Edge-connectivity of undirected and directed hypergraphs

58 Chapter 3. Edge-connectivity augmentation of hypergraphs
For a partition F, let s(F) denote the number of special singleton members of F. A critical
partition F is called proper if s(F) ≥ 3. Critical partitions have the following properties:
Claim 3.16. If F = {V1, . . . , Vl} is a critical partition, then 2l − 2 ≥ m(V ), thus s(F) ≥
m(Vl). In particular, s(F) ≥ 2 for every critical partition, and the partition is proper if
m(Vl) ≥ 3.
Proof. The partition is critical and
m(V ) ≤ ν
since p-fullness implies that l > ν.
m(V )
ν
l − 2
+ 1
ν − 1
is an integer, so
≤ 2ν + 2(l − ν − 1) = 2l − 2,
= 2ν + ν
(l − ν − 1) ≤
ν − 1
Claim 3.17. A partition {V1, . . . , Vl} is critical if and only if l > ν, l−1
ν−1
> m(V )
ν
− 1,
p(V1) = 1, and p(V1 ∪ Vi) ≥ 1 (i = 2, . . . , l). If the partition is critical and U is the union
of some partition members such that V1 ⊆ U and V2 ∩ U = ∅, then p(U) = 1.
Proof. Let {V1, . . . , Vl} be a partition with the above properties, and let U = ∪i∈IVi where
1 ∈ I and |I| ≤ l −1. We can use inequality (3.7) for the sets {V1 ∪Vi : i ∈ I} to show that
p(U) > 0. The symmetry of p implies that p(V − U) > 0 for all such U, so the partition is
p-full. If V1 ⊆ U and V2 ∩ U = ∅, then (3.6) gives 1 = p(V1) ≤ p(U) ≤ p(V − V2) = 1.
Claim 3.18. Let e be a ν-hyperedge which satisfies (3.14)–(3.16). Then m e and p e satisfy
(3.11) if and only if
e ⊆ X for any member X of any proper critical partition. (3.17)
Proof. If e ⊆ X for a member X of a critical partition, then the partition remains p-full
after the splitting off of e, hence the partition becomes deficient. To see the converse,
observe that only critical partitions can become deficient partitions after the splitting-off.
If e ⊆ X for every member X of a given critical partition F, then it is easy to see that
there exists a set U entered by e which is the union of some members of F including
exactly 1 special singleton member. According to Claim 3.17, p(U) = 1, so p e (U) = 0 and
F is not p-full after the splitting-off. It remains to show that if e ⊆ X for a member X
of a non-proper critical partition, then some set violates (3.14) or (3.15). But m(X) ≤ 2
according to Claim 3.16, so 0 = m e (X) < p(X) = p e (X) which by Claim 3.15 implies that
(3.14) or (3.15) is violated for some subset of X.