Edge-connectivity of undirected and directed hypergraphs

66 Chapter 3. Edge-connectivity augmentation of hypergraphs
We prove sufficiency by showing that if a p-full partition F = {V1, . . . , Vl} violates (3.23)
while conditions (3.24) and (3.25) are satisfied, then an appropriate E ′ 0 violates (3.26). Let
E ′ 0 be the set of hyperedges in E0 that enter at least one member of F, and let H ′ 0 := (V, E ′ 0).
Then dH ′ (U) ≤ k − 1 if U is the union of some (not all) members of F, since F is p-full.
0
The partition F violates (3.23), so (ν − 1)γ < l − 1 ≤ c(H0 − E ′ 0) − 1.
We claim that |E ′ 0| = k − 1, which then implies that (3.26) is violated. By (3.24), lk −
l
i=1 dH ′ 0 (Vi) = lk − l
i=1 dH0(Vi) ≤ νγ ≤ ν
ν−1 (l − 2) ≤ 2l − 4, from which l
i=1 dH ′ 0 (Vi) ≥
(k − 2)l + 4. This implies that |E ′ 0| ≥ k − 1, and there are at least 4 members of F (say
V1, V2, V3, V4), for which dH ′ 0 (Vi) = k − 1. We can assume that iH ′ 0 (V1 ∪ V2) ≤ iH ′ 0 (Vi ∪ Vj)
for every distinct i, j ∈ {1, 2, 3, 4}.
If iH ′ 0 (V1∪V2) > 0, then dH ′ 0 (V1∪V2) ≥ dH ′ 0 (V1)−iH ′ 0 (V1∪V2)+iH ′ 0 (V2∪V3)+iH ′ 0 (V2∪V4) ≥
k, contradicting the p-fullness of F. So iH ′ 0 (V1 ∪ V2) = 0, in which case there are k − 1
hyperedges in E ′ 0 that enter each of V1, V2, and V1 ∪ V2. Suppose that E ′ 0 contains a
hyperedge besides these k − 1, which enters a partition member Vi. Then dH ′ 0 (V1 ∪ Vi) ≥ k,
which would contradict the p-fullness of F. This proves that |E ′ 0| = k − 1.
3.4.5 Algorithmic aspects
It might be argued that Theorems 3.14 and 3.29 are not good characterizations, since it
is not possible to check in polynomial time whether a given partition is p-full, hence it
cannot be decided whether it is a deficient partition or not. Indeed, it is well known that
it is NP-complete to decide for given graph G = (V, E) whether there is a set X ⊆ V
with dG(X) ≥ k. Let F be the partition of V composed of singleton members, and let
p(X) := (k − dG(X)) + for ∅ = X ⊂ V and p(∅) = p(V ) := 0. Now F is a p-full partition
if and only if dG(X) < k for every X ⊆ V .
If a partition has at least one member Vi with p(Vi) = 1, then its deficiency can be checked
using the characterization in Claim 3.17. But in general, deciding whether a partition is
p-full or not is NP-complete. However, it is easy to see that if p(Vi) ≥ 2 for every member
of a deficient partition, then at least one of the partition members violates (3.9) (and the
partition violates (3.21) in case of Theorem 3.29). This means that Theorems 3.14 and
3.29 give good co-NP characterizations.
The proof of Theorem 3.14 provides a polynomial algorithm for the degree specified
problem if maximizing oracles are available for every set function of the form p − dH,
where H is an arbitrary hypergraph. In this case, a feasible splitting-off operation can be
found in polynomial time using the procedure described in the proof. However, we have to
introduce a kind of “multiple splitting-off” to ensure that the number of splitting-off steps
is polynomial.