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

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 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 2 / 23

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 ⊆ 2 V 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. 3 / 23

On the Cycle Structures of Hypergraphs 1. Cycle of hypergraph Let E be a hypergraph. Definition A cycle of E is a sequence ( e 0 , e 1 , · · · , e k − 1 ) in E satisfying that ∀ i ∈ Z k and ∀ e ∈ E , ( S i − 1 ∪ S i ∪ S i +1 ) \ e � = ∅ . where S i = e i ∩ e i +1 , for i ∈ Z k . 4 / 23

On the Cycle Structures of Hypergraphs 1. Cycle of hypergraph A simple cycle of E is a sequence ( e 0 , e 1 , · · · , e k − 1 ) in E satisfying that ∀ three different subscripts i , j , l ∈ Z k and ∀ e ∈ E , we have ( S i ∪ S j ∪ S l ) \ e � = ∅ . Sizes of the cycles k − 1 Let C = ( e 0 , e 1 , · · · , e k − 1 ) be a cycle of E , S ( C ) = � | S i | is the i =0 size of C . 5 / 23

On the Cycle Structures of Hypergraphs 2. Equivalences and extensions of cycles Let C = ( e 0 , e 1 , · · · , e k − 1 ) be a cycle of E . If there exists i ∈ Z k and e ′ i ∈ E such that S i − 1 ∪ S i ⊂ e ′ i then obviously, C ′ = ( e 0 , · · · , e i − 1 , e ′ i , e i +1 , · · · , e k − 1 ) is a cycle of E . We say that C ′ and C are equivalent if e i − 1 ∩ e ′ i = S i − 1 and e ′ i ∩ e i +1 = S i 6 / 23

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 ( e i − 1 ∩ e ′ i ) ∪ ( e ′ i ∩ e i +1 ) ⊃ S i − 1 ∪ S i . 7 / 23

On the Cycle Structures of Hypergraphs 2. Equivalences and extensions of cycles Second kind extension and contraction If there is e ∈ Z k and a sequence P = e ′ 1 , e ′ 2 , · · · , e ′ t in E with t ≥ 3 satisfying the following conditions: 1. e i = e ′ 1 , e i +1 = e ′ t . 2. e ′ j ∩ e ′ j +1 ⊃ S i , 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 ′ = ( e 0 , e 1 , · · · , e i − 1 , e ′ 1 , e ′ 2 , · · · , e ′ t , e i +2 , · · · , e k − 1 ) is a cycle of E . Then we say that C ′ is a second kind extension of C and C is a second kind contraction of C ′ . 8 / 23

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 of E . Extension, contraction and equivalence are all called homologous transformations of cycles of hypergraphs. Homologous cycles Let C 1 , C 2 be two cycles of E . We say that C 1 and C 2 are homologous if one can be obtained from other by using a sequence of homologous transformations. 9 / 23

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 C 1 and C 2 in D , C 1 and C 2 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 . 10 / 23

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 . 11 / 23

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 ∗ ). 12 / 23

On the Cycle Structures of Hypergraphs 3. Intersection closure semilattice of hypergraph Theorem 1 Let C = ( e 0 , e 1 , · · · , e k − 1 ) be a cycle of E , if e i and e i +1 are in same component of F ( S i ) − S i . 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. 13 / 23

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 = ( x 0 , x 1 , · · · , x m − 1 ). A node x i is called to be maximal(minimal) if x i ≥ x i − 1 and x i ≥ x i +1 ( x i ≤ x i − 1 and x i ≤ x i +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. 14 / 23

On the Cycle Structures of Hypergraphs 3. Intersection closure semilattice of hypergraph Σ-cycle Let C = ( x 0 , x 1 , x 2 , · · · , x 2 k − 2 , x 2 k − 1 ) be a cycle in G ( E ∗ ). x 0 , x 2 , · · · , x 2 k − 2 are maximal nodes, and x 1 , x 3 , · · · , x 2 k − 2 , x 2 k − 1 are minimal nodes. C is called a Σ-cycle in G ( E ∗ ), if the following conditions are satisfied. For i ∈ Z k , 1. { x ∈ E ∗ : x ≥ x 2 i − 2 } and { x ∈ E ∗ : x ≥ x 2 i } is respectively contained in different components of F ( x 2 i − 1 ) − x 2 i − 1 . 2. { x ∈ E ∗ : x ≥ x 2 i } ∩ { x ∈ E ∗ : x ≥ x 2 i +4 } = ∅ . 3. x 2 i = x 2 i − 1 ∨ x 2 i +1 . 4. y 1 ∈ { x ∈ E ∗ : x ≥ x 2 i − 2 } , y 2 ∈ { x ∈ E ∗ : x ≥ x 2 i } , we have that x 2 i − 1 = y 1 ∧ y 2 We also defined normal cycle in G ( E ∗ ). 15 / 23

On the Cycle Structures of Hypergraphs 4. Relations between cycles of E and cycles of G ( E ∗ ) A cycle C = ( e 0 , e 1 , · · · , e k − 1 ) of E corresponds to a cycle C in G ( E ∗ ). Let x 2 i − 1 = S i and x 2 i = x 2 i − 1 ∨ x 2 i +1 for i ∈ Z k . Then we have a cycle C in G ( E ∗ ) with x 0 , x 2 , · · · , x 2 k − 2 as maximal nodes and x 1 , x 3 , · · · , x 2 k − 1 as minimal nodes. 16 / 23

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 . 17 / 23

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 i 1 , C 0 i 2 , · · · , C 0 ip ∈ { C 0 1 , C 0 2 , · · · , C 0 t } such that in the field F 2 , C 0 i 1 + C 0 i 2 + · · · + 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. 18 / 23

On the Cycle Structures of Hypergraphs 4. Relations between cycles of E and cycles of G ( E ∗ ) Let C 1 , C 2 , · · · , C r be r cycles of E . C i corresponds to the cycle C i in G ( E ∗ ). We say that C 1 , C 2 , · · · , C r are dependent if there exist C i 1 , C i 2 , · · · , C iq ∈ { C 1 , C 2 , · · · , C r } such that in the field F 2 , α C 0 j , where C 0 1 , C 0 2 , · · · , C 0 C i 1 + C i 2 + · · · + C iq = ∅ or � α are j =1 independent 0-cycles. Otherwise C 1 , C 2 , · · · , C r are independent. 19 / 23

On the Cycle Structures of Hypergraphs 5. Some results Theorem 2 Any two cycles in a HC -class are dependent. Theorem 3 Let { C 1 , C 2 , · · · , C t } be a set of cycles of E . C ′ i is homologous to C i for 1 ≤ i ≤ t . Then { C 1 , C 2 , · · · , C t } 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. 20 / 23

On the Cycle Structures of Hypergraphs 5. Some results Theorem 4 [Wang and Lee] The maximum number of independent maximal cycles of E is x ∈E ∗ ( c + ( x ) − 1) 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 x ∈E ∗ ( c + ( x ) − 1) α ( E ) = 1 + � 21 / 23