Edge-connectivity of undirected and directed hypergraphs

Section 7.3. (k, l)-partition-connectivity augmentation 123
(ii) If F1 is a composition of Z ⊆ V and it contains a set W ⊆ Z such that F2 contains a
nontrivial partition {Z1, . . . , Zt} of W , then replace W in F1 by the sets Z1, . . . , Zt,
and replace Z1, . . . , Zt in F2 by W .
After a finite number of steps, none of the above operations are applicable; let (F ′ 1, F ′ 2) be
the pair obtained at that point. Then it follows from Claim 2.2 that F ′ 1 must be a partition
of some X ′ ⊆ V , and operation (ii) guarantees that F ′ 1 is a refinement of F ′ 2 if F ′ 2 = ∅. If
(7.17) holds, then the value QX(F1, F2) does not decrease during the above two operations.
For (i) this follows from (7.16), since the symmetry or monotone decreasing property of p
implies that QW ({W1, . . . , Ws}, {Z1, . . . , Zt}) ≤ Q∅({V −W1, . . . , V −Ws, Z1, . . . , Zt}, ∅) ≤
0. For (ii), it follows because p(W ) + t i=1 p(V − Zi) ≤ p(V − W ) + t i=1 p(Zi), again due
to the symmetry or the monotone decreasing property of p.
Remark. The following example shows that (7.17) itself is not sufficient in Theorem 7.9.
Let V = {v1, v2, v3, v4}, E = {v1v2, v1v3, v1v4}. Let p = 1 on the sets {v2}, {v3}, {v4} and
on their complement; let p = 0 on all other sets. If we allow only the addition of graph
edges, we need at least 2 of them for a feasible orientation (two edges suffice, since after
adding v2v3 and v3v4 the graph has a strong orientation) but (7.17) requires only γ ≥ 1,
since the only deficient partitions are {{vi}, V − vi} (i = 2, 3, 4).
7.3 (k, l)-partition-connectivity augmentation
If k ≥ l, then the set function pkl defined in (6.3) is non-negative, monotone decreasing,
and crossing supermodular, therefore Theorems 7.4 and 7.9 can be applied when it is the
requirement function. Since by Theorem 6.12 a hypergraph is (k, l)-partition-connected
for k ≥ l if and only if it has a (k, l)-edge-connected orientation, we obtain the following:
Corollary 7.10. Let H0 = (V, E0) be a hypergraph, m : V → Z+ a degree specification,
0 ≤ γ ≤ m(V )/2 an integer, and k ≥ l non-negative integers. There exists a hypergraph
H with γ hyperedges such that H0 + H is (k, l)-partition-connected and dH(v) = m(v) for
all v ∈ V if and only if the following hold for every nontrivial partition F of V :
γ ≥ (|F| − 1)k + l − eH0(F) , (7.20)
min
i m(V − Xi) ≥ (|F| − 1)k + l − eH0(F) . (7.21)
In addition, H can be chosen so that
m(V )
m(V )
≤ |e| ≤
γ
γ
for every e ∈ E. (7.22)