Edge-connectivity of undirected and directed hypergraphs

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56 Chapter 3. Edge-connectivity augmentation of hypergraphs account) must be connected, therefore it needs to have at least l−1 ν−1 partition is called a deficient partition if l − 1 ν − 1 m(V ) > . ν hyperedges. A p-full Theorem 3.14. Let p : 2 V → Z+ be a symmetric, positively crossing supermodular set function, ν ≥ 2 an integer, and m : V → Z+ a degree specification such that ν | m(V ). There is a ν-uniform hypergraph H covering p such that dH(v) = m(v) for every v ∈ V if and only if the following are true: m(X) ≥ p(X) for every X ⊆ V , (3.9) m(V ) ≥ p(X) ν for every X ⊆ V , (3.10) There are no deficient partitions. (3.11) Proof. The necessity of the conditions is easily verifiable. We prove sufficiency using induction on |V | + m(V ). If m(V ) = ν, conditions (3.9) and (3.10) are clearly sufficient, so we can assume that m(V ) ≥ 2ν. A ν-uniform hypergraph is called feasible if it matches the degree specification and covers p. First we show that if there is a set X ⊆ V such that m(X) = p(X) = 1 and |X| > 1, then there exists a feasible hypergraph. The contraction of X leads to a modified problem: V ′ := V − X + vX, m ′ (v) := m(v) if v ∈ V ′ − vX, m ′ (vX) := 1, p ′ (Y ) := p(Y ) if vX /∈ Y , and p ′ (Y ) := p((Y −vX)∪X) if vX ∈ Y . Conditions (3.9)–(3.11) are satisfied by m ′ and p ′ , and p ′ is symmetric and positively crossing supermodular, so by induction there is a ν-uniform hypergraph H ′ = (V ′ , E ′ ) with degree vector m ′ , that covers p ′ . This hypergraph naturally defines a ν-uniform hypergraph H = (V, E) with degree vector m; we claim that H covers p. Suppose that some Y ⊆ V is deficient, i.e. dH(Y ) < p(Y ). Then p(Y ) ≥ 1 and m(Y ) ≥ 2, so Y ⊆ X. Furthermore, Y must separate X, otherwise there would be a corresponding deficient set in the contracted problem. If m(X ∩ Y ) > 0, then we may assume that V − (X ∪ Y ) = ∅, because otherwise 0 = m(X − Y ) < p(Y ) = p(X − Y ). Using (2.2), p(X ∪ Y ) ≥ p(X) + p(Y ) − p(X ∩ Y ) ≥ p(Y ) > dH(Y ) = dH(X ∪ Y ), so X ∪ Y would be deficient. If m(X ∩ Y ) = 0, then p(Y − X) ≥ p(X) + p(Y ) − p(X − Y ) ≥ p(Y ) > dH(Y ) = dH(Y − X) according to (2.9), so Y − X would be deficient. From now on it is assumed that if m(X) = p(X) = 1 for some X ⊆ V , then |X| = 1; these singletons are called special singletons. The set of special singletons is denoted by S, and it is considered as a subset of V . We define an operation called splitting-off, which is an analogue of the splitting-off operation for graphs. A ν-hyperedge e can be split off from (p, m) if |e ∩ v| ≤ m(v) for
Section 3.4. Covering by uniform hypergraphs 57 every v ∈ V . For such a hyperedge let m e (v) := m(v) − |e ∩ v|, (3.12) p e (p(X) − 1) (X) := + if e enters X, (3.13) p(X) otherwise. We say that (m e , p e ) is obtained from (m, p) by splitting off the hyperedge e. A splitting-off operation is feasible if (3.9), (3.10), and (3.11) are true for m e and p e . It is easy to see that p e is symmetric and positively crossing supermodular; so after the execution of a feasible splitting-off, by induction there exists a ν-uniform hypergraph H ′ with degree vector m e that covers p e . By adding the hyperedge e to H ′ we obtain a feasible hypergraph H. The rest of the proof consists of showing that a feasible splitting-off always exists. We define the following families of sets: B1 := {∅ = X ⊆ V : m(X) − p(X) ≤ ν − 2, p(Y ) < p(X) ∀Y ⊂ X}, B2 := {X ⊆ V : m(X) − p(X) = ν − 1, p(X) > 0, p(Y ) ≤ p(X) ∀Y ⊂ X}, m(V ) m(V ) B3 := {X ⊆ V : p(X) = , p(Y ) < ∀Y ⊂ X}. ν ν Let e be a ν-hyperedge that can be split off, and suppose that m e and p e violates (3.9) because there is a set X entered by e such that p(X) − 1 > m(X) − |e ∩ X|. Notice that m(X ′ ) − |e ∩ X ′ | ≤ m(X) − |e ∩ X| for every X ′ ⊆ X. Thus there must be a set X ′ ∈ B1 such that X ′ ⊆ X and p(X ′ ) − 1 > m(X ′ ) − |e ∩ X ′ |. Similarly, if m e and p e violates (3.9) because there is a set X such that e ⊆ X and p(X) = m(X) − (ν − 1), then either there is a set X ′ ∈ B2 such that X ′ ⊆ X and e ⊆ X ′ , or there is a set X ′ ∈ B1 such that X ′ ⊆ X and p(X ′ ) − 1 > m(X ′ ) − |e ∩ X ′ |. A similar argument shows that if m e and p e violate (3.9) then there is a set X ∈ B3 such that e ⊆ V − X. We obtained the following: Claim 3.15. The inequalities (3.9) and (3.10) hold for m e and p e if and only if |e ∩ X| ≤ m(X) − p(X) + 1 for every X ∈ B1, (3.14) |e ∩ X| ≤ ν − 1 for every X ∈ B2, (3.15) |e ∩ X| ≥ 1 for every X ∈ B3. (3.16) Furthermore, if m e (X) of X such that either X ′ ∈ B1 and it violates (3.14), or X ′ ∈ B2 and it violates (3.15). In order to formulate necessary and sufficient conditions for m e and p e to satisfy (3.11), we call a p-full partition {V1, . . . , Vl} critical if l − 1 ν − 1 > m(V ) ν − 1.
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Edge-connectivity of undirected and
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4 Contents 6 Hypergraph orientation
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Page 8 and 9: 8 Chapter 1. Introduction and preli
Page 10 and 11: 10 Chapter 1. Introduction and prel
Page 28 and 29: 28 Chapter 2. Submodular functions
Page 48 and 49: 48 Chapter 3. Edge-connectivity aug
Page 70 and 71: 70 Chapter 4. Connectivity augmenta
Page 84 and 85: 84 Chapter 5. Partition-connectivit
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
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106 Chapter 6. Hypergraph orientati
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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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