Edge-connectivity of undirected and directed hypergraphs

Section 3.4. Covering by uniform hypergraphs 53
Figure 3.1: A p-full partition with respect to p = 3−dH0: the components after the deletion
of 2 hyperedges
supermodular set function. This more abstract setting provided a better insight into the
combinatorial structure underlying the augmentation problem.
If there is a partition F = {V1, . . . , Vl} such that p(∪i∈IVi) > 0 for every ∅ = I ⊂
{1, . . . , l}, then the number of edges of the covering graph is at least l − 1, since the edges
must connect the members of the partition. Such partitions are called p-full and they play
an important role in the solution of the problem. For example, when p(X) = k − dH0(X) if
∅ = X ⊂ V , then the components considered in (3.2) form a p-full partition (as illustrated
on Figure 3.1). Benczúr and Frank proved the following theorem:
Theorem 3.12 (Benczúr, Frank [9]). Let p : 2 V → Z+ be a symmetric, positively
crossing supermodular set function. There is a graph with γ edges that covers p if and
only if the following hold:
2γ ≥
p(Z) for every partition F of V , (3.3)
Z∈F
γ ≥ |F| − 1 for every p-full partition F. (3.4)
The core of the proof is an extension of the splitting-off operation to this abstract setting,
in order to provide a solution for the corresponding degree-specified problem. Note that in
this theorem, in contrast to the skew supermodular case, the requirement of symmetry is
important: one cannot make a non-symmetric p symmetric by taking max{p(X), p(V −X)}
because this set function is not necessarily positively crossing supermodular.
3.4 Covering by uniform hypergraphs
3.4.1 Covering symmetric crossing families
Another generalization of Cheng’s result due to Fleiner and Jordán [20] concerns the cov-
ering of symmetric crossing families by uniform hypergraphs. This includes the problem