Edge-connectivity of undirected and directed hypergraphs

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138 Index in-degree of node, 19 induced edge, 11 induced hyperarc, 19 induced hyperedge, 17 intersecting sets, 27 intersecting supermodular set function, 34 laminar family, 27 local edge-connectivity, 12 maximizing oracle, 45 minimizer, 88 minimizing oracle, 45 minimum cardinality problem, 48 mixed hypergraph, 104, 125 modular set function, 30 node induced cost, 12 one-way system, 44 orientation constraints, 104 orientation of graph, 11 orientation of hypergraph, 19 oriented sub-hypergraph, 104 out-degree of node, 19 partition-connectivity, 83 path in directed hypergraph, 19 path in hypergraph, 17 positively crossing supermodular, 35 proper critical partition, 58 rank of hypergraph, 17 regular family, 24 requirement function, 100 semi-parallel hyperarcs, 106 separating a set, 13 set function, 12 skew supermodular set function, 37 spanning hypertree, 86 spanning sub-hypergraph, 84 special singletons, 56 splitting-off, 48, 56 Steiner tree, 14, 93 strong Hall condition, 85 strongly connected dir. hypergraph, 19 submodular flow, 44 subpartition, 22 sum of matroids, 32 sum of two families, 22 tail node, 19 TDI system, 23 tight partition, 102 total size, 50 tree-composition, 28 tree-reducible hypergraph, 85 tree-representation, 27 uncrossing a family, 29 uncrossing operation, 29 underlying graph, 11 underlying hypergraph, 19 uniform directed hypergraph, 19 upper truncation, 35 wooded hypergraph, 85
Notation Z+ the set of non-negative integers (x) + for x ∈ R: max{x, 0} x > 0 for x ∈ R n : x ≥ 0, x = 0 x ≫ 0 for x ∈ R n : xi > 0 (i = 1, . . . , n) X ⊆ V X is a subset of V (possibly X = V ) X ⊂ V X is a proper subset of V X − Y for X, Y ⊆ V : {v ∈ V : v ∈ X, v /∈ Y } χX m(X) for X ⊆ V : the characteristic function of the set X for X ⊆ V and m : V → R: v∈X m(v) dH(v) for a hypergraph H = (V, E) and v ∈ V : the degree of v in H ϱD(v) for a dir. hypergraph D = (V, A): the in-degree of v in D δD(v) the out-degree of v in D ∆H(X) {e ∈ E : e enters X} ∆ − D (X) {a ∈ A : a enters X} ∆ + D (X) {a ∈ A : a enters V − X} dH(X) |∆H(X)| dE(X) dH(X) for H = (V, E) ϱD(X) |∆ − D (X)| δD(X) |∆ + D (X)| iH(X) the number of hyperedges of H induced by X ϱx(Z) for D = (V, A) and x : A → R: φx(Z) δx(Z) − ϱx(Z) a∈∆ − D st-set a set X ⊆ V for which s /∈ X and t ∈ X (Z) x(a) λH(s, t) local edge-connectivity between s and t in H λD(s, t) local edge-connectivity from s to t in D χe(v) for v ∈ V and a hyperedge e: the multiplicity of v in e |e ∩ X| for X ⊆ V and a hyperedge e: χe(X) ∪(E) for H = (V, E): {v ∈ V : χe(v) > 0 for some e ∈ E} 139
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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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48 Chapter 3. Edge-connectivity aug
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70 Chapter 4. Connectivity augmenta
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84 Chapter 5. Partition-connectivit
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86 Chapter 5. Partition-connectivit
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Page 132 and 133: 132 Bibliography [11] E. Cheng, Edg
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Page 136 and 137: 136 Bibliography [66] A. Schrijver,
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