## Convex drawings of Planar Graphs and the Order Dimension of 3-Polytopes (2000)

Venue: | ORDER |

Citations: | 33 - 13 self |

### BibTeX

@ARTICLE{Felsner00convexdrawings,

author = {Stefan Felsner},

title = {Convex drawings of Planar Graphs and the Order Dimension of 3-Polytopes},

journal = {ORDER},

year = {2000},

volume = {18},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.

### Citations

382 | How to draw a graph
- Tutte
- 1963
(Show Context)
Citation Context ...see e.g. [1]). The method of Schnyder's still gives the strongest result regarding the size of a grid embedding arbitrary planar graphs. Xin He [5] gives a comprehensive history of the problem. Tutte =-=[14, 15]-=- shows that every 3-connected planar graph G admits a strictly convex drawing, i.e. a drawing such that the boundary of every face is a strictly convex polygon. Actually this result can be obtained as... |

361 |
Lectures on Polytopes
- Ziegler
- 1995
(Show Context)
Citation Context ...vex polygon. Actually this result can be obtained as an easy consequence of much older theorems of Steinitz and of the Koebe circle packing theorem, a reviews of these connections is given by Ziegler =-=[17-=-]. Again the early approaches to convex drawings give no reasonable guarantee on resolution. We dene triorientations of the edges of 3-connected planar graphs so that they resemble the 3-tree decompos... |

197 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ....e., on a grid of polynomial size, was raised by Rosenstiehl and Tarjan [8]. Schnyder [9] shows how to construct a barycentric representation which yields an embedding on the (2n 5) (2n 5) grid. In [=-=1-=-0] Schnyder improved on hissrst result and shows 4 September 2000 1 Let G be a planar graph with n vertices, then G has a straight line drawing with its vertices embedded on the (n 2) (n 2) grid. Th... |

156 |
Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins
- Trotter
- 1992
(Show Context)
Citation Context ...n equals its order dimension. The dimension of a graph is just the interval dimension of its incidence order. For additional information on this background we suggest looking at Trotter's monograph [=-=1-=-2]. If G is a graph containing a cycle then dim(G) 3. It is also easy to construct a realizer consisting of 3 linear orders for the cycle Cn , n 3. The dimension of the complete graph K 5 is 4, but ... |

124 |
On straight lines representation of planar graphs
- Fáry
- 1948
(Show Context)
Citation Context ...n Schnyder had for his tree decompositions concerns straight line drawings of planar graphs. The existence of straight line embeddings for planar graphs was independently proven by Wagner [16], Fary [=-=3]-=- and Stein [11]. The question whether every planar graph has a straight line embedding of reasonable resolution, i.e., on a grid of polynomial size, was raised by Rosenstiehl and Tarjan [8]. Schnyder ... |

97 |
Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung
- Wagner
- 1936
(Show Context)
Citation Context ...d application Schnyder had for his tree decompositions concerns straight line drawings of planar graphs. The existence of straight line embeddings for planar graphs was independently proven by Wagner =-=[16-=-], Fary [3] and Stein [11]. The question whether every planar graph has a straight line embedding of reasonable resolution, i.e., on a grid of polynomial size, was raised by Rosenstiehl and Tarjan [8]... |

75 |
Partially ordered sets. In
- Trotter
- 1995
(Show Context)
Citation Context ...t 3. Proof. The easier part of the theorem is to show that dim(G) 3 implies that G is planar. The proof of this implication is actually due to Babai and Duus, the argument can be found in [12] and [1=-=3]-=-. We show that every planar graph G admits a realizer fL 1 ; L 2 ; L 3 g. By monotonicity we may assume that G is a maximal planar graph, i.e., a triangulation. The Schnyder 13 labeling of G induces t... |

66 |
Rectilinear planar layouts and bipolar orientations of planar graphs
- Rosenstiehl, Tarjan
- 1986
(Show Context)
Citation Context ...16], Fary [3] and Stein [11]. The question whether every planar graph has a straight line embedding of reasonable resolution, i.e., on a grid of polynomial size, was raised by Rosenstiehl and Tarjan [=-=-=-8]. Schnyder [9] shows how to construct a barycentric representation which yields an embedding on the (2n 5) (2n 5) grid. In [10] Schnyder improved on hissrst result and shows 4 September 2000 1 Let... |

