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 subsets of V which covers each member of ℱ, where 𝒳 ∈ ℱ is said to be covered by Y ∈ 𝒦 if X ∩ Y ≠ Ø ≠ 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|), consisting 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 representation of minimum cuts due to Dinitz et al.
