Edge-connectivity of undirected and directed hypergraphs

48 Chapter 3. Edge-connectivity augmentation of hypergraphs
3.1.1 Edge-connectivity augmentation of graphs
Initial deep results on the edge-connectivity augmentation of graphs are due to Lovász
[55] and to Watanabe and Nakamura [71], on the minimum number of edges needed to
be added to a graph to make it k-edge-connected (this is called the minimum cardinality
problem). Watanabe and Nakamura gave the following characterization:
Theorem 3.1 (Watanabe, Nakamura [71]). Let G0 = (V, E0) be a graph, and k ≥ 2
an integer. G0 can be made k-edge-connected by adding at most γ new edges if and only if
(k − dG0(Z)) ≤ 2γ for every subpartition F of V .
Z∈F
Not that the theorem does not hold for k = 1, but in that case even the minimum cost
problem is solvable in polynomial time. For k ≥ 2 the minimum cost problem is NP-
complete. Watanabe and Nakamura showed that a minimum cardinality augmentation
can be obtained in polynomial time by repeatedly increasing the edge-connectivity of the
graph by one using the minimum number of edges. However, this algorithm is not strongly
polynomial.
Frank [29] gave the first strongly polynomial algorithm for this problem. The algorithm
relies on the following result concerning degree specified augmentation:
Theorem 3.2. Let G0 = (V, E0) be a graph, k ≥ 2 an integer, and m : V → Z+ a degree
specification such that m(V ) is even. There is a graph G such that dG(v) = m(v) for every
v ∈ V and G0 + G is k-edge-connected if and only if
m(X) ≥ k − dG0(X) for every ∅ = X ⊂ V .
Furthermore, Frank proved in [29] that the local edge-connectivity augmentation of
graphs (i.e. when for every pair u, v ∈ V there is a local edge-connectivity requirement
r(u, v)) can also be solved in strongly polynomial time. These proofs use the so-called
splitting-off technique to solve the degree specified problem. Splitting off at a given node
s ∈ V means deleting edges st1 and st2, and adding a new edge t1t2. A sequence of splitting-
off operations that isolate the node s is called a complete splitting at s. The splitting-off
operation was originally introduced by Lovász [55] and subsequently developed further
by Mader ([58], [59]) and others. Here we cite the result of Mader [58] on splitting-off
preserving local edge-connectivity.
Theorem 3.3 (Mader [58]). Let G = (V + s, E) be a connected graph, where d(s) = 3
and there is no cut-edge or loop incident to s. Then there are edges st1 ∈ E and st2 ∈ E
such that for the graph G ′ = (V, E − {st1, st2} + {t1t2}), λG ′(u, v) = λG(u, v) for every
u, v ∈ V .