Edge-connectivity of undirected and directed hypergraphs

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92 Chapter 5. Partition-connectivity of hypergraphs described in the previous sections. Indeed, let us assign a sufficiently large weight N to every e ∈ E0, and let E ∗ be the set of hyperedges with positive weight, and H ∗ = (V, E ∗ ). Then finding a maximum weight feasible augmentation amounts to finding a maximum weight independent set in the matroid MkH ∗. The following theorem gives a min-max formula on the corresponding degree-specified augmentation problem, when we allow only the addition of graph edges: Theorem 5.10. Let H0 = (V, E0) be a hypergraph, m : V → Z+ a degree specification with m(V ) even, and k a positive integer. There exists an undirected graph G = (V, E) such that H0 + G can be covered by k hyperforests and dG(v) = m(v) for every v ∈ V if and only if m(X) − + m(V ) ≤ k(|X| − 1) − iH0(X) for every ∅ = X ⊆ V . (5.6) 2 m(V ) Proof. Clearly, iG(X) ≥ (m(X) − ) 2 + , so the requirement iH0+G(X) ≤ k(|X| − 1) implies the necessity of condition (5.6). We prove sufficiency by induction on m(V ). By Theorem 5.9, a hypergraph can be covered by k hyperforests if and only if iH0(X) ≤ k(|X| − 1) for every non-empty subset X of V ; hence we may assume that m(V ) ≥ 2. Let v ∈ V be an arbitrary node with m(v) > 0. A set X is called tight if (5.6) holds with equality. Let F1 be the family that consists of the tight sets X for which m(X) ≤ m(V )/2 and v ∈ X, and let F2 be the family of tight sets X for which m(X) ≥ m(V )/2 and v /∈ X. The union of two members of F1 is also in F1, since otherwise by the supermodularity of iH0 the intersection would violate (5.6); similarly, the intersection of two sets in F2 is in F2, since otherwise by the supermodularity of iH0 their union would violate (5.6). Let X1 be the maximal member of F1, and X2 the minimal member of F2. Then v ∈ X1 − X2 and m(X1) ≤ m(X2), so there is a node u ∈ X2 − X1 with m(u) > 0. Let m ′ be defined by decreasing m(u) and m(v) by 1, and H ′ 0 defined by adding the graph edge uv to H0. The node u was chosen so that no member of F1 contains both u and v, and every member of F2 contains u. From this it is easy to see that (5.6) holds for m ′ and H ′ 0, therefore H ′ 0 can be augmented by adding a graph G ′ with degree-specification m ′ such that H ′ 0 + G ′ can be covered by k forests. This means that G := G ′ + {uv} is a good augmenting graph for H0.
Section 5.3. Disjoint Steiner trees 93 5.3 Disjoint Steiner trees Let G = (V, E) be an undirected graph with a so-called terminal set W ⊆ V . In Chapter 1, we defined a Steiner tree for W as a subtree G ′ = (V ′ , E ′ ) of G such that W ⊆ V ′ . The disjoint Steiner trees problem consists of finding k edge-disjoint Steiner trees in G. When W = V , this corresponds to the existence of k disjoint spanning trees and Theorem 1.5 provides a characterization. When |W | = 2, a minimal Steiner tree is a path connecting the two terminal nodes, and Theorem 1.1 gives an answer. However, as it was mentioned in Chapter 1, for general W it is NP-complete to decide whether G contains k edge-disjoint Steiner trees for W . This means that deriving sufficient conditions for the existence of k disjoint Steiner trees may be of some interest. One type of possible sufficient condition is high edge-connectivity in W . Kriesell [53] proposed the conjecture that 2k-edge-connectivity in W is sufficient for the existence of k edge-disjoint Steiner trees. The conjecture is open even for an arbitrary constant multiple of k instead of 2. Jain, Mahdian, and Salavatipour proved the following: Theorem 5.11 (Jain et al. [46]). If G is l-edge-connected in W then G contains ⌊α|W |l⌋ edge-disjoint Steiner trees for W , where αi can be defined recursively by α2 = 1, αi = αi−1 − α2 i−1 4 for i > 2. In the following we show using Corollary 5.8 that if V − W is stable, then 3k-edge- connectivity in W of G is sufficient for the existence of k edge-disjoint Steiner trees. Theorem 5.12. Let G = (V, E) be an undirected graph and W ⊂ V a subset of nodes so that U := V − W is stable and G is (3k)-edge-connected in W . Then G contains k edge-disjoint Steiner trees for W . Proof. We use induction on the value µG := u∈U (dG(u) − 3) + . Suppose first that µG is zero, that is, the degree of each node in U is at most 3. We may assume that W is also stable for otherwise each edge induced by W can be subdivided by a new node. Such an operation may add new nodes of degree two to U but it does not affect (3k)-edge-connectivity in W and k disjoint Steiner trees in the new graph determine k disjoint Steiner trees in G. Let H = (W, E) be the hypergraph corresponding to the bipartite graph G = (U, W ; E), i.e. for each node u ∈ U there is a corresponding hyperedge of H consisting of the neigh- bours of u in G. As the degree of each node of U is at most 3, the rank of H is at most 3. For any ∅ = X ⊂ W let X ′ denote the set of those nodes of U which have at least one neighbour in X and at most one neighbour in W − X in the graph G. Since every degree
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Edge-connectivity of undirected and
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4 Contents 6 Hypergraph orientation
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6 Contents
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8 Chapter 1. Introduction and preli
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10 Chapter 1. Introduction and prel
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28 Chapter 2. Submodular functions
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Page 42 and 43: 42 Chapter 2. Submodular functions
Page 48 and 49: 48 Chapter 3. Edge-connectivity aug
Page 70 and 71: 70 Chapter 4. Connectivity augmenta
Page 84 and 85: 84 Chapter 5. Partition-connectivit
Page 96 and 97: 96 Chapter 6. Hypergraph orientatio
Page 98 and 99: 98 Chapter 6. Hypergraph orientatio
Page 100 and 101: 100 Chapter 6. Hypergraph orientati
Page 116 and 117: 116 Chapter 7. Combined augmentatio
Page 130 and 131: 130 Chapter 8. Concluding remarks w
Page 132 and 133: 132 Bibliography [11] E. Cheng, Edg
Page 134 and 135: 134 Bibliography [38] S. Fujishige,
Page 136 and 137: 136 Bibliography [66] A. Schrijver,
Page 138 and 139: 138 Index in-degree of node, 19 ind
Page 140 and 141: 140 Index h(a) the head node of the
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142 Index
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Edge-connectivity of undirected and directed hypergraphs

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