Edge-connectivity of undirected and directed hypergraphs

Chapter 5
Partition-connectivity of hypergraphs
5.1 Introduction
In Chapter 1, the notion of k-partition-connectivity was introduced for graphs. In this chap-
ter, we discuss an extension of this notion to hypergraphs. While 1-partition-connectivity
for graphs is simply equivalent to connectivity, the extension for hypergraphs defines a dif-
ferent type of connectivity property. In this section we define this extension and describe
some of its basic characteristics. The results of the chapter are joint results with András
Frank and Matthias Kriesell [37].
5.1.1 A different approach to connectivity
Recall that an equivalent characterization of connectivity of graphs is that in order to
dismantle the graph into t + 1 components, one has to delete at least t edges. The k-
partition-connectivity of graphs can also be defined from this point of view: in order to
dismantle the graph into t + 1 components, one has to delete at least kt edges. We start
with the introduction of this type of characterization for hypergraphs..
A hypergraph H = (V, E) is called partition-connected if one has to delete at least t
hyperedges to dismantle it into t + 1 components for every t. An equivalent formulation
is that eH(F) ≥ |F| − 1 holds for every partition F of V . Partition-connectivity clearly
implies connectivity; contrary to the graph case, however, a connected hypergraph is not
necessarily partition-connected (a partition-connected hypergraph must have at least |V |−1
hyperedges).
Theorem 1.5 of Tutte characterized graphs that can be decomposed into k connected
spanning subgraphs. In light of the previous remarks, at least two different generaliza-
tions are possible for hypergraphs. We say that a sub-hypergraph (V, E ′ ) of a hypergraph
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