circuit rank of , which is the dimension of the cycle space. Each nonempty subset of cycles has a vector sum that represents the boundary of the union of the bounded faces in the subset, which cannot be empty (the union includes at least one bounded face and excludes the unbounded face, so there must be some edges separating them). Therefore, there is no subset of cycles whose vectors sum to zero, which means that all the cycles are linearly independent. As a linearly independent set of the same size as the dimension of the space, this collection of cycles must form a basis. Necessity of planarity when a 2-basis exists provided the following simple argument for the other direction of the characterization, based on Wagner's theorem characterizing the planar graphs by forbidden minors. As O'Neill observes, the property of having a 2-basis is preserved under graph minors: if one contracts an edge, the same contraction may be performed in the basis vectors, if one removes an edge that has a nonzero coordinate in a single basis vector, then that vector may be removed from the basis, and if one removes an edge that has a nonzero coordinate in two basis vectors, then those two vectors may be replaced by their sum (modulo two). Additionally, if is a cycle basis for any graph, then it must cover some edges exactly once, for otherwise its sum would be zero (impossible for a basis), and so can be augmented by one more cycle consisting of these singly-covered edges while preserving the property that every edge is covered at most twice. However, the complete graph has no 2-basis: is six-dimensional, each nontrivial vector in has nonzero coordinates for at least three edges, and so any augmented basis would have at least 21 nonzeros, exceeding the 20 nonzeros that would be allowed if each of the ten edges were nonzero in at most two basis vectors. By similar reasoning, the complete bipartite graph has no 2-basis: is four-dimensional, and each nontrivial vector in has nonzero coordinates for at least four edges, so any augmented basis would have at least 20 nonzeros, exceeding the 18 nonzeros that would be allowed if each of the nine edges were nonzero in at most two basis vectors. Since the property of having a 2-basis is minor-closed and is not true of the two minor-minimal nonplanar graphs and ,