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## Pfaffian graphs, t-joins, and crossing numbers

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Citations: | 6 - 0 self |

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716 |
Matching Theory,
- Lovasz, Plummer
- 2009
(Show Context)
Citation Context ...ertices of odd degree. Perfect matching are other example of a T -join, where T = V (G). Since their introduction T -joins have been extensively studied (see for example [15], sections 6.5 and 6.6 of =-=[10]-=-, [3], section 2 of [2]). By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of [0, 1], not containing vertices in their interior... |

192 |
The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice
- Kasteleyn
- 1961
(Show Context)
Citation Context ... exists an orientation D of G such that every central cycle is oddly oriented in D, in which case we say that D is a Pfaffian orientation of G. Pfaffian orientations have been introduced by Kasteleyn =-=[4, 5, 6]-=-. He demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time. Pfaffian bipartite graphs were characterized in terms of forbidden subgraphs by Little [9]. A structu... |

131 |
Dimer statistics and phase transitions.
- Kasteleyn
- 1963
(Show Context)
Citation Context ... exists an orientation D of G such that every central cycle is oddly oriented in D, in which case we say that D is a Pfaffian orientation of G. Pfaffian orientations have been introduced by Kasteleyn =-=[4, 5, 6]-=-. He demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time. Pfaffian bipartite graphs were characterized in terms of forbidden subgraphs by Little [9]. A structu... |

78 | Geometric topology in dimension 2 and 3 - Moise - 1977 |

73 | Permanents, pfaffian orientations, and even directed circuits.
- Robertson, Seymour, et al.
- 1999
(Show Context)
Citation Context ...time. Pfaffian bipartite graphs were characterized in terms of forbidden subgraphs by Little [9]. A structural characterization of Pfaffian bipartite graphs was given by Robertson, Seymour and Thomas =-=[14]-=- and independently by McCuaig [11]. They also provided a polynomial time algorithm for recognition of Pfaffian bipartite graphs. The problem of recognition of Pfaffian bipartite graphs is equivalent t... |

68 |
Graph theory and crystal physics, in Graph theory and theoretical physics
- Kasteleyn
- 1967
(Show Context)
Citation Context ... exists an orientation D of G such that every central cycle is oddly oriented in D, in which case we say that D is a Pfaffian orientation of G. Pfaffian orientations have been introduced by Kasteleyn =-=[4, 5, 6]-=-. He demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time. Pfaffian bipartite graphs were characterized in terms of forbidden subgraphs by Little [9]. A structu... |

56 |
Toward a theory of crossing numbers.
- Tutte
- 1970
(Show Context)
Citation Context ...ph has a subgraph isomorphic to a subdivision of K5 or K3,3. One can therefore easily deduce the following well-known theorem from Theorem 4.2 and Kuratowski’s theorem. Theorem 4.3. (Hanani [1],Tutte =-=[17]-=-) Let Γ be a drawing of a nonplanar graph G in the plane. Then there exist distinct non-adjacent edges e, f ∈ E(G) such that cr(e, f) is odd. Acknowledgment. I would like to thank my advisor Robin Tho... |

51 | Combinatorial optimization: Packing and covering
- Cornuéjols
- 2001
(Show Context)
Citation Context ...erfect matching are other example of a T -join, where T = V (G). Since their introduction T -joins have been extensively studied (see for example [15], sections 6.5 and 6.6 of [10], [3], section 2 of =-=[2]-=-). By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of [0, 1], not containing vertices in their interiors. Edges are permitted ... |

47 |
Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs
- Vazirani, Yannakakis
- 1989
(Show Context)
Citation Context ...bipartite graphs. The problem of recognition of Pfaffian bipartite graphs is equivalent to many interesting problems, e.g. the Pólya permanent problem [13], the even cycle problem for directed graphs =-=[18]-=- and the problem of determining which real square matrices are sign non-singular [7]. No satisfactory characterization is known for general Pfaffian graphs. While attempting to find such a characteriz... |

46 |
Matchings in graphs on non-orientable surfaces.
- Tesler
- 2000
(Show Context)
Citation Context ...an if and only if there exists a drawing of G in the plane such that cr(M) is even for every perfect matching M of G. The “if” part of this theorem was known to Kasteleyn [6] and was proved by Tesler =-=[16]-=-; however our proof of this part is different. We derive Theorem 3.1 from a more general result. To state it we need a definition. Let Γ be a drawing of a graph G in the plane. We say that S ⊆ E(G) is... |

37 |
Bounds for generalized thrackles
- Cairns, Nikolayevsky
- 2000
(Show Context)
Citation Context ...r we characterize Pfaffian graphs in terms of their drawings in the plane. By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of =-=[0, 1]-=-, not containing vertices in their interiors. Edges are permitted to intersect, but there are only finitely many intersections and each intersection is a crossing. For edges e, f of a graph G drawn in... |

36 | Criticality for multicommodity flows
- Seymour
(Show Context)
Citation Context ...n, where T is the set of all vertices of odd degree. Perfect matching are other example of a T -join, where T = V (G). Since their introduction T -joins have been extensively studied (see for example =-=[15]-=-, sections 6.5 and 6.6 of [10], [3], section 2 of [2]). By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of [0, 1], not contain... |

