Path (graph theory)
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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.
The same concepts apply in a directed graph, with the edges being directed from each vertex to the following one. Often the terms directed path and directed cycle are used in this 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 there does not exist any edge that connects 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. In the graph shown, (1, 2, 5, 1, 2, 3) is a path of length 5, and (5, 2, 1) is a simple path of length 2.
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.
