Edge-connectivity of undirected and directed hypergraphs

Section 6.2. Orientation of graphs 97
both hold for every partition F of V . If p is monotone decreasing or symmetric then it
suffices to require the validity of (6.1).
The necessity of these conditions is obvious if we recall (1.7), and that eG(F) = eG(co(F))
if G is a graph (but this is not true for hypergraphs!). The result on (k, l)-edge-connected
orientations follows from this theorem by considering the following set function for an
arbitrary node s ∈ V :
⎧
⎪⎨
k if ∅ = X ⊆ V − s,
pkl(X) := l
⎪⎩ 0
if s ∈ X ⊂ V ,
otherwise.
(6.3)
It is easy to see that pkl is non-negative, monotone decreasing if k ≥ l, and crossing
supermodular.
For orientations satisfying local edge-connectivity requirements, the problem is much
more difficult. Let G = (V, E) be a graph, and let r : V 2 → Z+ be a local edge-connectivity
requirement function for which r(v, v) = 0 (v ∈ V ). If r is symmetric, i.e. r(u, v) =
r(v, u) for every u, v ∈ V , then an obvious necessary condition for the existence of a good
orientation is that
λG(u, v)
r(u, v) ≤
2
for every u, v ∈ V .
The orientation theorem of Nash-Williams [61] states that this condition is sufficient for
symmetric r:
Theorem 6.4 (Nash-Williams [61], strong form). Let G = (V, E) be an arbitrary
graph. There exists an orientation G = (V, E) of G such that
λG(u, v)
λG (u, v) ≥
for every u, v ∈ V ,
2
and
|ϱ G (v) − δ G (v)| ≤ 1 for every v ∈ V .
This theorem settles the case when r is symmetric. However, deciding whether there is
an orientation satisfying a general local edge-connectivity requirement r is NP-complete.
The following is a sketch of the reduction of 3-SAT (see Figure 6.1).
Consider a collection C of clauses, and construct the following graph G. For every pair
{x, x} of complementary literals, create two nodes vx and vx, and an edge vxvx. For each
clause c ∈ C, add nodes sc, tc, wc, zc; for each literal y ∈ c, add edges vysc, vytc, vywc, and