Edge-connectivity of undirected and directed hypergraphs

120 Chapter 7. Combined augmentation and orientation
Condition (7.14) is equivalent to the inequality maxX⊆V q(X) ≤ σ; let us assume that this
holds. We may observe that if a degree-specification m : V → Z+ satisfies m(Y ) ≥ q(Y )
for every Y ⊆ V and m(V ) = σ, then m satisfies (7.1)–(7.3). Indeed, (7.1) for a partition
F follows if we consider m(Y ) ≥ q(Y ) with the choice of Y = V , F1 = F, and F2 = ∅ (
hence hY (F1) = 0 and hY (F2) = −1). Inequality (7.2) for a partition F and a member
X ∈ F is obtained by setting Y = V − X, F1 = F − {X} (a partition of Y ) and F2 = {Y }
(thus hY (F1) = 0 and hY (F2) = 0). Finally, (7.3) for a partition F, a subpartition F ′ ⊆ F
and X = ∪F ′ is obtained from m(Y ) ≥ q(Y ) with the settings Y = V − X, F1 = co(F ′ ),
and F2 = F − F ′ (in which case hY (F1) = |F1| − 1 and hY (F2) = 0).
By Theorem 7.1 the existence of such a degree-specification m implies the existence of
a hypergraph H that satisfies the requirements. To prove the existence of a good degree-
specification, we use the properties of a set function slightly different from q:
q ′ (X) := max{QX(F1, F2) : F1 and F2 are tree-compositions of X}.
Claim 7.6. The value QX(F1, F2) does not decrease if we remove a partition or a co-
partition of V from F1 or F2.
Proof. It is easy to see that if X ∩ Y = ∅, F X 1 , F X 2 are compositions of X, and F Y 1 , F Y 2 are
compositions of Y , then
QX(F X 1 , F X 2 ) + QY (F Y 1 , F Y 2 ) = QX∪Y (F X 1 + F Y 1 , F X 2 + F Y 2 ). (7.16)
The case Y = ∅ proves the claim, since q(V ) ≤ σ implies that q(∅) ≤ 0.
Claim 7.7. The set function q ′ is fully supermodular.
Proof. Let X, Y ⊆ V , and suppose that the maximum in the definition of q ′ is reached on
families F X 1 , F X 2 , and F Y 1 , F Y 2 , respectively. Let F1 := F X 1 + F Y 1 , F2 := F X 2 + F Y 2 . We
apply the following operations, as long as any of them is possible:
(i) If Z1, Z2 ∈ F1 are crossing, then replace them in F1 by Z1 ∩ Z2 and Z1 ∪ Z2.
(ii) If Z1, Z2 ∈ F2 are crossing, then replace them in F2 by Z1 ∩ Z2 and Z1 ∪ Z2.
Lemma 2.4 implies that after a finite number of steps, the resulting families F ′ 1 and
F ′ 2 become cross-free. By Lemma 2.3 F ′ i decomposes into a composition F X∩Y
i
of X ∩
Y and a composition F X∪Y
i of X ∪ Y (i = 1, 2); and all of these families are crossfree.
The crossing supermodularity of p implies that
Z∈F1
p(Z) ≤ Z∈F ′ p(Z) and
1