Edge-connectivity of undirected and directed hypergraphs

Section 4.4. (k, l)-edge-connectivity of directed hypergraphs 79
is k-rooted-connected, then it can be shrinked (meaning that we delete all but one tail-
nodes of every hyperarc) to a k-rooted connected digraph. For (k, l)-edge-connectivity no
similar statement is true; in fact, Bang-Jensen and Thomassé [5] observed the following:
Proposition 4.10 ([5]). It is NP-complete to decide whether a strongly connected directed
hypergraph can be shrinked to a strongly connected digraph.
Proof. Let D = (V, A) be a strongly connected digraph. We define a directed hypergraph
D ′ = (V, A) the following way: A contains one hyperarc av for every v ∈ V , namely
h(av) = v, and t(av) = {u ∈ V : uv ∈ A}. Then D ′ is strongly connected, and it can be
shrinked to a strongly connected digraph if and only if D contains a Hamiltonian cycle.
4.4.2 (k, l)-edge-connectivity augmentation
Let D0 = (V, A0) be a directed hypergraph, and let k, l be non-negative integers. We
are interested in the (k, l)-edge-connectivity augmentation of D0, either with the objective
of minimizing the total size of the new hyperarcs, or with the objective of adding the
minimum number of (r, 1)-hyperarcs . To solve these problems using the results in Section
4.3, we define the following set function:
⎧
⎪⎨
(k − ϱD0(X))
p(X) :=
⎪⎩
+ if ∅ = X ⊆ V − s,
(l − ϱD0(X)) + 0
if s ∈ X ⊂ V ,
if X = ∅ or X = V .
(4.10)
The set function p is positively crossing supermodular, so we can apply Theorems 4.8
and 4.9. If k ≥ l, then the conditions of the theorems can be simplified:
Claim 4.11. Let k ≥ l, and let p be the set function defined in (4.10). If
γ ≥
p(Z)
for every partition F of V , then
for every partition F.
Z∈F
(|F| − 1) γ ≥
p(V − Z)
Proof. Suppose that (t − 1)γ <
Z∈F p(V − Z) for a partition F = {X1, . . . , Xt} of V
with t ≥ 2. If p(V − Xt) = 0, then
Z∈F p(V − Z) = t−1
i=1 p(V − Xi) ≤ t−1
i=1 (p(Xi) +
p(V − Xi)) ≤ (t − 1)γ, by considering the two-member partitions {Xi, V − Xi}. So we can
Z∈F
assume that p(V −Xi) > 0 for every i, thus
Z∈F p(V −Z) = k+(t−1)l− t
i=1 δD0(Xi) ≤
k + (t − 1)l − t
i=1 ρD0(Xi) ≤ (t − 1)k + l − t
i=1 ρD0(Xi) ≤ t
i=1 p(Xi) ≤ γ.