Edge-connectivity of undirected and directed hypergraphs

Chapter 8
Concluding remarks
In these paragraphs we give a brief account of the open problems related to the various
topics discussed in the thesis. Most of these questions were mentioned in previous chapters;
they are collected here in order to indicate some possible directions for future research.
One of the new results described in the thesis is Corollary 3.30 on k-edge-connectivity
augmentation of hypergraphs by the addition of uniform hyperedges. This result followed
from Theorems 3.14 and 3.29, which addressed the more general problem of covering sym-
metric crossing supermodular set functions by uniform hypergraphs. The proofs included
an extension of the well-known splitting-off technique to hypergraphs. While the exis-
tence of a special splitting-off sequence was shown, it would be desirable to obtain a more
complete structural description of the feasible splitting-off operations. This might help in
dealing with problems with extra constraints, like augmentation with partition constraints,
or simultaneous augmentation of two hypergraphs.
For directed hypergraphs, the splitting-off method used in the proof of Theorem 4.7 is
essentially the same as the one described by Berg, Jackson, and Jordán in [7]. Theorem
4.7 shows that if mi(V ) ≤ mo(V ) then there is always a complete splitting-off preserving
k-edge-connectivity in V . However, this is not necessarily true if only digraph edges are
allowed as new edges. Therefore in this case it is natural to ask what is the maximum length
of a feasible splitting-off sequence. Berg, Jackson, and Jordán conjectured the following:
Conjecture 8.1. Let D = (V + s, A) be a directed hypergraph that is k-edge-connected
in V , and suppose that s is incident only with digraph edges and ϱ(s) ≥ δ(s). Then D
does not have a sequence of γ feasible (1, 1)-splittings at s if and only if there is a family
F = {X1, X2, . . . , Xt} of pairwise co-disjoint subsets of V for some 2 ≤ t ≤ 2γ + 1, such
that
t
ϱ(Xi) ≤ tk + (t − 1)γ − µ − 1,
i=1
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