Edge-connectivity of undirected and directed hypergraphs

The objective of the thesis is to discuss edge-connectivity and related connectivity conceptsin the context of undirected and directed hypergraphs. In particular, we focus on k-edge-connectivity and (k, l)-partition-connectivity of hypergraphs, and (k, l)-edge-connectivityof directed hypergraphs. A strong emphasis is placed on the role of submodularity in thestructural aspects of these problems.One area that is discussed extensively is connectivity augmentation. A min-max theoremis given on the minimum number of ν-hyperedges that have to be added to an initialhypergraph to make it k-edge-connected. Analogously, we prove a formula on the minimumnumber of (r, 1)-hyperarcs whose addition makes an initial directed hypergraph (k, l)-edge-connected. These problems (and most others in the thesis) are studied in the generalframework of covering supermodular set functions.We show that matroid techniques can be used in the description of (k, l)-partition-connected hypergraphs. This notion also leads to connectivity orientation problems forhypergraphs, and with these tools we prove characterizations of (k, l)-partition-connectivityand (k, l)-edge-connected orientations. An application concerning edge-disjoint Steinertrees is also given, as well as some new results on directed network design with orientationconstraints.The thesis is concluded with the study of a new class of connectivity augmentationproblems, in which the aim is to add hyperedges to an undirected (or mixed) hypergraphsuch that the resulting hypergraph has an orientation with specified connectivity properties.A special case of the described results is a solution of the (k, l)-partition-connectivityaugmentation problem.The above results are based on the papers [35], [36], [37], [51], and [52]. The thesis alsoincludes a new characterization of set functions defining base polyhedra.