Edge-connectivity of undirected and directed hypergraphs

88 Chapter 5. Partition-connectivity of hypergraphs
A subpartition where the minimum is attained will be called a minimizer of (5.3). Let
N := {Z1, . . . , Zt} be a minimizer of (5.3) so that |N | is as small as possible.
We claim that (∪(Zi)) ∩ (∪(Zj)) = ∅ holds whenever 1 ≤ i < j ≤ t. Indeed, (∪(Zi)) ∩
(∪(Zj)) = ∅ would imply | ∪ (Zi ∪ Zj)| ≤ | ∪ (Zi)| + | ∪ (Zj)| − 1, and hence bH(Zi ∪ Zj) ≤
bH(Zi) + bH(Zj). This is however impossible since by replacing Zi and Zj in N by their
union we would obtain another minimizer subpartition N ′ of E for which |N ′ | < |N |.
We claim furthermore that e ⊆ ∪(Zi) for every hyperedge e ∈ E − ∪ t i=1Zi. If we had
e ⊆ ∪(Zi), then by replacing Zi by Z ′ i := Zi + e we would obtain a subpartition N ′ of E
for which
Z∈N ′
bH(Z) + |E − ∪Z∈N ′Z| < Z∈N bH(Z) + |E − ∪Z∈N Z| contradicting the
fact that N is a minimizer.
Let F be the following partition of V . For each member Zi of N , let ∪(Zi) be a
member of F, and for each element v ∈ V − ∪ t i=1 ∪ (Zi), let {v} be a member of F. By
the claims above eE(F) = |E − ∪ t i=1Zi|. Thus rH(E) = t
i=1 bH(Zi) + |E − ∪ t i=1Zi| =
t
i=1 (| ∪ (Zi)| − 1) + eE(F) = |V | − |F| + eE(F), as required.
Corollary 5.5. The rank of the circuit matroid of a hypergraph H = (V, E) is |V | − 1 (i.e.
H contains a tree-reducible sub-hypergraph) if and only if H is partition-connected.
Proof. By definition rH(E) ≤ |V | − 1 and it follows from (5.2) that equality holds precisely
if |V | − |F| + eH(F) ≥ |V | − 1 holds for every partition F of V , that is, eH(F) ≥ |F| − 1,
which in turn is equivalent to the partition-connectivity of H.
5.2.3 Decomposition into partition-connected sub-hypergraphs
By Corollary 5.5, a hypergraph can be decomposed into k partition-connected sub-hyper-
graphs if and only if it contains k edge-disjoint tree-reducible sub-hypergraphs. This can
also be described as a matroid problem. For a positive integer k let MkH denote the sum
of k copies of the matroid MH, as defined in Theorem 2.7 of Edmonds. This matroid is
defined on the set of hyperedges and a subset of hyperedges is independent if it can be
decomposed into k wooded sub-hypergraphs.
Theorem 5.6. The rank-function rkH of the matroid MkH is given by the following for-
mula:
rkH(X ) = min{k(|V | − |F|) + eX (F) : F is a partition of V }. (5.4)
Proof. Again, it suffices to prove the formula for the special case X = E. Also, as Theorem
5.4 contained the special case k = 1, we may assume that k ≥ 2. As an independent set of