Edge-connectivity of undirected and directed hypergraphs

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94 Chapter 5. Partition-connectivity of hypergraphs in U is at most 3, we have dG(X ∪ X ′ ) = dH(X) and hence the (3k)-edge-connectivity of G in W implies the (3k)-edge-connectivity of H. By Corollary 5.8, H can be decomposed into k connected spanning sub-hypergraphs, thus U can be partitioned into k disjoint subsets U1, . . . , Uk so that W ∪ Ui induces a connected subgraph Gi = (W ∪Ui, Ei) of G for each i = 1, . . . , k. By choosing one spanning tree from each Gi, we obtain the required edge-disjoint Steiner trees of G. Suppose now that µG is positive and that the theorem holds for each graph G ′ with µG ′ < µG. Let s ∈ U be a node with dG(s) ≥ 4. If there is a cut-edge e of G, then the elements of W belong to the same component of G − e as G is at least (3k)-edge- connected in W and then we may discard the other component of G−e without destroying (3k)-edge-connectivity in W . Therefore we may assume that G is 2-edge-connected. By Theorem 3.3 of Mader on splitting-off preserving local edge-connectivity, there are two edges e1 = v1s, e2 = v2s in E so that the local edge-connectivities in V − s do not decrease if we replace e1 and e2 by a new edge v1v2. In particular, the resulting graph G ′ remains (3k)-edge-connected in W . By induction there are k edge-disjoint Steiner trees for W in G ′ . If one of these trees contains the split-off edge v1v2, we replace it by e1 and/or e2 in order to obtain a Steiner tree of G. Therefore we have proved the existence of k edge-disjoint Steiner trees for W in G. It should be noted that if the degree of each node in U is even, then 2k-edge-connectivity in W implies the existence of k edge-disjoint Steiner trees for W , even without the restric- tion that U must be stable. This can be seen by observing that in this case Theorem 3.3 of Mader can always be applied since we can assume that G is 2-edge-connected, and the splitting-off preserves the parity of the degrees, therefore the degree of a node in U will never be 3. As far as algorithmic aspects are concerned, Edmonds’ matroid partition algorithm may be used to compute a decomposition of a hypergraph into k partition-connected sub-hypergraphs or to compute a deficient partition to show that such a decomposition does not exist. Therefore Theorem 5.12 can be used for an approximation algorithm to compute the maximum number τ of disjoint Steiner trees when V − W is stable. W is clearly τ-edge-connected in G, so by Theorem 5.12 we can compute τ/3 disjoint Steiner trees.
Chapter 6 Hypergraph orientation 6.1 Introduction In Chapter 4 we saw that many results on the edge-connectivity of digraphs are extendable to directed hypergraphs. It is a natural question to ask whether similar extensions are possible for connectivity orientation problems. For example, when does a hypergraph have an orientation that is (k, l)-edge-connected? This Chapter is devoted to questions of this type, and we also present some orientation results that are new even for graphs. In Section 6.2 we briefly review known connectivity orientation results for graphs. In particular, we describe the relation between high partition-connectivity of a graph and the existence of highly edge-connected orientations. One of the observations in Section 6.3 is that a similar relation exists in the case of hypergraphs. The reason behind the similarities is the link between orientation problems and submodularity, which can be established for hypergraphs just like for graphs. In [50], Khanna, Naor and Shepherd proposed a new framework, called network design with orientation constraints, that successfully integrated network design problems like minimum cost k-rooted-connected sub-digraphs, and orientation problems like k-rooted- connected orientation of a mixed graph. In Section 6.4 we extend their result to hypergraphs, and show that their formulation actually defines a TDI system, which gives rise to new min-max formulas. Orientations with local edge-connectivity properties define an exciting class of problems. For graphs we have the beautiful theorem of Nash-Williams [61] (Theorem 6.4) on well- balanced orientations; a major open question of the field is how to generalize this theorem to hypergraphs. The new results in Section 6.5 are rather simple and do not really mean a step forward in that direction; however, they do have some nice corollaries, including a new characterization of (2k + 1)-edge-connected graphs, and characterization of (k, l)- 95
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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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28 Chapter 2. Submodular functions
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Page 44 and 45: 44 Chapter 2. Submodular functions
Page 46 and 47: 46 Chapter 2. Submodular functions
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Page 70 and 71: 70 Chapter 4. Connectivity augmenta
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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
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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Edge-connectivity of undirected and directed hypergraphs

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