Edge-connectivity of undirected and directed hypergraphs

20 Chapter 1. Introduction and preliminaries
is true for every directed hypergraph D and subsets X, Y ⊆ V :
ϱD(X) + ϱD(Y ) = ϱD(X ∩ Y ) + ϱD(X ∪ Y ) + dD(X, Y ), (1.4)
δD(X) + δD(Y ) = δD(X ∩ Y ) + δD(X ∪ Y ) + dD(V − X, V − Y ). (1.5)
Theorem 1.2 extends naturally to directed hypergraphs:
Proposition 1.12. In a directed hypergraph D = (V, A), there exist k edge-disjoint paths
from node s to node t if and only if ϱD(X) ≥ k for every st-set X.
Proof. Suppose that ϱD(X) ≥ k for every st-set X ⊆ V . To reduce the problem to the
digraph case, a new node va is added to V for every hyperarc a ∈ A, and the hyperarc a
is replaced by edges uva for every u ∈ t(a), and an edge vah(a); let D ′ = (V ′ , A ′ ) be the
obtained digraph. There is a one-to-one correspondence between the paths from s to t in
D and the paths from s to t in D ′ , and edge-disjointness is preserved. By Theorem 1.2,
the maximum number of edge-disjoint paths from s to t is
min{ϱD ′(X′ ) : X ′ is an st-set in V ′ }.
For such an X ′ , let X := X ′ ∩ V ; then k ≤ ϱD(X) ≤ ϱD ′(X′ ).
As a consequence, local edge-connectivity can be defined similarly as for digraphs: for
distinct nodes s, t ∈ V , λD(s, t) is the maximal number of edge-disjoint paths from s to t.
The following is true on global connectivity:
Proposition 1.13. For a directed hypergraph D = (V, A) and a positive integer k, the
following are equivalent:
(i) λD(u, v) ≥ k for every pair u, v ∈ V of distinct nodes.
(ii) ϱD(X) ≥ k holds for every non-empty proper subset X of V .
(iii) D remains strongly connected if we delete any k − 1 edges.
A directed hypergraph D is called k-edge-connected if the above hold for D. Given
S, T ⊆ V , D is called k-edge-connected from S to T if λD(s, t) ≥ k for every distinct s ∈ S
and t ∈ T .
Like Menger’s theorem, Theorem 1.8 of Edmonds can be easily extended to directed
hypergraphs. Given a set S ⊆ V , a directed hypergraph D = (V, A) is connected from S if
every node v ∈ V is reachable from some s ∈ S.