Cyclesfrom

Cyclesfrom is a concept in graph theory describing a method or function that enumerates directed cycles in a graph that originate from a specified set of starting vertices. The idea focuses on identifying cycles that can be reached from chosen sources, and it can be defined for directed graphs as well as adapted for undirected graphs by considering cycles up to symmetry.

Formal definitions and variants of cyclesfrom vary, but a common framing is: given a directed graph G = (V,E) and a starting set S ⊆ V, cyclesfrom(G,S) outputs sequences (v0, v1, ..., vk-1) where v0 ∈ S, each edge vi → v(i+1) ∈ E (indices modulo k), and the cycle satisfies certain repetition constraints (typically simple cycles where all vertices are distinct). Implementations often use depth-first search with a stack to avoid duplicates, sometimes incorporating Johnson-like enumeration ideas, and may prune searches using strongly connected components or other structural properties.

Algorithmic notes include scalability considerations: cyclesfrom can enumerate all elementary cycles from the specified starting points, or a restricted subset based on length, repetition allowances, or isomorphism avoidance. The worst-case complexity is exponential, so practical use depends on constraints and data structure optimizations.

Applications of cyclesfrom span network analysis, circuit testing, dependency and workflow evaluation, chemical graph theory, and routing scenarios. By focusing on a designated source set, cyclesfrom helps identify feedback loops, potential bottlenecks, and cycle-based behaviors in complex systems.

Example: in a digraph with edges A→B, B→C, C→A, A→D, D→A and starting set S = {A}, cyclesfrom would report cycles A→B→C→A and A→D→A, subject to the specified cycle constraints. See also cycle detection, Johnson’s algorithm, and cycle basis concepts.