Edge-connectivity of undirected and directed hypergraphs

16 Chapter 1. Introduction and preliminaries
A digraph D is called k-rooted-connected from root s if it has the above properties. It
is easy to see that a graph G has a k-rooted-connected orientation from a given node
s if and only if it is k-partition-connected. Indeed, if G is k-partition-connected, then
by Proposition 1.9 it contains k edge-disjoint spanning trees, which can be oriented so
as to obtain k edge-disjoint arborescences rooted at s. Conversely, if D is a k-rooted-
connected orientation of G, then by Proposition 1.10 it contains k edge-disjoint spanning
arborescences, thus G contains k edge-disjoint spanning trees.
A common generalization of k-edge-connectivity and k-rooted-connectivity of digraphs
can be formulated easily. Given a fixed root node s and positive integers k, l, a digraph D
is called (k, l)-edge-connected from root s if it contains k edge-disjoint paths from s to any
other node, and l edge-disjoint paths to s from any other node. Equivalently, ϱD(X) ≥ k
for every non-empty subset X of V − s, and ϱD(X) ≥ l for every proper subset X of V
containing s. Obviously (k, k)-edge-connectivity corresponds to k-edge-connectivity, while
(k, 0)-edge-connectivity corresponds to k-rooted-connectivity.
It should be noted that a graph G has a (k, l)-edge-connected orientation from a given
root s if and only if it has a (k, l)-edge-connected orientation from any root. To see this,
suppose that D is a (k, l)-edge-connected orientation of G with root s1, and s2 is the desired
root. It suffices to see the case k ≥ l (since reversing all edges of D switches the role of k
and l). By definition there are k edge-disjoint paths from s1 to s2. Let us reverse the edges
on k − l of these paths. Then ϱD(X) decreases by k − l if X is an s1s2-set, it increases
by k − l if X is an s2s1-set, and remains unchanged otherwise. So the new orientation is
(k, l)-edge-connected from s2.
Chapter 6 will discuss (k, l)-edge-connected orientations in more detail.
1.3 Hypergraphs and directed hypergraphs
Since the thesis discusses hypergraphs and directed hypergraphs from the perspective of
connectivity and orientations, this introduction focuses mainly on these themes.
1.3.1 Connectivity of hypergraphs
Graphs are considered with the possibility of loops and parallel edges, and the same ap-
proach is used for hypergraphs. A hypergraph is denoted as H = (V, E), where V is the
set of nodes and E is the set of hyperedges. Hyperedges are considered to be multisets. To
a hyperedge e we associate the characteristic function χe : V → Z+, i.e. χe(v) equals the
multiplicity of the node v in the hyperedge e. Some special notations will be used when
describing the relation of a hyperedge e and a set X: