Path (graph theory)

Snake-in-the-box_and_Hamiltonian_path

Trail_but_not_path

Path (graph theory)

In graph theory, a path is a sequence of vertices (or nodes) connected by edges (or arcs) in a graph. A path is defined as a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. Paths are fundamental concepts in the study of graphs and are used to describe various properties and structures within a graph.

Types of Paths

There are several types of paths in graph theory, each with specific characteristics:

• Simple Path: A path that does not repeat any vertices, except possibly the first and last vertices if the path is a cycle.
• Elementary Path: A path that does not repeat any vertices at all.
• Hamiltonian Path: A path that visits each vertex exactly once.
• Eulerian Path: A path that visits each edge exactly once.

Path Length

The length of a path is the number of edges in the path. For example, in a path with vertices \(v_1, v_2, v_3, \ldots, v_k\), the length of the path is \(k-1\).

Directed and Undirected Paths

Paths can be classified based on whether the graph is directed or undirected:

• Directed Path: A path in a directed graph where all edges have a direction, and the path follows the direction of the edges.
• Undirected Path: A path in an undirected graph where edges do not have a direction.

Applications

Paths are used in various applications within computer science, mathematics, and related fields. Some common applications include:

• Shortest path problem: Finding the shortest path between two vertices in a graph.
• Network flow: Analyzing the flow of resources through a network.
• Routing: Determining the optimal path for data transmission in networks.

Related Concepts

• Graph (discrete mathematics)
• Vertex (graph theory)
• Edge (graph theory)
• Cycle (graph theory)
• Connected graph
• Tree (graph theory)
• Graph traversal

See Also

• Graph theory
• Shortest path problem
• Hamiltonian path
• Eulerian path
• Cycle (graph theory)
• Graph traversal

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