Abstract: "Let G be an undirected k-edge connected graph. In this paper we give a representation for all edge cuts with capacity not exceeding roughly (6/5)k. This generalizes the cactus representation (Dinits et al) for all minimum cuts. Karger proved that the number of cuts within a multiplicative factor α of the connectivity is at most O(n[superscript 2α]). We improve this bound to O(n2) for α=6/5. An important corollary of our result is a proof with new insights to the Lovaśz splitting theorem. A splitting of the edge pair us and vs at vertex s means replacing the two edges by uv. A splitting is admissible if it preserves the minimum local edge connectivity of the graph apart from vertex s. In other words, we may split us and vs if there is no set of degree minimum or minimum+1 containing both u and v; this property can be checked by our representation. Our new technique makes it possible to derive structure results of admissible pairs; among others we can show that splittable pairs form a connected graph unless the degree of s is odd or equals 4. We believe that by using our representation it will be possible to improve on results using the cactus or splittings. One such result is the edge augmentation problem where one needs a minimum cardinality edge set which increases the connectivity of the graph. The algorithm of Naor, Gusfield and Martels applies the cactus representation to solve this problem; Frank's algorithm relies on the splitting theorems."