Edge-connectivity of undirected and directed hypergraphs

24 Chapter 1. Introduction and preliminaries
Again, if M is 0-1 valued and y is integral, then it can be regarded as a characteristic
function of a family of sets. Let us cite another version of Farkas’ Lemma that will be used
later. For a vector z, z ≫ 0 means that every coordinate of z is strictly greater than 0,
while z > 0 means that z ≥ 0 and at least one coordinate is strictly greater than 0.
Lemma 1.16. Let M ∈ Q m×n be a matrix. The system
is solvable if and only if the system
has no solution.
1.4.3 Compositions
{x ∈ Q n : Mx ≫ 0, x ≥ 0}
{y ∈ Q m : yM ≤ 0, y > 0}
Most of the families that appear in the thesis are so-called compositions. A family F of
sets is a composition of a set X ⊆ V if the value
Z∈F χZ(v) − χX(v) is the same for every
v ∈ V . If X ∩ Y = ∅, F1 is a composition of X, and F2 is a composition of Y , then F1 + F2
is a composition of X ∪ Y . A composition of V is called a regular family; note that these
are also compositions of ∅. If F is regular, then so is co(F).
For a set X ⊆ V and a family F that is a composition of X, we define the height of F
with respect to X:
hX(F) :=
χZ(v) − χX(v) for an arbitrary v ∈ V . (1.11)
Z∈F
We will omit the subscript X if it is unambiguous; in fact, the only ambiguity is that if
F is a regular family, then hV (F) = h∅(F) − 1. Note also that if F is a partition, then
hV (F) = 0, if it is a co-partition then hV (F) = |F| − 2 and if F = ∅, then hV (F) = −1!
If H = (V, E) is a hypergraph and F is a regular family, then the value eH(F) defined
in (1.6) can be expressed in a simpler form:
Moreover, the following holds for regular families:
eH(F) = h∅(F)|E| −
iH(X). (1.12)
Proposition 1.17. If F1 and F2 are regular families, then eH(F1+F2) = eH(F1)+eH(F2).
X∈F