When Do Short Cycles Generate the Cycle Space?

Let G = (V, E) be a graph with arbitrary (perturbed) edge weights and let C(e) denote the shortest cycle containing the edge e. It is easy to show that the cycles in {C(e) | e ∈ E} are not only independent (over GF(2)) but are also contained in the cycle basis of minimum weight. We characterize, in several ways, those graphs for which {C(e) | e ∈ E} is a cycle basis (hence, the cycle basis of minimum weight) for every perturbed edge weighting. For example, these are the planar graphs such that no dual graph has two non-adjacent nodes connected by three internally node-disjoint paths. Another characterization shows that these graphs can be obtained from cycles, bonds, and K4′s by a special type of 2-sum operation; this leads to a linear time recognition algorithm for this class.