Augmenting hypergraphs by edges of size two

Abstract

We give a good characterization for the minimum number of edges of size two whose addition to a given hypergraph H makes it k-edge-connected. Our result extends a previous theorem by E. Cheng [4] for the case when H is already (k - 1)-edge-connected. We also describe a strongly polynomial algorithm to find both a minimum cardinality augmentation, and a minimum cost augmentation where the cost of an edge is equal to the sum of the weights of its endvertices, for some given weight function on V(H). Our proof is based on a new ‘splitting’ theorem for hypergraphs for the case when the special vertex s is only incident with edges of size two. This theorem generalizes a ‘splitting’ result of Lovász for graphs.