Edge-connectivity of undirected and directed hypergraphs

22 Chapter 1. Introduction and preliminaries
1.4 Families of sets
Most of the min-max theorems presented in the thesis contain conditions involving special
kinds of families of sets, and the proofs depend heavily on the manipulation of these
families. Some basic notations are presented here, while families with special structures
will be discussed in Chapter 2.
1.4.1 Basic definitions
Given a finite ground set V , a family of sets is a collection of (not necessarily distinct)
subsets of V . The empty family is denoted by ∅. An example is a partition of a set X
which is a family of pairwise disjoint sets whose union is X. A subpartition of X is a
partition of some X ′ ⊆ X; a partition of V is sometimes simply called a partition. For a
family F, we use the notation co(F) := {V − X : X ∈ F}. If F is a partition, then co(F)
is called a co-partition.
The multiplicities of the sets in a family are always taken into account unless otherwise
noted; for example, for a set function p and a family F,
X∈F p(X) counts the value of each
set as many times as its multiplicity in F. To a family F we associate the characteristic
function χF : 2 V → Z+, i.e. χF(X) equals the multiplicity of the set X in F.
The notation F1 ⊆ F2 is used for χF1 ≤ χF2. The sum of two families F1 and F2,
denoted by F1 + F2, is the family with characteristic function χF1 + χF2.
Let H = (V, E) be a hypergraph, and F a family of subsets of V . We define
It is easy to see that
eH(F) :=
eH(F) = max
e∈E
X∈F
max |{X ∈ F : u ∈ X, e ⊆ X}|. (1.6)
u∈e
ϱ H (X) : H is an orientation of H
. (1.7)
If F is a partition, then eH(F) is the number of hyperedges that are not induced by any
member of the partition (these are called cross-hyperedges).
1.4.2 Duality
One area where families of sets appear naturally is the theory of duality. We do not give a
proper introduction to duality here, just mention a few facts and special formulations that
will be needed later.