Orienting cycle elements in orientable rotation systems Original Research Article

The natural orientation that facial walks inherit from one of the two senses of an orientable rotation system cannot be extended to all cycles of the embedding, even for genus zero. We explore the limits of such extensions, which we call bifurcating cycle elements (of the cycle space of the embedded graph). We apply this purely combinatorial theory to show the structure (previously known by homology) of the cycle space of an embedded graph.