Edge-connectivity of undirected and directed hypergraphs

Section 4.3. Covering supermodular functions by directed hypergraphs 77
if and only if
hold for every partition F of V .
γ ≥
p(X), (4.7)
X∈F
rγ ≥
p(V − X), (4.8)
X∈F
(|F| − 1) γ ≥
p(V − X) (4.9)
X∈F
Proof. The necessity of the conditions can be seen easily. To prove sufficiency, one can
construct degree specifications mi and mo that satisfy the conditions of Theorem 4.5. Let
us define the set function p ′ by p ′ (X) := p(X) for X ⊂ V and p ′ (V ) := γ. Then p ′ is
positively crossing supermodular. Since (4.7) and (4.9) hold for p ′ , by applying Theorem
2.13 to p ′ we get a nonnegative integer vector mi such that mi(X) ≥ p(X) for all X ⊆ V ,
and mi(V ) = γ.
To construct mo let m ′ o be a nonnegative vector satisfying m ′ o(V − X) ≥ p(X) for
all X ⊆ V , which is minimal in the sense that for every v ∈ V with m ′ o(v) > 0 there
exists a set X for which v /∈ X and m ′ o(V − X) = p(X) (such a set is called tight). Let
B = {X1, X2, . . . , Xl} be a family of minimum cardinality which for every node v with
m ′ o(v) > 0 contains a tight set not containing v. Then B is cross-free, since we could
replace a crossing pair Xi, Xj by their intersection according to (2.2). If the family is
composed of co-disjoint sets, then
m ′ o(V ) =
l
m ′ o(V − Xi) =
i=1
l
p(Xi) ≤ rγ
by (4.8). If there are two disjoint sets Xi and Xj in B, then
m ′ o(V ) ≤ m ′ o(V − Xi) + m ′ o(V − Xj) = p(Xi) + p(Xj) ≤ γ.
Thus we can obtain an out-degree specification mo by increasing m ′ o on an arbitrary node
to obtain mo(V ) = rγ. By Theorem 4.5 there is a directed (r, 1)-hypergraph with degrees
mi and mo that covers p.
The following example demonstrates that condition (4.9) cannot be left out. Let V =
{v1, v2, v3}, p({v1, v2}) = p({v1, v3}) = p({v2, v3}) := 2 and p(X) := 0 for the other sets,
r := 3, γ := 2. Conditions (4.7) and (4.8) are satisfied, but (4.9) is not, and there is no
directed (3, 1)-hypergraph of 2 hyperarcs covering p. When r = 1, i.e. we add digraph
edges, (4.9) follows from (4.8).
i=1