Edge-connectivity of undirected and directed hypergraphs

Abstract
The objective of the thesis is to discuss edge-connectivity and related connectivity concepts
in 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-connectivity
of directed hypergraphs. A strong emphasis is placed on the role of submodularity in the
structural aspects of these problems.
One area that is discussed extensively is connectivity augmentation. A min-max theorem
is given on the minimum number of ν-hyperedges that have to be added to an initial
hypergraph to make it k-edge-connected. Analogously, we prove a formula on the minimum
number 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 general
framework 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 for
hypergraphs, and with these tools we prove characterizations of (k, l)-partition-connectivity
and (k, l)-edge-connected orientations. An application concerning edge-disjoint Steiner
trees is also given, as well as some new results on directed network design with orientation
constraints.
The thesis is concluded with the study of a new class of connectivity augmentation
problems, in which the aim is to add hyperedges to an undirected (or mixed) hypergraph
such 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-connectivity
augmentation problem.
The above results are based on the papers [35], [36], [37], [51], and [52]. The thesis also
includes a new characterization of set functions defining base polyhedra.
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