Edge-connectivity of undirected and directed hypergraphs

62 Chapter 3. Edge-connectivity augmentation of hypergraphs
Proof. We can assume that |K ∗ | ≥ 3, otherwise there are no proper critical partitions.
Claim 3.22 implies that K ∗ is either disjoint from or subset of any non-singleton set in B1,
so the construction of e ′ gives that |e ′ ∩ K ∗ | is the minimum of the following three values:
|K ∗ |, (3.18)
ν − 1, (3.19)
min
K∗ (m(X) − p(X) + 1). (3.20)
⊆X∈B1
Suppose that a proper critical partition F becomes deficient after the splitting off of e.
Let Vi be the member of F for which e ⊆ Vi; then |K ∗ ∩ Vi| ≥ |e ′ ∩ K ∗ | > 0. If K ∗ ⊆ Vi
and m(Vi − K ∗ ) > 0, then s(F) < m(Vi), given that by Claim 3.23 the special singleton
members are in the same component of G whose size is at most |K ∗ |. This contradicts the
criticality of F according to Claim 3.16.
If K ∗ ⊆ Vi and m(Vi − K ∗ ) = 0, then the set Z := {v ∈ Vi : v ∈ e} is a subset
of K ∗ , hence p({u, v}) = 1 for every u, v ∈ Z. Let v ∈ Z; by using (3.7) on the sets
{{v, u} : u ∈ Z − v} we get that p(Z) > 0. But then m e (Z) < p e (Z), so by Claim 3.15 a
subset of Z violates (3.14) or (3.15).
Now suppose that K ∗ ⊆ Vi, and let FS denote the S-refinement of F. By Claim 3.23,
K ∗ is the set of special singletons that are members of FS (and thus the special singleton
members of F are in K ∗ ). If e ′ ⊆ K ∗ , then there are at least ν − 1 nodes in Vi that
are special singleton members of FS, so |FS|−1
ν−1
≥ |F|+(ν−1)−1
ν−1
criticality of F, which means that FS would violate (3.11).
≥ |F|−1
ν−1
+ 1 > m(V )
ν
by the
If |e ′ ∩ K ∗ | is determined by (3.20), then there is a set X ∈ B1 such that K ∗ ⊆ X,
Vi ⊆ X, and |K ∗ ∩ Vi| ≥ m(X) − p(X) + 1. Let U denote the union of the members of
F − {Vi} that are not special singletons. Then s(F) = |K ∗ − U| − |K ∗ ∩ Vi| ≤ |K ∗ −
U| − m(X) + p(X) − 1 < p(X) − m(X ∩ U). By Claim 3.16, this would imply that
m(Vi) < p(X) − m(X ∩ U); we show that this is impossible. If U ⊆ X, then m(Vi) ≥
p(Vi−X) = p(V −X) = p(X) ≥ p(X)−m(X ∩U). If U ⊆ X, then by using (2.2) on X and
V1 ∪ U we get that m(Vi) ≥ p(Vi − X) = p(U ∪ X) ≥ p(X) + p(V1 ∪ U) − p(X ∩ (V1 ∪ U)) ≥
p(X) + 1 − m(X ∩ (V1 ∪ U)) = p(X) − m(X ∩ U).
Lemma 3.24 implies that condition (3.17) can be ignored when choosing an appropriate
node w ∈ W . We have already seen that e = e ′ + w ∈ Q is a necessary condition for the
feasibility of the splitting-off. In addition to that, e ⊆ X must hold for every set X ∈ B2;
however, if e ′ ⊆ X or there is a set Y ∈ B1 such that X ⊆ Y and m(Y ) − p(Y ) = ν − 2,
then this automatically follows from e = e ′ + w ∈ Q. So we call a set X ∈ B2 critical if
e ′ ⊆ X and there is no Y ∈ B1 such that X ⊆ Y and m(Y ) − p(Y ) = ν − 2. A ν-hyperedge