Edge-connectivity of undirected and directed hypergraphs

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80 Chapter 4. Connectivity augmentation of directed hypergraphs s Figure 4.3: Counterexample for Claim 4.11 for k = 1, l = 3. Inequality (4.9) on the indicated copartition gives γ ≥ 2, while Z∈F p(Z) ≤ 1 for every partition F. tion: The counterexample on Figure 4.3 shows that the claim is not necessarily true if k < l. Using Claim 4.11, Theorem 4.8 implies the following on minimum total size augmenta- Theorem 4.12. Let D0 = (V, A0) be a directed hypergraph, let k ≥ l be non-negative integers, and let p be the set function defined in (4.10). The minimum total size σ of a directed hypergraph D for which D0 + D is (k, l)-edge-connected from root s is σ = σ1 + σ2, where σ1 = max p(X) : F is a partition of V , Z∈F σ2 = max max p(X), p(V − X) : F is a partition of V . Z∈F Z∈F Theorem 4.13. Let D0 = (V, A0) be a directed hypergraph, let k < l be non-negative integers, and let p be the set function defined in (4.10). The minimum total size σ of a directed hypergraph D for which D0 + D is (k, l)-edge-connected from root s is σ = σ1 + σ2, where 1 σ1 = max max p(X), |F| − 1 Z∈F ( p(V − X)) : F is a partition of V , Z∈F σ2 = max max p(X), p(V − X) : F is a partition of V . Z∈F Z∈F For minimum cardinality augmentation using (r, 1)-hyperedges, the following results are obtained from Theorem 4.9 and Claim 4.11:
Section 4.4. (k, l)-edge-connectivity of directed hypergraphs 81 Theorem 4.14. Let D0 = (V, A0) be a directed hypergraph, and let k ≥ l and r ≥ 1 be non-negative integers. D0 can be made (k, l)-edge-connected by the addition of γ new (r, 1)-hyperarcs if and only if γ ≥ p(X), X∈F rγ ≥ p(V − X) hold for every partition F of V , where p is the set-function defined in (4.10). X∈F Theorem 4.15. Let D0 = (V, A0) be a directed hypergraph, and let k < l and r ≥ 1 be non-negative integers. D0 can be made (k, l)-edge-connected by the addition of γ new (r, 1)-hyperarcs if and only if γ ≥ p(X), X∈F rγ ≥ p(V − X), X∈F (|F| − 1) γ ≥ p(V − X) hold for every partition F of V , where p is the set-function defined in (4.10). X∈F
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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 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
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130 Chapter 8. Concluding remarks w
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132 Bibliography [11] E. Cheng, Edg
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134 Bibliography [38] S. Fujishige,
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136 Bibliography [66] A. Schrijver,
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138 Index in-degree of node, 19 ind
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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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