Edge-connectivity of undirected and directed hypergraphs

100 Chapter 6. Hypergraph orientation
6.2.4 Parity constrained orientation
A very exciting new class of connectivity orientation problems is obtained if there are
additional requirements involving the parity of the in-degrees in the oriented graph. These
problems are outside the scope of this thesis, so we just mention a few results.
Let G = (V, E) be a graph. For a subset T ⊆ V , a T -odd orientation of G is an
orientation where T is exactly the set of nodes with odd in-degrees. It is easy to see that
a T -odd orientation can exist only if |T | + |E| is even; such sets are called G-even. Frank,
Jordán and Szigeti proved the following on k-rooted-connected T -odd orientations:
Theorem 6.8 (Frank et al. [33]). Let G = (V, E) be a graph with a fixed root node
s ∈ V , let k be a positive integer, and let T ⊆ V be a G-even subset. For a partition F of
V , let odd(F) denote the number of members of F for which |X ∩ T | − iG(X) − k is odd.
Then G has a k-rooted-connected T -odd orientation if and only if
for every partition F of V with |F| ≥ 2.
k(|F| − 1) + odd(F) ≤ eG(F)
It is open (even for k = 1) whether graphs with a k-edge-connected T -odd orientation
can be characterized. However, Frank and Z. Király [31] managed to characterize graphs
that have a k-edge-connected T -odd orientation for every G-even subset T .
Theorem 6.9 (Frank, Z. Király [31]). A graph G = (V, E) has a k-edge-connected T -
odd orientation for every G-even subset T if and only if it is (k + 1, k)-partition-connected.
The proof of this theorem is a very nice application of constructive characterizations.
6.3 Orientation with supermodular requirements
Now we turn to orientation problems featuring hypergraphs. This section presents hyper-
graph analogues of Theorems 6.2 and 6.3. We start by describing the framework used for
hypergraph connectivity orientation.
6.3.1 General framework
Hypergraph orientation problems can be formulated in basically the same way as their
graph counterparts. Let H = (V, E) be a hypergraph, and p : 2 V → Z ∪ {−∞} a set func-
tion, called the requirement function. We consider the problem of finding an orientation of
H that covers p. For example, if p is the set function pkl defined in (6.3), then this amounts