55 |
Convex maps
- Stein
- 1951
(Show Context)
Citation Context ... for his tree decompositions concerns straight line drawings of planar graphs. The existence of straight line embeddings for planar graphs was independently proven by Wagner [16], Fary [3] and Stein [=-=11]-=-. The question whether every planar graph has a straight line embedding of reasonable resolution, i.e., on a grid of polynomial size, was raised by Rosenstiehl and Tarjan [8]. Schnyder [9] shows how t... |

53 |
Planar graphs and poset dimension
- Schnyder
- 1989
(Show Context)
Citation Context ...every plane triangulation admits a special decomposition of its interior edges into three trees. Based on these Schnyder 3-tree decompositions he proved two beautiful theorems about planar graphs. In =-=[9-=-] Schnyder characterized planar graphs in terms of the order dimension of their incidence order: The dimension of the incidence order of vertices and edges of a graph G is at most 3 () G is planar. T... |

24 |
Tutte, Convex representations of graphs
- T
- 1960
(Show Context)
Citation Context ...see e.g. [1]). The method of Schnyder's still gives the strongest result regarding the size of a grid embedding arbitrary planar graphs. Xin He [5] gives a comprehensive history of the problem. Tutte =-=[14, 15]-=- shows that every 3-connected planar graph G admits a strictly convex drawing, i.e. a drawing such that the boundary of every face is a strictly convex polygon. Actually this result can be obtained as... |

17 |
3-orientations and Schnyder 3-tree-decompositions. Master’s Thesis. Freie Universität
- Brehm
- 2000
(Show Context)
Citation Context ...a planar graph with n vertices, then G has a straight line drawing with its vertices embedded on the (n 2) (n 2) grid. The proof is constructive and the embedding be computed in O(n) time (see e.g. [=-=1]-=-). The method of Schnyder's still gives the strongest result regarding the size of a grid embedding arbitrary planar graphs. Xin He [5] gives a comprehensive history of the problem. Tutte [14, 15] sho... |

12 | The order dimension of planar maps
- Brightwell, Trotter
- 1997
(Show Context)
Citation Context ...edges of a graph G is at most 3 () G is planar. This striking result recently found applications even in algebra [6]. Schnyder's Theorem has found several extensions: Brightwell and Trotter proved in =-=[2]-=- that the incidence order of vertices, edges and faces of a planar map has dimension at most 4. The proof of this result is inductive and required them tosrst establish the following theorem. Theorem ... |

11 |
Grid embedding of 4-connected plane graphs
- He
- 1997
(Show Context)
Citation Context ...onstructive and the embedding be computed in O(n) time (see e.g. [1]). The method of Schnyder's still gives the strongest result regarding the size of a grid embedding arbitrary planar graphs. Xin He =-=[5]-=- gives a comprehensive history of the problem. Tutte [14, 15] shows that every 3-connected planar graph G admits a strictly convex drawing, i.e. a drawing such that the boundary of every face is a str... |

7 | Posets and planar graphs
- Felsner, Trotter
(Show Context)
Citation Context ...osrst establish the following theorem. Theorem 1. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. Felsner and Trotter =-=[4]-=- used Schnyder's Theorem to give a characterization of outerplanar graphs in terms of order dimension. That paper should also give a good overview of other results on the dimension of graphs. The seco... |

5 |
On the order dimension of convex polytopes
- Reuter
- 1990
(Show Context)
Citation Context ...s not hard to see that all critical pairs of L(P ) are min-max pairs, so that by the above remarks the two concepts of dimension coincide. The next theorem is a lower bound which was proved by Reuter =-=[-=-7] in the context of Ferrer's dimension. Theorem 5. If P is a d-polytope with d 2, i.e., a polytope whose ane hull is d dimensional, then dim(P ) d + 1. Proof. The proof is by induction on d. If d =... |

2 | Sturmfels: Monomial ideals and planar graphs
- Miller, Bernd
- 1999
(Show Context)
Citation Context ...nsion of their incidence order: The dimension of the incidence order of vertices and edges of a graph G is at most 3 () G is planar. This striking result recently found applications even in algebra [=-=6]-=-. Schnyder's Theorem has found several extensions: Brightwell and Trotter proved in [2] that the incidence order of vertices, edges and faces of a planar map has dimension at most 4. The proof of this... |