Coverings and structure of crossing families

Let F be a symmetric and crossing family of subsets of V. Our first result is a min-max equality for the size of a smallest family K: of k-element sub sets of V which covers each member of F, where X is an element of F is said to be covered by Y is an element of K if X boolean AND Y not equal 0 not e qual Y - X. Our formula generalizes, among others, a recent result of Cheng on optimally augmenting the edge-connectivity of a hypergraph by one. The second problem we consider is to find a compact representation of F. We prove that there exists a so-called hypercactus K of size O(\V\), consisti ng of cycles and (hyper)edges arranged in a tree-like manner, and a mapping from V to V(K) in such a way that there is a bijection between the minimum cuts of K and the members of F. If F corresponds to the minimum cuts of a k-edge-connected graph then K reduces to the well-known cactus representati on of minimum cuts due to Dinitz et al.