:Directed cycle.svg|frame|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.

== Different types of path graphs ==
The same concepts apply both to [[undirected graph]]s and [[directed graph]]s, 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 a cycle with no repeated vertices or edges aside from the necessary repetition of the start and 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. 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, without repetition, of the graph is known as a [[Hamiltonian cycle]].

A cycle with just one edge removed in the corresponding [[spanning tree]] of the original graph is known as a Fundamental 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]]
*[[Cycle space]]
*[[Path (graph theory)]]
*[[Caterpillar tree]]
*[[Cycle graph]]
*[[Complete graph]]
*[[Null graph]]
*[[Path decomposition]]

== References ==

*{{cite book
| author = [[John Adrian Bondy|Bondy, J. A.]]; [[U. S. R. Murty|Murty, U. S. R.]]
| title = Graph Theory with Applications
| year = 1976
| publisher = North Holland
| isbn = 0-444-19451-7
| pages = 12–21
| url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html}}
*{{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
| 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
| isbn = 0-521-28881-9}}
*{{cite book
| author = [[Bernhard Korte|Korte, Bernhard]]; [[László Lovász|Lovász, László]]; Prömel, Hans Jürgen; [[Alexander Schrijver|Schrijver, Alexander]] (Eds.)
| title = Paths, Flows, and VLSI-Layout
| publisher = Algorithms and Combinatorics 9, Springer-Verlag
| year = 1990
| isbn = 0-387-52685-4}}

==External links==
* {{MathWorld | urlname=PathGraph | title=Path Graph }}

[[Category:Graph theory objects]]
[[Category:Graph connectivity]]
[[Category:Trees (graph theory)]]
[[Category:Parametric families of graphs]]

[[cs:Cesta (graf)]]
[[es:Camino (teoría de grafos)]]
[[fr:Chaîne (graphe)]]
[[he:מסלול (תורת הגרפים)]]
[[ja:道 (グラフ理論)]]
[[pl:Ścieżka (teoria grafów)]]
[[pt:Caminho (teoria dos grafos)]]
[[ru:Путь (теория графов)]]
[[uk:Шлях (теорія графів)]]
[[ur:رستہ (نظریہ مخطط)]]
[[zh:道路 (图论)]]

formulasearchengine - User contributions [en]

Turboencabulator