On the Cycle Structures of Hypergraphs Jianfang Wang Academy of - - PowerPoint PPT Presentation

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On the Cycle Structures of Hypergraphs On the Cycle Structures of Hypergraphs Jianfang Wang Academy of Mathematics and System Science, Chinese Academy of Science. Beijing 100190, China. 1 / 23 On the Cycle Structures of Hypergraphs 1. Cycle

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On the Cycle Structures of Hypergraphs

On the Cycle Structures of Hypergraphs

Jianfang Wang

Academy of Mathematics and System Science, Chinese Academy of Science. Beijing 100190, China.

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On the Cycle Structures of Hypergraphs

  • 1. Cycle of hypergraph
  • 2. Equivalences and extensions of cycles
  • 3. Intersection closure semilattice of hypergraph
  • 4. Relations between cycles of E and cycles of G(E∗)
  • 5. Some results

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On the Cycle Structures of Hypergraphs

  • 1. Cycle of hypergraph

Hypergraphs are families of subsets of finite sets.

Definition

Let V be a finite set, E ⊆ 2V is called a hypergraph on V , if ∀e ∈ E, e = ∅ and E = V . We constructed a cycle structure system of hypergraphs. Now, we give main parts of the system.

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On the Cycle Structures of Hypergraphs

  • 1. Cycle of hypergraph

Let E be a hypergraph.

Definition

A cycle of E is a sequence (e0, e1, · · · , ek−1) in E satisfying that ∀i ∈ Zk and ∀e ∈ E, (Si−1 ∪ Si ∪ Si+1)\e = ∅. where Si = ei ∩ ei+1, for i ∈ Zk.

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On the Cycle Structures of Hypergraphs

  • 1. Cycle of hypergraph

A simple cycle of E is a sequence (e0, e1, · · · , ek−1) in E satisfying that ∀ three different subscripts i, j, l ∈ Zk and ∀e ∈ E, we have (Si ∪ Sj ∪ Sl)\e = ∅.

Sizes of the cycles

Let C = (e0, e1, · · · , ek−1) be a cycle of E, S(C) =

k−1

  • i=0

|Si| is the size of C.

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On the Cycle Structures of Hypergraphs

  • 2. Equivalences and extensions of cycles

Let C = (e0, e1, · · · , ek−1) be a cycle of E. If there exists i ∈ Zk and e′

i ∈ E such that Si−1 ∪ Si ⊂ e′ i then

  • bviously, C ′ = (e0, · · · , ei−1, e′

i, ei+1, · · · , ek−1) is a cycle of E.

We say that C ′ and C are equivalent if ei−1 ∩ e′

i = Si−1 and

e′

i ∩ ei+1 = Si

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On the Cycle Structures of Hypergraphs

  • 2. Equivalences and extensions of cycles

First kind extension and contraction

We say that C ′ is a first kind extension of C and C is a first kind contraction of C ′ if (ei−1 ∩ e′

i) ∪ (e′ i ∩ ei+1) ⊃ Si−1 ∪ Si.

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On the Cycle Structures of Hypergraphs

  • 2. Equivalences and extensions of cycles

Second kind extension and contraction

If there is e ∈ Zk and a sequence P = e′

1, e′ 2, · · · , e′ t in E with

t ≥ 3 satisfying the following conditions:

  • 1. ei = e′

1, ei+1 = e′ t.

  • 2. e′

j ∩ e′ j+1 ⊃ Si, for 1 ≤ j ≤ t − 1.

  • 3. any segment of P does not form a cycle of the hypergraph

E′ = {e′

1, e′ 2, · · · , e′ t}.

  • 4. C ′ = (e0, e1, · · · , ei−1, e′

1, e′ 2, · · · , e′ t, ei+2, · · · , ek−1) is a cycle

  • f E.

Then we say that C ′ is a second kind extension of C and C is a second kind contraction of C ′.

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On the Cycle Structures of Hypergraphs

  • 2. Equivalences and extensions of cycles

Maximal cycles

A cycle of E which has no any extension is called a maximal cycle

  • f E.

Extension, contraction and equivalence are all called homologous transformations of cycles of hypergraphs.

Homologous cycles

Let C1, C2 be two cycles of E. We say that C1 and C2 are homologous if one can be obtained from other by using a sequence

  • f homologous transformations.

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On the Cycle Structures of Hypergraphs

  • 2. Equivalences and extensions of cycles

Homologous cycle class

A set D of cycles of E is called to be a homologous cycle class, simply HC-class, if

  • 1. for any pair of cycles C1 and C2 in D, C1 and C2 are

homologous.

  • 2. for a cycle C in D and a cycle C ′ of E, if C ′ is homologous to

C, then C ′ ∈ D. Thus a HC-class of E is an equivalent class of cycles of E. So all HC-classes form a partition of the set of all cycles of E.

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On the Cycle Structures of Hypergraphs

  • 3. Intersection closure semilattice of hypergraph

Definition

The intersection closure semilattice E∗ of E is defined as follows: (a) E ⊆ E∗ (b) x, y ∈ E∗ implies x ∩ y ∈ E∗ It is easy to test that (E∗, ⊆) is a meet-semilattice in which x ≤ y ⇐ ⇒ x ⊆ y.

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On the Cycle Structures of Hypergraphs

  • 3. Intersection closure semilattice of hypergraph

Hassen digraph and Hassen graph of E∗

If x < y and there exists no z ∈ E∗ such that x < z < y, then we say y covers x, write x ⋖ y. Hassen digraph, write D(E∗), of E∗ is defined V (D(E∗)) = E∗. A(D(E∗)) = {(x, y) : x, y ∈ E∗ and x ⋖ y}. Hassen graph of E∗ is the basic graph of D(E∗).

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On the Cycle Structures of Hypergraphs

  • 3. Intersection closure semilattice of hypergraph

Theorem 1

Let C = (e0, e1, · · · , ek−1) be a cycle of E, if ei and ei+1 are in same component of F(Si) − Si. Then C has an extension. where F(x) = G(E∗)[{y ∈ E∗ : y ≥ x}] and denote by c+(x) the number of components of F(x) − x, for x ∈ E∗. The elements of E∗ is called nodes.

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On the Cycle Structures of Hypergraphs

  • 3. Intersection closure semilattice of hypergraph

Let C be a cycle of G(E). We represent C by the sequence of nodes of C. C = (x0, x1, · · · , xm−1). A node xi is called to be maximal(minimal) if xi ≥ xi−1 and xi ≥ xi+1(xi ≤ xi−1 and xi ≤ xi+1). Obviously, the maximal nodes and minimal nodes appears alternately on C. The number of maximal nodes equals the number of minimal nodes. The number is called the width of C, write ω(C) . We call C as an 0-cycle if ω(C) = 1.

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On the Cycle Structures of Hypergraphs

  • 3. Intersection closure semilattice of hypergraph

Σ-cycle

Let C = (x0, x1, x2, · · · , x2k−2, x2k−1) be a cycle in G(E∗). x0, x2, · · · , x2k−2 are maximal nodes, and x1, x3, · · · , x2k−2, x2k−1 are minimal nodes. C is called a Σ-cycle in G(E∗), if the following conditions are satisfied. For i ∈ Zk,

  • 1. {x ∈ E∗ : x ≥ x2i−2} and {x ∈ E∗ : x ≥ x2i} is respectively

contained in different components of F(x2i−1) − x2i−1.

  • 2. {x ∈ E∗ : x ≥ x2i} ∩ {x ∈ E∗ : x ≥ x2i+4} = ∅.
  • 3. x2i = x2i−1 ∨ x2i+1.
  • 4. y1 ∈ {x ∈ E∗ : x ≥ x2i−2}, y2 ∈ {x ∈ E∗ : x ≥ x2i}, we have

that x2i−1 = y1 ∧ y2 We also defined normal cycle in G(E∗).

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On the Cycle Structures of Hypergraphs

  • 4. Relations between cycles of E and cycles of G(E∗)

A cycle C = (e0, e1, · · · , ek−1) of E corresponds to a cycle C in G(E∗). Let x2i−1 = Si and x2i = x2i−1 ∨ x2i+1 for i ∈ Zk. Then we have a cycle C in G(E∗) with x0, x2, · · · , x2k−2 as maximal nodes and x1, x3, · · · , x2k−1 as minimal nodes.

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On the Cycle Structures of Hypergraphs

  • 4. Relations between cycles of E and cycles of G(E∗)

A maximal cycle of E corresponds to a Σ-cycle in G(E∗). Conversely, a Σ-cycle in G(E∗) corresponds to a family of equivalent maximal cycles of E. A maximal simple cycle of E corresponds to a normal cycle in G(E∗). Conversely, a normal cycle in G(E∗) corresponds to a family of equivalent maximal simple cycles of E.

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On the Cycle Structures of Hypergraphs

  • 4. Relations between cycles of E and cycles of G(E∗)

Let C 0

1 , C 0 2 , · · · , C 0 t be t 0-cycles in G(E∗).

If there exist C 0

i1, C 0 i2, · · · , C 0 ip ∈ {C 0 1 , C 0 2 , · · · , C 0 t } such that

in the field F2, C 0

i1 + C 0 i2 + · · · + C 0 ip forms a cycle C in G(E∗) and

there exists a cycle C of E which corresponds to C, then we say that C 0

1 , C 0 2 , · · · , C 0 t are dependent 0-cycles.

Otherwise they are independent 0-cycles.

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On the Cycle Structures of Hypergraphs

  • 4. Relations between cycles of E and cycles of G(E∗)

Let C1, C2, · · · , Cr be r cycles of E. Ci corresponds to the cycle Ci in G(E∗). We say that C1, C2, · · · , Cr are dependent if there exist Ci1, Ci2, · · · , Ciq ∈ {C1, C2, · · · , Cr} such that in the field F2, Ci1 + Ci2 + · · · + Ciq = ∅ or

α

  • j=1

C 0

j , where C 0 1 , C 0 2 , · · · , C 0 α are

independent 0-cycles. Otherwise C1, C2, · · · , Cr are independent.

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On the Cycle Structures of Hypergraphs

  • 5. Some results

Theorem 2

Any two cycles in a HC-class are dependent.

Theorem 3

Let {C1, C2, · · · , Ct} be a set of cycles of E. C ′

i is homologous to

Ci for 1 ≤ i ≤ t. Then {C1, C2, · · · , Ct} is independent if and only if {C ′

1, C ′ 2, · · · , C ′ t} is independent.

So independence of cycles of E is invariant for homologous transformations.

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On the Cycle Structures of Hypergraphs

  • 5. Some results

Theorem 4 [Wang and Lee]

The maximum number of independent maximal cycles of E is 1 +

x∈E∗(c+(x) − 1)

From theorem 3 and theorem 4, we immediately obtain the following theorem.

Theorem 5 [Wang and Yan]

The maximum number of independent cycles of E is α(E) = 1 +

x∈E∗(c+(x) − 1)

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On the Cycle Structures of Hypergraphs

  • 5. Some results

Theorem 5 holds for graphs. For a graph G, α(G) is equal to cyclomatic number of G. For plane graph, the formula equivalent to Euler formula. So the theorem 5 shows that the cycle structure system of hypergraphs established above is scientific.

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On the Cycle Structures of Hypergraphs

  • 5. Some results

Thanks for your attention!

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