cyclindependent

Cyclindependent is a concept in graph theory describing a particular independence property among cycles in a graph. A collection C of simple cycles in a graph G is called cyclindependent if every cycle c in C contains at least one edge that is not contained in any other cycle of C. In other words, each cycle has a private edge that no other cycle in the collection uses.

This private-edge condition has a clear algebraic consequence. The incidence vectors of the cycles in a cyclindependent set are linearly independent over the two-element field GF(2). Therefore, cyclindependent families provide a readily verifiable route to constructing independent sets within the cycle space of a graph, which is the vector space formed by edge-disjoint sums of cycles modulo 2.

Origin and usage: The term cyclindependent was coined in discussions of cycle bases and network decomposition, with attention to identifying structurally distinct cyclic components in complex networks. The concept is used to simplify analysis in contexts such as electrical grids, transportation systems, and molecular graphs, where overlapping cycles can complicate computations.

Properties: Key properties include the requirement of a private edge for each cycle and the automatic implication of linear independence in the GF(2) cycle space. The size of a cyclindependent family is bounded by the graph’s edge set, and constructing large cyclindependent collections depends on the graph’s topology.

Examples and applications: In a ladder-like graph, one can select face cycles that each rely on a frame edge not present in other cycles to obtain a cyclindependent set. Practically, identifying cyclindependent cycles aids modular analysis and fault isolation in networks.

See also: cycle space, cycle basis, independent set, graph theory.