Edge-connectivity of undirected and directed hypergraphs

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130 Chapter 8. Concluding remarks where µ = min{γ, |{sv ∈ A : v /∈ ∩ t i=1Xi}|}. Let us return to undirected hypergraphs. We have shown in Chapter 5 how matroid- theoretic tools applied on hypergraphic matroids can be used to prove results like Theorem 5.7 on the decomposition of a hypergraph into k partition-connected sub-hypergraphs. This result in turn easily implies Theorem 5.12 which gives a sufficient condition for the existence of k edge-disjoint Steiner trees for W when V − W is stable. Kriesell [53] conjectures that 2k-edge-connectivity in W is sufficient, even in the case when V − W is not necessarily stable. To date, no constant c is known such that ck-edge-connectivity in W guarantees the existence of k edge-disjoint Steiner trees. Another open problem for graphs that is related to partition-connectivity is the constructive characterization of (k, l)-partition-connected graphs, i.e. Conjecture 6.7. This conjecture would follow from Conjecture 6.6 on the constructive characterization of (k, l)- edge-connected digraphs. Since by Theorem 6.11 there is a similar relation between (k, l)- partition-connectivity and (k, l)-edge-connectivity for hypergraphs, we might obtain similar constructive characterizations for hypergraphs as well. One of the exciting questions in graph connectivity orientation is the solvability of orientation problems with parity constraints. Known results were cited in Chapter 6; a basic open problem is the characterization of graphs which have a strongly connected orientation where the in-degree of every node is odd. Of course, similar questions can be asked for hypergraphs. A problem where the answer is known for graphs but not for hypergraphs is local edge- connectivity orientation with symmetric demands. For example, no characterization is known for hypergraphs which have a k-edge-connected orientation that is at the same time l-edge-connected in a given subset W (l > k). For graphs, this problem is solved by Theo- rem 6.4 of Nash-Williams, whose proof makes use of the basic properties of Eulerian graphs; so it might be interesting to find an appropriate definition of “Eulerian hypergraphs” that is useful in this context. One can also extend orientation problems to mixed graphs and hypergraphs. Instances where we obtain tractable problems are the framework of directed network design with orientation constraints, introduced by Khanna, Naor, and Shepherd (an extended hypergraph version is described in Theorem 6.14), and Theorem 6.5 of Frank. The latter can be seen as a special case of the combined augmentation and orientation problem solved in Theorem 7.13. This theorem gives a characterization for a degree-specified augmentation problem; we do not yet know how to solve the corresponding minimum cardinality problem.
Bibliography [1] J. Bang-Jensen, H. Gabow, T. Jordán, and Z. Szigeti, Edge-connectivity augmentation with partition constraints, SIAM J. Discrete Mathematics 12 No. 2 (1999), 160–207. [2] J. Bang-Jensen, B. Jackson, Augmenting hypergraphs by edges of size two, in: Con- nectivity Augmentation of Networks: Structures and Algorithms, Mathematical Pro- gramming (ed. A. Frank) Ser. B, Vol. 84 No. 3 (1999), 467–481. [3] J. Bang-Jensen, T. Jordán, Edge-connectivity augmentation preserving simplicity, SIAM J. Discrete Math. Vol. 11 No. 4 (1998), 603–623. [4] J. Bang-Jensen, A. Frank, B. Jackson, Preserving and increasing local edge-connectivity in mixed graphs, SIAM J. Discrete Math. 8 No. 2 (1995), 155–178. [5] J. Bang-Jensen, S. Thomassé, Decompositions and orientations of hypergraphs, IMADA Preprint 2001-10. [6] A. A. Benczúr, D.R. Karger, Quadratic-time algorithms for edge-connectivity augmentation and splitting off, J. Alg 37 (2000), 2–36. [7] A. Berg, B. Jackson, T. Jordán, Edge splitting and connectivity augmentation in directed hypergraphs, Discrete Mathematics, to appear [8] A. Berg, B. Jackson, T. Jordán, Highly edge-connected detachments of graphs and digraphs, J. Graph Theory Vol. 43, No. 3 (2003), 67–77. [9] A. Benczúr, A. Frank, Covering symmetric supermodular functions by graphs, in: Con- nectivity Augmentation of Networks: Structures and Algorithms, Mathematical Pro- gramming (ed. A. Frank), Ser. B, Vol. 84, No. 3 (1999), 483–503. [10] F. Boesch, R. Tindell, Robbins’s theorem for mixed multigraphs, Am. Math. Monthly 87 (1980), 716–719. 131
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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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48 Chapter 3. Edge-connectivity aug
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70 Chapter 4. Connectivity augmenta
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