cycletypes

Cycletypes are a framework used primarily in graph theory and combinatorics to classify the various kinds of directed or undirected cycles that can occur within a graph. The classification is based on properties such as length, vertex repetition, edge direction and nesting. The most common cycle types include: simple cycles, where all vertices are distinct and the cycle is closed; repeated cycles, where a vertex or a set of vertices repeat within the same cycle; directed versus undirected cycles, depending on whether the edges have a direction that must be followed; and complex cycles, which are combinations of simple or repeated cycles that share vertices or edges.

In directed graphs, cycle types are important for algorithms that detect deadlocks, partitions, or topological sorting, because the presence of a directed cycle implies a non‑acyclic structure. Simple directed cycles of length three are called directed triangles and are fundamental motifs in network analysis. Repeated directed cycles can appear in systems with feedback loops, such as in biological regulatory networks. Undirected cycle types are used in maze generation, route planning and in the study of Eulerian circuits, where every edge of a cycle is visited exactly once.

The cycletype classification also appears in permutation group theory, where the cycle type of a permutation is a partition of the number of elements that specifies the lengths of the disjoint cycles. This concept is central to the cycle index of a permutation group and to Polya counting problems. In operations research, cycle types in integer programming help to identify cyclic patterns in schedules. Overall, cycletypes provide a concise vocabulary for describing the basic building blocks of cyclic structures across many areas of mathematics and computer science.