## Lattice Structures from Planar Graphs (2004)

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Venue: | Elec. J. Comb |

Citations: | 21 - 6 self |

### BibTeX

@ARTICLE{Felsner04latticestructures,

author = {Stefan Felsner},

title = {Lattice Structures from Planar Graphs},

journal = {Elec. J. Comb},

year = {2004},

volume = {15},

pages = {15}

}

### OpenURL

### Abstract

The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

### Citations

406 | Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms 9:223–252
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(Show Context)
Citation Context ...o happen quite fast in many processes of the described kind, only few of these processes have been analyzed satisfactorily. For more on this subject we recommend the work of Propp and Wilson [12] and =-=[14]-=-. 3.1 Eulerian orientations Let G be a plane graph, such that every vertex v has even degree d(v). An Eulerian orientation of G is an orientation with indeg(v) =outdeg(v) for every vertex v. Hence, th... |

196 |
Embedding planar graphs on the grid
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- 1990
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Citation Context ... each of the three special vertices ai. The closure G σ ∞ of a suspension G σ is obtained by adding a new vertex v∞, thisvertexisusedassecond endpoint for the three half-edges of G σ . Schnyder=-= [15], [16]-=- introduced edge orientations and equivalent angle labelings for planar triangulations. He used this structures for a remarkable characterization of planar graphs in terms of order dimension. The inci... |

53 |
Planar graphs and poset dimension
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- 1989
(Show Context)
Citation Context ...ch of the three special vertices a i . The closure G 1 of a suspension G is obtained by adding a new vertex v 1 , this vertex is used as second endpoint for the three half-edges of G . Schnyder [14=-=]-=-, [15] introduced edge orientations and equivalent angle labelings for planar triangulations. He used this structures for a remarkable characterization of planar graphs in terms of order dimension. Th... |

33 | Convex drawings of planar graphs and the order dimension of 3-polytopes
- Felsner
- 2001
(Show Context)
Citation Context ...research that lead to Theorem 1. The proof of the general theorem given in the first part of this paper is widely based on ideas that are already contained in the cited proofs of the special case. In =-=[2]-=- the concept of Schnyder labelings was generalized to 3-connected planar graphs. It was also shown that like the original concept the generalization yields strong applications in the areas of dimensio... |

22 |
Orientations bipolaires
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- 1994
(Show Context)
Citation Context ...ed by Miller [9] and further investigated in [3]. The set of Schnyder woods of a planar triangulation has the structure of a distributive lattice. This was independently shown by Brehm [1] and Mendez =-=[10]-=-. My original objective was to generalize this and prove that the set of Schnyder woods of a 3-connected planar graph also has a distributive lattice structure. The theory developed to this aim turned... |

19 |
Planar graphs and poset dimension, Order 5
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Citation Context ...ace to each of the three special vertices ai. The closure G σ ∞ of a suspension G σ is obtained by adding a new vertex v∞, thisvertexisusedassecond endpoint for the three half-edges of G σ . Sc=-=hnyder [15]-=-, [16] introduced edge orientations and equivalent angle labelings for planar triangulations. He used this structures for a remarkable characterization of planar graphs in terms of order dimension. Th... |

18 | The lattice structure of flow in planar graphs - Khuller, Naor, et al. - 1993 |

17 |
3-orientations and Schnyder 3-tree-decompositions. Master’s Thesis. Freie Universität
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- 2000
(Show Context)
Citation Context ...s were discovered by Miller [9] and further investigated in [3]. The set of Schnyder woods of a planar triangulation has the structure of a distributive lattice. This was independently shown by Brehm =-=[1]-=- and Mendez [10]. My original objective was to generalize this and prove that the set of Schnyder woods of a 3-connected planar graph also has a distributive lattice structure. The theory developed to... |

17 | Generating random elements of a finite distributive lattice, Electron
- Propp
- 1997
(Show Context)
Citation Context ...bserved to happen quite fast in many processes of the described kind, only few of these processes have been analyzed satisfactorily. For more on this subject we recommend the work of Propp and Wilson =-=[12]-=- and [14]. 3.1 Eulerian orientations Let G be a plane graph, such that every vertex v has even degree d(v). An Eulerian orientation of G is an orientation with indeg(v) =outdeg(v) for every vertex v. ... |

