## Three-Dimensional Tutte Embedding (1995)

Venue: | in Proc. 26th Southeastern International Conf. on Combinatorics, Graph Theory and Computing, vol. 107 of Cong. Numer |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Chilakamarri95three-dimensionaltutte,

author = {Kiran Chilakamarri and Nathaniel Dean and Michael Littman},

title = {Three-Dimensional Tutte Embedding},

booktitle = {in Proc. 26th Southeastern International Conf. on Combinatorics, Graph Theory and Computing, vol. 107 of Cong. Numer},

year = {1995},

pages = {129--140}

}

### OpenURL

### Abstract

Conditions are given for a graph to have a convex representation in three dimensions. This extends Tutte's barycentric embedding. 1 Introduction Tutte [7] proved that a 3-connected planar graph has a nice embedding in the plane in the sense that it can be drawn so that the boundary of every face is a convex polygon and every edge is a straight line segment. This is called a convex embedding or convex representation. More details on Tutte embedding are given in the next section. The main objective of this work is to visualize nice drawings of nonplanar graphs in three-dimensional space and to introduce some ideas for future research. Now we describe some basic notation to be used throughout this paper. Let G = (V; E) represent a finite graph with vertex set V and edge set E. A cycle C in G is a collection of distinct vertices v 1 ; v 2 ; : : : ; v n such that v i is adjacent to v i+1 for 1 i n, where the indices are considered modulo n. If there are no edges in G between any tw...

### Citations

386 | How to draw a graph
- Tutte
- 1963
(Show Context)
Citation Context ...ersity Providence, RI 02912 May 5, 1995 Abstract Conditions are given for a graph to have a convex representation in three dimensions. This extends Tutte's barycentric embedding. 1 Introduction Tutte =-=[7]-=- proved that a 3-connected planar graph has a nice embedding in the plane in the sense that it can be drawn so that the boundary of every face is a convex polygon and every edge is a straight line seg... |

263 | Convex Polytopes
- GruÌˆnbaum
(Show Context)
Citation Context ... A graph G is d-polytopal if there is a d-polytope P such that G is isomorphic to G(P ). The following theorems of Steinitz and Tutte are of interest for this paper. Theorem A (Steinitz, see Grunbaum =-=[3]-=-) A graph G is 3-polytopal if and only if G is planar and 3-connected. Theorem B (Tutte [7]) If G is 3-connected planar graph, then the facial cycles in any plane embedding of G are those cycles which... |

102 |
A survey of matrix theory and matrix inequalities
- Marcus, Newman
- 1992
(Show Context)
Citation Context ...to the xy-plane corresponds to the Tutte embedding of G on the plane with the vertices of C fixed on a circle. 3. z i ? 0 for i = 1; 2; : : : ; n. For the proof we need a theorem of L'evy-Desplanques =-=[4]-=- (page 146) and its variation, both of which are stated below. Theorem D Let A = (a ij ) be a complex n \Theta n matrix and ja ii j ? P n j=1 ja ij j \Gamma ja ii j for i = 1; 2; : : : ; n. Then A is ... |

25 |
Tutte, Convex Representations of Graphs
- T
- 1960
(Show Context)
Citation Context ...ar and 3-connected. Theorem B (Tutte [7]) If G is 3-connected planar graph, then the facial cycles in any plane embedding of G are those cycles which are chordless and nonseparating. Theorem C (Tutte =-=[6]-=-) Every 3-connected planar graph has a convex representation and any properly chosen convex polygon can play the role of outer cycle. The Tutte embedding or barycentric mapping is defined as follows. ... |

8 |
NETPAD: An interactive graphics system for network modeling and optimization
- Dean, Mevenkamp, et al.
- 1992
(Show Context)
Citation Context ...such embedding of a graph in a higher dimensional space is called a Tutte embedding or barycentric mapping. In the next section we show embeddings of several graphs. We used the software tools NETPAD =-=[2]-=- and XGobi [1] to construct and view the embeddings in 3-space. 3 Examples Much is known about drawing planar graphs, and so we feel it is important to focus on drawing nonplanar graphs. In particular... |

7 |
XGobi: a dynamic graphics program implemented in X with a link to S
- Swayne, Cook
- 1990
(Show Context)
Citation Context ... of a graph in a higher dimensional space is called a Tutte embedding or barycentric mapping. In the next section we show embeddings of several graphs. We used the software tools NETPAD [2] and XGobi =-=[1]-=- to construct and view the embeddings in 3-space. 3 Examples Much is known about drawing planar graphs, and so we feel it is important to focus on drawing nonplanar graphs. In particular, observe that... |

2 |
Embeddings and Minors, Handbook of Combinatorics, Edited by R. L Graham at el
- Thomassen
- 1995
(Show Context)
Citation Context ...gative answer to a question posed by Tutte [7]. The reader is referred to any standard text book on Graph Theory for the undefined terms. Several of the terms may also be found in Thomassen's article =-=[5]-=-. 2 The Tutte Embedding The graph G(P ) of a d-polytope P (i.e., P is the convex hull of a finite set of points in ! d ) is the graph consisting of the vertices and the edges of P . If d = 3, then an ... |