Edge-connectivity of undirected and directed hypergraphs

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58 Chapter 3. Edge-connectivity augmentation of hypergraphs For a partition F, let s(F) denote the number of special singleton members of F. A critical partition F is called proper if s(F) ≥ 3. Critical partitions have the following properties: Claim 3.16. If F = {V1, . . . , Vl} is a critical partition, then 2l − 2 ≥ m(V ), thus s(F) ≥ m(Vl). In particular, s(F) ≥ 2 for every critical partition, and the partition is proper if m(Vl) ≥ 3. Proof. The partition is critical and m(V ) ≤ ν since p-fullness implies that l > ν. m(V ) ν l − 2 + 1 ν − 1 is an integer, so ≤ 2ν + 2(l − ν − 1) = 2l − 2, = 2ν + ν (l − ν − 1) ≤ ν − 1 Claim 3.17. A partition {V1, . . . , Vl} is critical if and only if l > ν, l−1 ν−1 > m(V ) ν − 1, p(V1) = 1, and p(V1 ∪ Vi) ≥ 1 (i = 2, . . . , l). If the partition is critical and U is the union of some partition members such that V1 ⊆ U and V2 ∩ U = ∅, then p(U) = 1. Proof. Let {V1, . . . , Vl} be a partition with the above properties, and let U = ∪i∈IVi where 1 ∈ I and |I| ≤ l −1. We can use inequality (3.7) for the sets {V1 ∪Vi : i ∈ I} to show that p(U) > 0. The symmetry of p implies that p(V − U) > 0 for all such U, so the partition is p-full. If V1 ⊆ U and V2 ∩ U = ∅, then (3.6) gives 1 = p(V1) ≤ p(U) ≤ p(V − V2) = 1. Claim 3.18. Let e be a ν-hyperedge which satisfies (3.14)–(3.16). Then m e and p e satisfy (3.11) if and only if e ⊆ X for any member X of any proper critical partition. (3.17) Proof. If e ⊆ X for a member X of a critical partition, then the partition remains p-full after the splitting off of e, hence the partition becomes deficient. To see the converse, observe that only critical partitions can become deficient partitions after the splitting-off. If e ⊆ X for every member X of a given critical partition F, then it is easy to see that there exists a set U entered by e which is the union of some members of F including exactly 1 special singleton member. According to Claim 3.17, p(U) = 1, so p e (U) = 0 and F is not p-full after the splitting-off. It remains to show that if e ⊆ X for a member X of a non-proper critical partition, then some set violates (3.14) or (3.15). But m(X) ≤ 2 according to Claim 3.16, so 0 = m e (X) < p(X) = p e (X) which by Claim 3.15 implies that (3.14) or (3.15) is violated for some subset of X.
Section 3.4. Covering by uniform hypergraphs 59 It suffices to show the existence of a ν-hyperedge e for which χe ≤ m and which satisfies (3.14), (3.15), (3.16), and (3.17). First we consider only (3.14) and (3.16): Q :={e ∈ Z V + : χe ≤ m; |e| = ν; |e ∩ X| ≤ m(X) − p(X) + 1 ∀X ∈ B1; |e ∩ X| ≥ 1 ∀X ∈ B3}. Observe that p(X) > 0 for every X ∈ B1 ∪ B2 ∪ B3, so inequalities (2.2) and (2.9) can be used for these sets. Claim 3.19. The family B1 ∪ B3 is laminar. The sets in B3 are pairwise disjoint, and if X ∈ B1 and Y ∈ B3 are not disjoint, then X ⊆ Y . Proof. If X, Y ∈ B1 ∪ B3, and X − Y, Y − X, X ∩ Y = ∅, then p(X) ≤ p(X − Y ) or p(Y ) ≤ p(Y − X) by (2.9), which contradicts the definition of B1 and B3. If X ∈ B1 and Y ∈ B3, then p(Y ) ≥ p(X), so Y ⊂ X according to the definition of B1. Claim 3.20. Q is non-empty. Proof. We use the following algorithm to find an element e of Q: 1. Construct a hyperedge e ′ by choosing one node from every X ∈ B3. 2. Fix an arbitrary ordering of the nodes of V , and consider them one by one. Increase the multiplicity in e ′ of each node to the maximum value with which the obtained hyperedge does not violate conditions of type (3.14) and its size is at most ν. It follows from Claim 3.19 that this algorithm finds a hyperedge e ∈ Q if and only if the following hold: (i) |B3| ≤ ν, (ii) m(V − ∪ t i=1Xi) + t i=1 (m(Xi) − p(Xi) + 1) ≥ ν for every sub-partition {X1, . . . , Xt}. The first condition holds since m(X) ≥ m(V ) ν for every X ∈ B3, and they are disjoint by Claim 3.19. The second condition is clearly true if t ≥ ν. If t < ν, then m(V − ∪Xi) + (m(Xi) − p(Xi) + 1) ≥ m(V − ∪Xi) + (m(Xi) − last inequality being valid because m(V ) ≥ ν. m(V ) ν m(V ) + 1) = (ν − t) + t ≥ ν, the ν Obviously, if a ν-hyperedge e can be feasibly split off, then it is in Q. The converse is generally not true; however, it turns out to be true when B3 = ∅: Lemma 3.21. If B3 = ∅, then any ν-hyperedge e ∈ Q can be feasibly split off.
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Edge-connectivity of undirected and
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4 Contents 6 Hypergraph orientation
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6 Contents
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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108 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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Edge-connectivity of undirected and directed hypergraphs

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