Edge-connectivity of undirected and directed hypergraphs

112 Chapter 6. Hypergraph orientation
The end of the proof is the same as for Theorem 6.11. We define the partition FY :=
{Y } ∪ {{v} : v ∈ V − Y } for every set Y ⊂ V . The conditions of the theorem imply that
x(Y ) = |E| − x(V − Y ) = |E| + p ′ (Y ) −
Z∈FY
≥ |E| − eH(FY ) + p ′ (Y ) = iH(Y ) + p ′ (Y ).
p ′ (Z)
Thus x(Y ) ≥ iH(Y )+p ′ (Y ) for every set Y ⊆ V , and by Lemma 6.10 there is an orientation
with in-degree vector x that covers p ′ , hence it covers p.
6.5.2 Characterization of (k, l)-partition-connectivity (k < l)
An interesting corollary of Theorem 6.16 is a characterization of (k, l)-partition-connected
hypergraphs for k < l.
Corollary 6.19. A hypergraph H = (V, E) is (k, l)-partition-connected for k < l if and
only if for every pair s, t ∈ V it has a k-edge-connected orientation with l edge-disjoint
paths from s to t.
Proof. It is easy to check that if for every pair s, t ∈ V the hypergraph H has a k-edge-
connected orientation with l edge-disjoint paths from s to t, then eH(F) ≥ k(|F| − 1) + l
must hold for every nontrivial partition F, since we can always choose s and t to be in
different members of the partition.
To prove the other direction, let H be a (k, l)-partition-connected hypergraph. Then
Theorem 6.16 implies that for every pair s, t ∈ V it has a k-edge-connected orientation
with l edge-disjoint paths from s to t.
Corollary 6.19 gives a method (albeit a relatively complicated one) for showing someone
that a given hypergraph is (k, l) partition-connected for some k < l. For every pair of
nodes s, t ∈ V , we can give a k-edge-connected orientation that contains l edge-disjoint
paths from s to t, and this property can be verified in polynomial time.
6.5.3 Characterization of (2k + 1)-edge-connected graphs
A simple observation shows that for graphs the condition of Theorem 6.16 can be further
simplified: instead of partitions, it suffices to consider cut conditions.
Theorem 6.20. Let G = (V, E) be an undirected graph, with s, t ∈ V designated nodes,
and let k, k1, k2 be positive integers for which k1, k2 ≥ k. Then G has a k-edge-connected
orientation such that there are k1 edge-disjoint paths from s to t and k2 edge-disjoint paths