Edge-connectivity of undirected and directed hypergraphs

128 Chapter 7. Combined augmentation and orientation
for every X ⊆ V and every tree-composition F of X. Using (1.12) and the fact that
eH0(F) = eH0(F + {V − X}), this is equivalent to the condition of the theorem.
From here we can follow the line of the proof of Theorem 7.1. Let H ′ 0 be the orientation
of H ′ 0 satisfying (7.5)–(7.7), and let H0 denote the directed hypergraph obtained from H ′ 0
by deleting the node z. Let mi(v) be the multiplicity of the edge zv in H ′ 0, and mo(v) the
multiplicity of the edge vz in H ′ 0. We define the set function q(X) = (p(X) − ϱ H0 (X))+ ;
then q is positively crossing supermodular. As in the proof of Theorem 7.1, we can apply
Theorem 4.7 (with the mi, mo and q defined above) to obtain a directed hypergraph D
whose underlying undirected hypergraph H is a good augmentation of H0. Theorem 4.7
also ensures that (7.25) can be satisfied. This concludes the proof of Theorem 7.13.