pseudocycle

Pseudocycle is a term used in graph theory to describe a closed walk in a graph that does not revisit any edge. This is distinct from a cycle, which is a closed walk that does not revisit any vertex except for the start and end vertex. A pseudocycle can, therefore, pass through vertices multiple times.

For example, consider a graph with vertices A, B, C, and D, and edges (A,B), (B,C), (C,A), and (B,D). A walk from A to B, then to C, then back to A, and finally to B would be a pseudocycle. The path is A-B-C-A-B. The edges traversed are (A,B), (B,C), (C,A), and (A,B) in reverse. Note that the edge (A,B) is traversed twice, once in each direction, which is permissible in a pseudocycle. The vertex B is also revisited.

A key property of pseudocycles is their relationship to the concept of edge connectivity. In undirected graphs, a pseudocycle corresponds to a closed trail. The existence of pseudocycles can be relevant in problems involving network flow and pathfinding where edge usage constraints are important. The term pseudocycle is not as widely used as "cycle" or "trail," and sometimes the concept is described using other terminology depending on the specific context or field of study. It is important to clarify the precise definition when encountering this term to avoid confusion with standard graph-theoretic structures.