Edge-connectivity of undirected and directed hypergraphs

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124 Chapter 7. Combined augmentation and orientation Note that Corollary 7.10 has some interest even in the very special case when H0 = ∅. A result of Edmonds [16] states that a degree-sequence m1, . . . , mn is realizable by a k-edgeconnected graph if and only if n i=1 mi is even, and mi ≥ k for every i. Corollary 7.10 implies the following similar result: given k ≥ l, a degree-sequence m1, . . . , mn is realizable by a (k, l)-partition-connected graph if and only if n i=1 mi is at least 2k(n − 1) + 2l, it is even, and mi ≥ k + l for every i. When l = 0, this implies the following tiny result (which is not difficult to prove directly either): If a degree-sequence is realizable by a k-edge-connected graph with at least k(n−1) edges, then it is also realizable by a graph containing k edge-disjoint spanning trees. On minimum cardinality augmentation, Theorem 7.9 gives the following: Corollary 7.11. Let H0 = (V, E0) be a hypergraph, σ ≥ 0, 0 ≤ γ ≤ σ, and k ≥ l non- negative integers. There is a hypergraph H with γ hyperedges of total size σ such that H0 + H is (k, l)-partition-connected if and only if the following two conditions are met: (i) γ ≥ (|F| − 1)k + l − eH0(F) for every nontrivial partition F, (ii) σ ≥ |F1|k + |F2|l − eH0 (F1 + co(F2)) whenever F1 and F2 are partitions of some ∅ = X ⊂ V and F1 is a refinement of F2. In addition, H can be chosen so that σ ≤ |e| ≤ γ σ γ for every e ∈ E. (7.23) Remark. The graph on Figure 7.2 shows that the second condition in Corollary 7.11 cannot be simplified. We need to add at least 2 edges to the graph to have a (3, 1)-edge- connected orientation from root s, but the simplest evidence for this is the family indicated on the figure (consisting of the round sets and the complements of the square sets), whose deficiency is 3, while a new edge can enter at most 2 sets. The figure on the right shows that the addition of 2 edges is sufficient (to see that the digraph is (3, 1)-edge-connected, observe that it contains 3 edge-disjoint out-arborescences from s, and also an in-arborescence to s). Let ν ≥ 2 be an integer. Because of (7.23), Corollary 7.11 gives a characterization for minimum cardinality augmentation with ν-hyperedges. A simple observation shows that a characterization of minimum total size augmentation can also be derived. It is easy to see that we get the best bound if γ = ⌊σ/2⌋:
Section 7.4. More general requirement functions 125 s s Figure 7.2: (3, 1)-partition-connectivity augmentation by graph edges. Condition (ii) in Corollary 7.11 is tight on the indicated family (the non-rounded rectangles represent their complement). The picture on the right shows a (3, 1)-edge-connected orientation of the augmented graph. Corollary 7.12. Let H0 = (V, E0) be a hypergraph, σ ≥ 0 and k ≥ l non-negative integers. There is a hypergraph H of total size σ such that H0 + H is (k, l)-partition-connected if and only if the following two conditions are met: (i) σ 2 ≥ (|F| − 1)k + l − eH0(F) for every nontrivial partition F, (ii) σ ≥ |F1|k + |F2|l − eH0 (F1 + co(F2)) whenever F1 and F2 are partitions of some X ⊂ V and F1 is a refinement of F2. In addition, there is an optimal H that contains only graph edges and at most one 3- hyperedge. 7.4 More general requirement functions In this section we illustrate what happens if instead of considering only non-negative crossing supermodular requirement functions, we allow the broader class of positively crossing supermodular functions. We were only able to solve the degree-specified problem in this case, and the characterizations are not elegant. However, the result provides an extension of combined augmentation and orientation to mixed graphs and mixed hypergraphs, so it may be of some interest. 7.4.1 Mixed hypergraphs The orientation of mixed graphs was the main motivation behind Theorem 6.5 of Frank [25], which extended Theorem 6.3. We intend to generalize Theorem 7.1 in a similar way. A mixed hypergraph M = (V ; E, A) contains a set E of hyperedges and a set A of hyperarcs;
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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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28 Chapter 2. Submodular functions
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48 Chapter 3. Edge-connectivity aug
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70 Chapter 4. Connectivity augmenta
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72 Chapter 4. Connectivity augmenta
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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,
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Edge-connectivity of undirected and directed hypergraphs

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