Edge-connectivity of undirected and directed hypergraphs

94 Chapter 5. Partition-connectivity of hypergraphs
in U is at most 3, we have dG(X ∪ X ′ ) = dH(X) and hence the (3k)-edge-connectivity of
G in W implies the (3k)-edge-connectivity of H.
By Corollary 5.8, H can be decomposed into k connected spanning sub-hypergraphs, thus
U can be partitioned into k disjoint subsets U1, . . . , Uk so that W ∪ Ui induces a connected
subgraph Gi = (W ∪Ui, Ei) of G for each i = 1, . . . , k. By choosing one spanning tree from
each Gi, we obtain the required edge-disjoint Steiner trees of G.
Suppose now that µG is positive and that the theorem holds for each graph G ′ with
µG ′ < µG. Let s ∈ U be a node with dG(s) ≥ 4. If there is a cut-edge e of G, then
the elements of W belong to the same component of G − e as G is at least (3k)-edge-
connected in W and then we may discard the other component of G−e without destroying
(3k)-edge-connectivity in W . Therefore we may assume that G is 2-edge-connected.
By Theorem 3.3 of Mader on splitting-off preserving local edge-connectivity, there are
two edges e1 = v1s, e2 = v2s in E so that the local edge-connectivities in V − s do not
decrease if we replace e1 and e2 by a new edge v1v2. In particular, the resulting graph G ′
remains (3k)-edge-connected in W . By induction there are k edge-disjoint Steiner trees for
W in G ′ . If one of these trees contains the split-off edge v1v2, we replace it by e1 and/or
e2 in order to obtain a Steiner tree of G. Therefore we have proved the existence of k
edge-disjoint Steiner trees for W in G.
It should be noted that if the degree of each node in U is even, then 2k-edge-connectivity
in W implies the existence of k edge-disjoint Steiner trees for W , even without the restric-
tion that U must be stable. This can be seen by observing that in this case Theorem 3.3
of Mader can always be applied since we can assume that G is 2-edge-connected, and the
splitting-off preserves the parity of the degrees, therefore the degree of a node in U will
never be 3.
As far as algorithmic aspects are concerned, Edmonds’ matroid partition algorithm
may be used to compute a decomposition of a hypergraph into k partition-connected
sub-hypergraphs or to compute a deficient partition to show that such a decomposition
does not exist. Therefore Theorem 5.12 can be used for an approximation algorithm to
compute the maximum number τ of disjoint Steiner trees when V − W is stable. W is
clearly τ-edge-connected in G, so by Theorem 5.12 we can compute τ/3 disjoint Steiner
trees.