Path (graph theory)

In mathematics, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex. The first vertex is called start vertex and the last vertex is called end vertex. The other vertices in the path are internal vertices.

A path is called a cycle if its start vertex is also its end vertex.

A path is called a simple path if none of the vertices in the path are repeated.

A cycle with no repeated vertices is called simple 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.

See also

• Glossary of graph theory
• Shortest path problem
• Traveling salesman problem