Edge-connectivity of undirected and directed hypergraphs

Section 1.3. Hypergraphs and directed hypergraphs 21
Proposition 1.14 ([36]). Let D = (V, A) be a directed hypergraph, and S1, . . . , Sk non-
empty subsets of V . For X ⊆ V , let f(X) denote the number of sets Si not disjoint from
X. Then D can be decomposed into directed sub-hypergraphs D1, . . . , Dk such that Di is
connected from Si if and only if
ϱD(X) ≥ k − f(X) for every ∅ = X ⊆ V .
Proof. We prove the theorem by induction on the number of hyperarcs of size at least 3.
If every hyperarc is a digraph edge, then we can use Theorem 1.8. Suppose that there is a
hyperarc a ∈ A with |a| > 2. Call a set ∅ = X ⊆ V tight if ϱD(X) = k − f(X). Note that
f(X) + f(Y ) ≥ f(X ∩ Y ) + f(X ∪ Y ),
for every X, Y ⊆ V , so (1.4) implies that if X ∩ Y = ∅ then the intersection and union
of tight sets is tight. Let F be the family of tight sets entered by a. If F = ∅ or F
has a unique maximal element X, then we can replace the hyperarc a by a directed edge
uh(a) where u is an arbitrary node in a − X, and use induction. If F has at least two
maximal elements, say X and Y , then a cannot enter X ∪ Y , since the union would also
be tight, which would contradict the maximality. But then dD(X, Y ) ≥ 1, so by (1.4)
ϱD(X ∩ Y ) + ϱD(X ∪ Y ) = ϱD(X) + ϱD(Y ) − dD(X, Y ) < 2k − f(X ∩ Y ) − f(X ∪ Y ), so
X ∩ Y or X ∪ Y would violate the condition.
As a consequence, the following are equivalent for a directed hypergraph (D = V, A)
and a fixed root node s ∈ V :
(i) D can be decomposed into k edge-disjoint spanning directed sub-hypergraphs that are
connected from s.
(ii) There are k edge-disjoint paths from s to any other node.
(iii) ϱD(X) ≥ k for every non-empty subset X of V − s.
A directed hypergraph D is called k-rooted-connected from root s if it has the above prop-
erties.
A directed hypergraph D is said to cover a set function p if ϱD(X) ≥ p(X) for every
X ⊆ V . Note that k-edge-connectivity and k-rooted-connectivity of directed hypergraphs
can be described as the covering of appropriate set functions.