## Drawing Stressed Planar Graphs in Three Dimensions (1995)

Venue: | In |

Citations: | 18 - 0 self |

### BibTeX

@INPROCEEDINGS{Eades95drawingstressed,

author = {Peter Eades and Patrick Garvan},

title = {Drawing Stressed Planar Graphs in Three Dimensions},

booktitle = {In},

year = {1995},

pages = {212--223},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples. Key words: graph, stressed graph, convex polyhedron, reciprocal polyhedron 1 Introduction It is well known that 3-connected planar graphs can be drawn as convex polyhedra. However, no practical algorithms exist to draw general 3-connected planar graphs as convex polyhedra. The two-dimensional (2D) drawing in Figure 1 is 3-connected and planar, and the corresponding polyhedron is drawn in Figure 2 as three different views. The 2D ...

### Citations

543 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...t if it is high. A method for drawing any graph in 3D using straight line edges such that no pair of edges cross is presented in [2]. This method does not restrict the drawing to the integer grid. In =-=[3]-=- a method is provided for drawing any graph in 3D using straight line edges such that no pair of edges cross. All vertices are located on the integer grid and, for a graph of n vertices, the required ... |

472 | Graph drawing by force-directed placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ...ster of small facets in the final polyhedron. In [5] there are drawings of some 3-connected planar graphs in 2D, and in some cases the result is similar to a projection of a 3D polyhedron into 2D. In =-=[6]-=- there are drawings of some 3-connected planar graphs in both 2D and 3D, however the resulting drawings are not polyhedral in the sense that for any particular face of a graph the vertices of that fac... |

394 | How to draw a graph
- Tutte
(Show Context)
Citation Context ...estrictions are placed on the outer stresses. [Note: Not every drawing of a 3-connected planar graph has a convex equilibrium stress.] An example of a restricted equilibrium stressed graph is a Tutte =-=[12]-=- drawing of a 3connected planar graph. This well known algorithm chooses a face of the graph and draws it as a convex polygon on the plane. The position of any internal vertex is then defined as the b... |

271 | Convex Polytopes
- Grünbaum
- 1967
(Show Context)
Citation Context ...e views of the 3D polyhedron corresponding to Figure 1 as produced by Algorithm 1. The polyhedron is isomorphic to a three frequency geodesic sphere. The following is a well-known theorem by Steinitz =-=[7]-=-: Theorem 1 (Steinitz) A graph G is the skeleton of a polyhedron P if and only if it is planar and 3-connected. This theorem guarantees that any 3-connected planar graph can be drawn as the vertices a... |

252 |
Regular Polytopes
- Coxeter
- 1973
(Show Context)
Citation Context ...of P , and with face-planes that are reciprocal to the vertices of P . Further, if P is convex and contains the origin, then the reciprocal polyhedron P will also be convex. (For more information see =-=[1]-=- and [8].) Algorithm 1 Input: A planar embedding of a 3-connected planar graph G Output: A strictly convex polyhedron P 1. If G does not contain a C 3 (a cycle of three edges), then replace G by its g... |

191 | Drawing Graphs Nicely Using Simulated Annealing
- Davison, Harel
- 1996
(Show Context)
Citation Context ...iprocation step. This polyhedron has many long thin facets clustered down one side, and so reciprocation about a single internal point results in a cluster of small facets in the final polyhedron. In =-=[5]-=- there are drawings of some 3-connected planar graphs in 2D, and in some cases the result is similar to a projection of a 3D polyhedron into 2D. In [6] there are drawings of some 3-connected planar gr... |

119 |
L.: Introduction to Combinatorial Mathematics
- Liu
(Show Context)
Citation Context ... this way, the maximum y-coordinate for a similar graph of n vertices can be found by solving the recurrence y i = 4y i\Gamma1 \Gamma y i\Gamma2 ; where y 0 = 1 and y 1 = 3. Using standard techniques =-=[10]-=-, the solution to this recurrence equation shows that y n = \Theta(k n ), where k is a constant and k ? 1. Therefore, the worst case resolution of a Tutte drawing is \Omega\Gamma k n ); k ? 1. 2 Theor... |

76 |
Planar Graphs: Theory and Algorithms
- Nishizeki, Chiba
- 1988
(Show Context)
Citation Context ... skeleton of a polyhedron is dual to the skeleton of a reciprocal of that polyhedron. 2 Theorem 4 Algorithm 1 requires O(n 3=2 ) time. Proof: Step 3 requires O(n 3=2 ) time to perform a Tutte drawing =-=[11]. Every ot-=-her step requires at most O(n) time. Therefore the overall time complexity is O(n 3=2 ). 2 Unfortunately, Algorithm 1 does not always produce drawings with good "resolution", where resolutio... |

21 |
A paradigm for robust geometric algorithms
- Hopcroft, Kahn
- 1992
(Show Context)
Citation Context ... with drawing polyhedra using Algorithm 1 has revealed a useful addition to our original algorithm. We conclude with some open problems in section 4. Note: some of the concepts in this paper are from =-=[9]. 2 The Al-=-gorithm In this section we review the concept of stressed graphs (see for example [9]) and their relation to polyhedra. We give a definition of "reciprocal" polyhedra and present the main dr... |

13 |
Acoptic polyhedra
- Grünbaum
(Show Context)
Citation Context ...nd with face-planes that are reciprocal to the vertices of P . Further, if P is convex and contains the origin, then the reciprocal polyhedron P will also be convex. (For more information see [1] and =-=[8]-=-.) Algorithm 1 Input: A planar embedding of a 3-connected planar graph G Output: A strictly convex polyhedron P 1. If G does not contain a C 3 (a cycle of three edges), then replace G by its graphtheo... |

9 | On the complexity of approximating and illuminating three-dimensional convex polyhedra
- Das, Goodrich
- 1995
(Show Context)
Citation Context ...f a polyhedron P if and only if it is planar and 3-connected. This theorem guarantees that any 3-connected planar graph can be drawn as the vertices and edges of a convex polyhedron. Some recent work =-=[4]-=- provides a linear-time algorithm for realizing any 3-connected triangulated planar graph as a convex polyhedron. However, [4] only deals with triangulated graphs. In this paper, we are concerned spec... |

2 |
Drawing the Complete Graph in 3-D with Straight Lines and Without Crossings
- Cahit
- 1994
(Show Context)
Citation Context ...that a planar graph is cluttered if this ratio is low or easy-to-look-at if it is high. A method for drawing any graph in 3D using straight line edges such that no pair of edges cross is presented in =-=[2]-=-. This method does not restrict the drawing to the integer grid. In [3] a method is provided for drawing any graph in 3D using straight line edges such that no pair of edges cross. All vertices are lo... |