Edge-connectivity of undirected and directed hypergraphs

Section 5.2. Tree-reducible hypergraphs 91
There is a difference in the structural properties of k-partition-connected hypergraphs,
and (k, l)-partition-connected hypergraphs in general. Namely, Theorem 5.7 implies that a
k-partition-connected hypergraph is always reducible to a k-partition-connected graph, in
the sense used in the definition of forest-reducibility. However, a (k, l)-partition-connected
hypergraph is not necessarily reducible to a (k, l)-partition-connected graph. For exam-
ple, the hypercircuits on Figure 5.2 are not reducible (of course there are also reducible
hypercircuits).
If we want to prove someone that a given hypergraph is k-partition-connected, we can do
so by showing its reduction to a k-partition-connected graph, which contains k edge-disjoint
spanning trees. But how can we prove that a given hypergraph is (k, l)-partition-connected?
An answer to this question will be given in Chapter 6.
5.2.5 Covering by hyperforests
The problems discussed so far in this chapter were generalizations of problems that involve
the packing of trees. It may be asked how problems related to covering with forests
generalize to hypergraphs. An extension of Theorem 1.6 of Nash-Williams on partitioning
a graph into k forests easily follows from Theorem 5.6.
Theorem 5.9. The edge-set E of a hypergraph H = (V, E) can be decomposed into k
hyperforests if and only if
holds for every nonempty subset X of V .
iH(X) ≤ k(|X| − 1) (5.5)
Proof. Since one hyperforest may contain at most |X| − 1 hyperedges induced by X, (5.5)
is necessary. To see the sufficiency, let F := {V1, . . . , Vt} be a partition of V . By (5.5) each
member Vi induces at most k(|Vi|−1) hyperedges, so the number eH(F) of cross-hyperedges
is at least |E| − t
j=1 k(|Vj| − 1) = |E| − k(|V | − t). Therefore k(|V | − |F|) + eH(F) ≥ |E|
holds for every partition F of V , which means that E is independent in the matroid MkH
by (5.4), and H decomposes into k hyperforests.
The problem of connectivity augmentation with respect to (k, l)-partition-connectivi-
ty will be discussed in Chapter 7. Here we address the much easier analogous problem
concerning covering with hyperforests: add hyperedges to a given hypergraph H0 = (V, E0)
such that the resulting hypergraph can still be decomposed into k hyperforests. It is a
simple observation that even the maximum weight problem is solvable (assuming that
the number of hyperedges with positive weight is polynomial), by the matroid techniques