Edge-connectivity of undirected and directed hypergraphs

54 Chapter 3. Edge-connectivity augmentation of hypergraphs
of adding ν-hyperedges (for a fixed ν) to a (k − 1)-edge-connected hypergraph to make it
k-edge-connected. A hypergraph H covers a family C if dH(X) > 0 for every X ∈ C. A
family C of sets is crossing if for any two crossing sets X, Y ∈ C, X ∩ Y and X ∪ Y are in
C. Note that if C is also symmetric, then X − Y and Y − X are in C as well. If we define
p(X) := 1 if X ∈ C and p(X) := −∞ otherwise, then p is crossing supermodular.
If a hypergraph H0 is (k − 1)-edge-connected, then the sets ∅ = X ⊂ V for which
k − dH0(X) = 1 form a symmetric crossing family, and a hypergraph H covers this family
if and only if H0 + H is k-edge-connected.
A partition F = {V1, . . . , Vl} is called a full partition if ∪i∈IVi ∈ C for every ∅ = I ⊂
{1, . . . , l}. In other words, it is full if it is p-full for the set function p defined above. Fleiner
and Jordán proved the following theorem:
Theorem 3.13 (Fleiner, Jordán [20]). Let C be a symmetric crossing family on the
ground set V , and ν ≥ 2 an integer. Then C can be covered by γ ν-hyperedges if and only
if
and
νγ ≥ max{|F| : F ⊆ C, F is a subpartititon of V },
(ν − 1)γ ≥ max{|F| − 1 : F is a full partititon}.
Note that the covering by ν-hyperedges is equivalent to the problem of covering by
hyperedges of size at most ν, since one can always take hyperedges of maximal size. The
proof of Fleiner and Jordán is different from the proofs of the results cited previously, in
that it does not solve a degree-specified result using some splitting-off technique, instead
it depends on an analysis of the structure of symmetric crossing families.
3.4.2 Covering symmetric supermodular functions by uniform
hypergraphs
The main result of this chapter is a common generalization of the above mentioned results
in [9] and [20] (Theorems 3.12 and 3.13), based on the approach of Benczúr and Frank. We
give a min-max formula on the minimum number of ν-hyperedges that can cover a given
symmetric, positively crossing supermodular set function. As in [9], the substantial part of
the proof is a solution of the degree-specified problem (i.e. when the degree of each node
v ∈ V is a prescribed value m(v)), which then easily leads to a min-max formula on the
minimum number of new hyperedges needed.