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Path (graph theory)

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Wikipedia

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.

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. 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.

References

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

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

Gibbons, A. (1985). Algorithmic Graph Theory. Cambridge University Press. 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.