Edge-connectivity of undirected and directed hypergraphs

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86 Chapter 5. Partition-connectivity of hypergraphs Figure 5.2: Some examples of hypercircuits A bipartite graph G = (U, V ; E) is called elementary if G is connected, |U| = |V |, and Γ(X) ≥ |X| + 1 holds for every ∅ = X ⊂ U (which is equivalent to requiring the inequality for nonempty proper subsets of V .) By a hypercircuit we mean a hypergraph the associated bipartite graph of which is elementary (see Figure 5.2). Note that in the special case when the hypergraph is a graph, this notion coincides with the usual notion of a graph circuit. By a hyperforest we mean a hypergraph H = (V, E) where there is no subset X ⊆ E for which H ′ = (∪(X ), X ) is a hypercircuit. This is equivalent to saying that H satisfies the strong Hall condition, or that there are at most |X| − 1 hyperedges of H induced by X for every ∅ = X ⊆ V . A hyperforest H = (V, E) is a spanning hypertree if ∪(E) = V and |E| = |V | − 1. Erdős had conjectured and Lovász [56] proved that the node set of a hyperforest can always be coloured by two colours so that every hyperedge contains nodes of both colours. What Lovász actually proved (in a more general form) was the following result (which can also be derived from Theorem 2.8 of Edmonds on matroid intersection): Theorem 5.2. A hypergraph H is a hyperforest if and only if it is wooded. This result clearly implies Erdős’ conjecture since the hypergraph can be reduced to a forest that is bipartite, and a two-colouring of its nodes gives the required two-colouring of the hypergraph. It should be mentioned that the proof of Lovász is constructive, so it gives an algorithm for deciding whether a hypergraph is wooded. (Note that the word wooded in Hungarian translates to erdős.) Let us turn our attention to the relation between wooded sub-hypergraphs and matroid theory. In Chapter 1 we already mentioned the well-known fact that the subforests of an undirected graph G = (V, E) form the family of independent sets of a matroid on the ground set E, called the circuit matroid of G. Lorea [54] extended this notion to hypergraphs. Theorem 5.3 (Lorea). Given a hypergraph H = (V, E), the sub-hypergraphs of H which are hyperforests (or equivalently, the wooded sub-hypergraphs of H) form the family of independent sets of a matroid on the ground set E.
Section 5.2. Tree-reducible hypergraphs 87 Proof. We define the set-function bH : 2 E → Z+ by bH(X ) = | ∪ (X )| − 1 (5.1) for a non-empty subset X ⊆ E, and bH(∅) := 0. Since | ∪ (X )| is a fully submodular function of the ground set E, bH is intersecting submodular, and it is obviously monotone increasing. Let us consider the matroid MbH defined in Theorem 2.16. In this a subset X ⊆ F is independent if |Z| ≤ bH(Z) holds for every subset Z ⊆ X . This is equivalent to requiring that the sub-hypergraph H ′ = (V, X ) should meet the strong Hall condition, that is, by Theorem 5.2, H ′ should be wooded. A matroid arising this way is called the circuit matroid of the hypergraph H and it is denoted by MH. We call a matroid which is the circuit matroid of a hypergraph a hypergraphic matroid. By choosing H to be the hypergraph consisting of a three-element ground set V and of four copies of V as hyperedges, we can observe that the uniform matroid U4,2 (the smallest non-binary matroid) is hypergraphic. It should be noted that unlike graphic matroids, the class of hypergraphic matroids is not closed under contraction. 5.2.2 Rank function For a hypergraph H = (V, E), let bH be the set function defined by (5.1) for ∅ = X ⊆ E. We can apply Theorem 2.16 of Edmonds to determine the rank-function of the circuit matroid MH. Theorem 5.4. The rank function rH of the circuit matroid of a hypergraph H is given by the following formula: rH(X ) = min{|V | − |F| + eX (F) : F is a partition of V }. (5.2) Proof. It suffices to prove the formula for the special case X = E since the value of eX (F) does not change if the hyperedges in E − X are deleted. Let H ′ = (V, E ′ ) be a wooded sub-hypergraph of H. Then, for every partition F of V , there are at most |V |−|F| hyperedges in H ′ which are induced by a member of F according to the strong Hall condition. Therefore |E ′ | cannot be bigger than |V | − |F| + eE ′(F), that is, rH(E) ≤ |V | − |F| + eE(F). We have to prove the existence of a partition for which equality holds. By Theorem 2.16, rH(E) = min{ t i=1 bH(Zi) + |E − (∪ t i=1Zi)| : {Z1, . . . , Zt} is a subpartition of E}. (5.3)
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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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28 Chapter 2. Submodular functions
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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
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Page 132 and 133: 132 Bibliography [11] E. Cheng, Edg
Page 134 and 135: 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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