simplecycle

A simple cycle is a closed path in a graph that visits every vertex at most once, with the starting vertex equal to the ending vertex. In an undirected simple graph, this implies at least three distinct vertices. The length of the cycle equals the number of edges in the path.

This notion distinguishes a simple cycle from a generic cycle or a closed walk, which may repeat vertices or edges. In directed graphs, a simple directed cycle must follow edge directions and also visit each vertex at most once before returning to the start.

Simple cycles can be found and studied with standard graph algorithms. Depth-first search can detect a cycle when a back edge to a previously visited vertex is encountered. A graph is acyclic if and only if no directed cycle exists. Algorithms exist to enumerate all simple cycles in a graph, such as Johnson's algorithm; practical limits apply because the number of simple cycles can be large.

Common applications include verifying acyclicity in data processing and scheduling, analyzing feedback loops in networks, and studying ring structures in chemistry, where a simple cycle represents a chemical ring. The term is widely used in graph theory and related disciplines.