15 | Planar graphs as minimal resolutions of trivariate monomial ideals
- Miller
(Show Context)
Citation Context ... Schnyder wood for the dual. 1 Introduction This work originated in the study of rigid embeddings of planar graphs and the connections with Schnyder woods. These connections were discovered by Miller =-=[9]-=- and further investigated in [3]. The set of Schnyder woods of a planar triangulation has the structure of a distributive lattice. This was independently shown by Brehm [1] and Mendez [10]. My origina... |

10 | On topological aspects of orientations
- Fraysseix, Mendez
(Show Context)
Citation Context ...aim turned out to work in a more general situation. In the first half of this paper we present a theory of α-orientations of a planar graph and show that they form a distributive lattice. As noted in=-= [4]-=- this result was already obtained in the thesis of Mendez [10]. Another source for related results is a paper of Propp [13] where he describes lattice structures in the dual setting. The cover relatio... |

10 |
On reorienting graphs by pushing down maximal vertices, Order 3
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Citation Context ...cribes lattice structures in the dual setting. The cover relations in Propp’s lattices are certain pushing-down operations. These operations were introduced by Mosesian and further studied by Pretze=-=l [11]-=- as reorientations of diagrams of ordered sets. The second part of the paper deals with special instances of the general result. In particular we find lattice structures on the following combinatorial... |

9 | Geodesic embeddings and planar graphs
- Felsner
- 2003
(Show Context)
Citation Context ...ntroduction This work originated in the study of rigid embeddings of planar graphs and the connections with Schnyder woods. These connections were discovered by Miller [9] and further investigated in =-=[3]-=-. The set of Schnyder woods of a planar triangulation has the structure of a distributive lattice. This was independently shown by Brehm [1] and Mendez [10]. My original objective was to generalize th... |

5 |
The duality conjecture in formal knot theory
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- 1986
(Show Context)
Citation Context ...al edge. The lattice structure on spanning trees of a planar graph has been the electronic journal of combinatorics 11 (2004), #R15 1sdiscovered in the context of knot theory by Gilmer and Litherland =-=[5]-=- and by Propp [13] as an example of his lattice structures. A closely related family of examples concerns lattices on matchings and more generally f-factors of plane bipartite graphs. To show that the... |

4 |
Lattice structure for orientations of graphs. Manuscript: www.math.wisc.edu/∼propp/orient.html
- Propp
- 1993
(Show Context)
Citation Context ...of a planar graph and show that they form a distributive lattice. As noted in [4] this result was already obtained in the thesis of Mendez [10]. Another source for related results is a paper of Propp =-=[13] w-=-here he describes lattice structures in the dual setting. The cover relations in Propp’s lattices are certain pushing-down operations. These operations were introduced by Mosesian and further studie... |

2 |
Generating random elements of distributive lattices
- Propp
- 1997
(Show Context)
Citation Context ...observed to happen quite fast in many processes of the described kind only few of these processes have been analyzed satisfactorily. For more on this subject we recommend the work of Propp and Wilson =-=[11]-=- and [13]. 11 3.1 Eulerian orientations Let G be a plane graph, such that every vertex v has even degree d(v). An Eulerian orientation of G is an orientation with indeg(v) = outdeg(v) for every vertex... |

1 |
A distributive lattice on the set of perfect matchings of a plane bipartite graph, Order 20
- Lam, Zhang
- 2003
(Show Context)
Citation Context ...rley’s bijection between spanning trees and matchings of square grids. The general correspondence between trees and matchings has been exploited in [6]. Another recent source for the above theorem i=-=s [8].-=- 3.5 Schnyder woods Let G be a plane graph and let a1,a2,a3 be three different vertices in clockwise order from the outer face of G. Thesuspension G σ of G is obtained by adding a half-edge that reac... |

1 | The lattice structure of in planar graphs - Khuller, Naor, et al. - 1993 |