Edge-connectivity of undirected and directed hypergraphs

80 Chapter 4. Connectivity augmentation of directed hypergraphs
s
Figure 4.3: Counterexample for Claim 4.11 for k = 1, l = 3. Inequality (4.9) on the
indicated copartition gives γ ≥ 2, while
Z∈F p(Z) ≤ 1 for every partition F.
tion:
The counterexample on Figure 4.3 shows that the claim is not necessarily true if k < l.
Using Claim 4.11, Theorem 4.8 implies the following on minimum total size augmenta-
Theorem 4.12. Let D0 = (V, A0) be a directed hypergraph, let k ≥ l be non-negative
integers, and let p be the set function defined in (4.10). The minimum total size σ of a
directed hypergraph D for which D0 + D is (k, l)-edge-connected from root s is σ = σ1 + σ2,
where
σ1 = max p(X) : F is a partition of V ,
Z∈F
σ2 = max max p(X),
p(V − X) : F is a partition of V .
Z∈F
Z∈F
Theorem 4.13. Let D0 = (V, A0) be a directed hypergraph, let k < l be non-negative
integers, and let p be the set function defined in (4.10). The minimum total size σ of a
directed hypergraph D for which D0 + D is (k, l)-edge-connected from root s is σ = σ1 + σ2,
where
1
σ1 = max max p(X),
|F| − 1
Z∈F
(
p(V − X)) : F is a partition of V ,
Z∈F
σ2 = max max p(X),
p(V − X) : F is a partition of V .
Z∈F
Z∈F
For minimum cardinality augmentation using (r, 1)-hyperedges, the following results are
obtained from Theorem 4.9 and Claim 4.11: