Edge-connectivity of undirected and directed hypergraphs

Section 6.3. Orientation with supermodular requirements 103
hyperarcs. To prove the inequality, define the partition FY := {Y } ∪ {{v} : v ∈ V − Y }
for every set Y ⊂ V . Using (6.4) on the partition FY , we get
x(Y ) = |E| − x(V − Y ) = |E| + p(Y ) −
Z∈FY
≥ |E| − eH(FY ) + p(Y ) = iH(Y ) + p(Y ).
p(Z)
It should be noted that using Theorem 2.11 of Fujishige, a short alternative proof of
Theorem 6.11 can be given. Define the set function q(X) := p(X) + iH(X); then q is
crossing supermodular. If F = {X1, . . . , Xt} is a partition of V , then, by (6.4) and (1.12),
q(Xi) = p(Xi) + iH(Xi) ≤ eH(F) + iH(Xi) = |E| = q(V ), and q(V − Xi) =
p(V −Xi)+ iH(V −Xi) ≤ eH(co(F))+ iH(V −Xi) = (t−1)|E| = (t−1)q(V ). Thus
Theorem 2.11 implies that if the conditions (6.4) hold, then there is an integral vector x :
V → Z satisfying x(V ) = q(V ) = |E| and x(Y ) ≥ q(Y ) = iH(Y ) + p(Y ) ≥ iH(Y ) ∀Y ⊆ V .
By Lemma 6.10, H has an orientation with in-degree vector x, and this orientation covers
p.
Remark. The proofs show that Theorem 6.11 is true under the weaker assumption that p is
non-negative and p+iH is crossing supermodular. If p is monotone decreasing or symmetric,
then the co-partition type constraints are unnecessary, since
X∈F p(X) ≥ X∈co(F) p(X)
and eH(F) ≤ eH(co(F)) for every partition F.
6.3.4 Characterization of (k, l)-partition-connectivity (k ≥ l)
By applying Theorem 6.11 to the set function pkl defined in (6.3), we get a new characteri-
zation of (k, l)-partition-connected hypergraphs (as defined in Chapter 5) if k ≥ l . In this
case the set function pkl is monotone decreasing, so by taking into account the previous
remark, we obtain the following:
Corollary 6.12. Let k ≥ l be non-negative integers. Then a hypergraph has a (k, l)-edge-
connected orientation if and only if it is (k, l)-partition-connected.
Note that in the special case when l = 0, this gives an alternative proof for Theorem
5.7, since by Proposition 1.14 a k-rooted-connected orientation can be decomposed into k
rooted-connected directed sub-hypergraphs.
Corollary 6.12 also gives us a method by which we can show someone that a given
hypergraph is (k, l)-partition-connected if k ≥ l. All we have to do is to show a (k, l)-edge-
connected orientation, which is a property that can be checked in polynomial time.