Augmenting Hypergraphs to Meet Local Connectivity Requirements

We consider aspects of vertex splitting and connectivity augmentation in hypergraphs. In the splitting problem we are given a hypergraph G = (V + s, E) in which s is only incident with edges of size two and asked to “split” s in such a way that we preserve the local-connectivity in the set V . We develop results that lead to a characterisation of the hypergraphs in which there is no such split. In the augmentation problem we are given a hypergraph H and a requirement function r : V 2 → N. We are asked to find a set of new edges F , such that in H + F the local connectivity between every pair of vertices x, y is at least r(x, y). We provide a sharp upper bound on the minimum cardinality of F , in the case when F must consist entirely of size-two edges, and describe how to produce an augmenting set that stays within this upper bound. We also provide a refinement of a result due to Szigeti that gives the minimum value of ∑ e∈F |e| such that the connectivity requirement is met in H + F . Here we show that there is always a set of edges that meets the connectivity requirement, meets Szigeti’s minimum, and contains at most one edge of size greater than two.