Edge-connectivity of undirected and directed hypergraphs

38 Chapter 2. Submodular functions
We call a set X significant if p(X) = p ∗ (X), and tight if p ∗ (X) + p ∗ (V − X) = 0. We
also introduce for X, Y ⊆ V the notation
p ∗
(X, Y ) := max p(Z) : F is an (X, Y )-composition . (2.18)
Z∈F
This function clearly has the following properties:
p ∗ (X, Y ) ≥ max{p ∗ (X) + p ∗ (Y ), p ∗ (X ∩ Y ) + p ∗ (X ∪ Y )}, (2.19)
p ∗ (Y ) ≥ p ∗ (X, Y ) + p ∗ (V − X). (2.20)
The main result of this section is the following theorem:
Theorem 2.17. If p(X) + p(Y ) ≤ p ∗ (X ∩ Y ) + p ∗ (X ∪ Y ) holds whenever X and Y are
significant and crossing, then p ∗ is fully supermodular.
Observe that if X and Y are not crossing, then p ∗ (X) + p ∗ (Y ) ≤ p ∗ (X ∩ Y ) + p ∗ (X ∪ Y )
automatically holds. So the condition of the theorem always holds for non-crossing X and
Y . Theorem 2.17 implies Proposition 2.15 (it is easy to prove using the uncrossing technique
that p ↑ (X) = p ∗ (X) for every X ⊆ V if p is crossing supermodular and p ∗ (V ) = 0).
Furthermore, it shows that instead of crossing supermodularity, it is sufficient to require
p(X) + p(Y ) ≤ p ↑ (X ∩ Y ) + p ↑ (X ∪ Y ) for every crossing X, Y .
For technical reasons we will prove a theorem that is a bit more general but less elegant
than Theorem 2.17. We say that a family G generates p∗ if
p ∗
(X) = max p ∗
(Z) : F is a composition of X, and its members are in G
Z∈F
for every X ⊆ V . Obviously the family of significant sets generates p ∗ .
Theorem 2.18. Suppose that a family G generates p ∗ , and p ∗ (X, Y ) = p ∗ (X ∩Y )+p ∗ (X ∪
Y ) holds for every pair X, Y ∈ G for which p ∗ (X, Y ) = p ∗ (X) + p ∗ (Y ). Then p ∗ is fully
supermodular.
Proof. The main difficulty of the proof is that the composition F of X that defines p ∗ (X)
in (2.16) can not always be cross-free (an example for this will be presented at the end
of the proof). We will show however that the family F can be assumed to be cross-free
restricted to minimal tight sets. First we prove with the help of a few preliminary claims
that p ∗ (X, Y ) = p ∗ (X ∩ Y ) + p ∗ (X ∪ Y ) is true for every tight X and arbitrary Y .
Claim 2.19. If X is tight, then p ∗ (X, Y ) = p ∗ (X) + p ∗ (Y ) for every Y ⊆ V .