Edge-connectivity of undirected and directed hypergraphs

70 Chapter 4. Connectivity augmentation of directed hypergraphs
so the following theorem of Frank [24] can be applied to solve the degree-constrained
problem:
Theorem 4.1 (Frank [24]). Let p : 2 V → Z+ be a positively crossing supermodular set
function; let mi : V → Z+ be an indegree-specification and mo : V → Z + an outdegree-
specification such that mi(V ) = mo(V ). There exists a digraph D = (V, A) such that
ϱD(v) = mi(v) and δD(v) = mo(v) ∀v ∈ V , and ϱD(X) ≥ p(X) ∀X ⊆ V if and only if
and
mi(X) ≥ p(X) for every X ⊆ V ,
mo(V − X) ≥ p(X) for every X ⊆ V .
It should be noted that for the p defined above this result can be obtained from the
directed splitting-off theorem of Mader [59], which is generalized by the above result of
Frank.
The degree-specified result easily implies the following theorem on minimum cardinality
augmentation:
Theorem 4.2 (Frank [24]). Let p : 2 V → Z+ be a positively crossing supermodular set
function, and γ a non-negative integer. There exists a digraph D = (V, A) with γ edges
that covers p if and only if
and
γ ≥
p(X) for every partition F of V ,
Z∈F
γ ≥
p(V − X) for every partition F of V .
Z∈F
The minimum cost k-edge-connectivity augmentation problem for digraphs is NP-comp-
lete even for k = 1, since it includes the problem of determining whether a digraph has a
Hamiltonian cycle.
For undirected graphs, the local edge-connectivity augmentation problem was solvable
[29] using Theorem 3.3 of Mader on splitting-off preserving local edge-connectivity. How-
ever, no such result can be expected for digraphs: local edge-connectivity augmentation of
directed graphs is NP-complete, even in the special case when the requirement is 1 between
the nodes of a specified subset T and 0 otherwise (see [29]).