34 |
Signsolvability revisited
- Klee, Ladner, et al.
- 1984
(Show Context)
Citation Context ...nt to many interesting problems, e.g. the Pólya permanent problem [13], the even cycle problem for directed graphs [18] and the problem of determining which real square matrices are sign non-singular =-=[7]-=-. No satisfactory characterization is known for general Pfaffian graphs. While attempting to find such a characterization I was able to obtain the following result. A self-contained proof of it will a... |

27 |
A characterization of convertible (0,1)-matrices
- Little
- 1975
(Show Context)
Citation Context ...leyn [4, 5, 6]. He demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time. Pfaffian bipartite graphs were characterized in terms of forbidden subgraphs by Little =-=[9]-=-. A structural characterization of Pfaffian bipartite graphs was given by Robertson, Seymour and Thomas [14] and independently by McCuaig [11]. They also provided a polynomial time algorithm for recog... |

18 | On Conway’s thrackle conjecture.
- Lovasz, Pach, et al.
- 1997
(Show Context)
Citation Context ...-adjacent edges e, f ∈ E(G) such that cr(e, f) is odd. Further applications of Theorem 3.1 are considered in [13] . Finally, let us note that parities of crossings have been studied in other contexts =-=[1, 9, 17]-=-. It might be interesting to analyze similarities and differences between methods employed in these papers and here. Acknowledgment. I would like to thank my advisor Robin Thomas for his guidance and ... |

13 | Pólya’s permanent problem
- McCuaig
(Show Context)
Citation Context ...re characterized in terms of forbidden subgraphs by Little [9]. A structural characterization of Pfaffian bipartite graphs was given by Robertson, Seymour and Thomas [14] and independently by McCuaig =-=[11]-=-. They also provided a polynomial time algorithm for recognition of Pfaffian bipartite graphs. The problem of recognition of Pfaffian bipartite graphs is equivalent to many interesting problems, e.g. ... |

10 |
A note on the parity of the number of crossings of a graph
- Kleitman
- 1976
(Show Context)
Citation Context ...rem 1.1 it suffices to prove that � |J ⊓ S| J∈J is even for any S ⊆ Æ(G). This is true by the definition of a nice set of T -joins. � We derive the next theorem from Lemma 4.1. Theorem 4.2. (Kleitman =-=[8]-=-) Let G = K2j+1 or G = K2j+1,2k+1 for some positive integers j and k. Then the parity of the total number of crossings of non-adjacent edges is independent of the choice of a drawing of G in the plane... |

7 |
theory and crystal physics, Graph Theory and Theoretical
- Graph
- 1967
(Show Context)
Citation Context ...). Thus we say that a graph with an arbitrary vertex-set is Pfaffian if it is isomorphic to a labeled graph that admits a Pfaffian orientation. Pfaffian orientations have been introduced by Kasteleyn =-=[5, 6, 7]-=-, who demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time, and The author was supported in part by NSF under Grant No. DMS-0200595. 1 � .s2 SERGUEI NORINE has ... |

6 |
statistics and phase transitions,
- Dimer
- 1963
(Show Context)
Citation Context ...). Thus we say that a graph with an arbitrary vertex-set is Pfaffian if it is isomorphic to a labeled graph that admits a Pfaffian orientation. Pfaffian orientations have been introduced by Kasteleyn =-=[5, 6, 7]-=-, who demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time, and The author was supported in part by NSF under Grant No. DMS-0200595. 1 � .s2 SERGUEI NORINE has ... |

5 | Drawing Pfaffian graphs
- Norine
(Show Context)
Citation Context ...ory characterization is known for general Pfaffian graphs. While attempting to find such a characterization I was able to obtain the following result. A self-contained proof of it will also appear in =-=[12]-=-. Theorem 3.1. A graph G is Pfaffian if and only if there exists a drawing of G in the plane such that cr(M) is even for every perfect matching M of G. The “if” part of this theorem was known to Kaste... |

2 |
Über weentlich unplättbare Kurven in dreidimensionalen Raume
- Chojnacki
(Show Context)
Citation Context ...ee for example [15], sections 6.5 and 6.6 of [10], [3], section 2 of [2]). By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of =-=[0, 1]-=-, not containing vertices in their interiors. Edges are permitted to intersect, but there are only finitely many intersections and each intersection is a crossing. For edges e, f of a graph G drawn in... |

2 |
Überweentlich unplättbare kurven in dreidimensionalen raume
- Hanani
- 1934
(Show Context)
Citation Context ... easily deduce the following well-known theorem from Theorem 4.2 and Kuratowski’s theorem. This observation most likely is not new.sPFAFFIAN GRAPHS, T-JOINS AND CROSSING NUMBERS 9 Theorem 4.3 (Hanani =-=[4]-=-,Tutte [17]). Let Γ be a drawing of a non-planar graph G in the plane. Then there exist distinct non-adjacent edges e, f ∈ E(G) such that cr(e, f) is odd. Further applications of Theorem 3.1 are consi... |

1 |
On the theory of Pfaffian orientations. II. T -joins, k-cuts, and duality of enumeration
- Galluccio, Loebl
- 1999
(Show Context)
Citation Context ...s of odd degree. Perfect matching are other example of a T -join, where T = V (G). Since their introduction T -joins have been extensively studied (see for example [15], sections 6.5 and 6.6 of [10], =-=[3]-=-, section 2 of [2]). By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of [0, 1], not containing vertices in their interiors. Ed... |