## Universal 3-Dimensional Visibility Representations for Graphs (1997)

Citations: | 21 - 6 self |

### BibTeX

@MISC{Alt97universal3-dimensional,

author = {Helmut Alt and Michael Godau and Sue Whitesides},

title = {Universal 3-Dimensional Visibility Representations for Graphs},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper studies 3-dimensional visibility representations of graphs in which objects in 3-d correspond to vertices and vertical visibilities between these objects correspond to edges. We ask which classes of simple objects are universal, i.e. powerful enough to represent all graphs. In particular, we show that there is no constant k for which the class of all polygons having k or fewer sides is universal. However, we show by construction that every graph on n vertices can be represented by polygons each having at most 2n sides. The construction can be carried out by an O(n ) algorithm. We also study the universality of classes of simple objects (translates of a single, not necessarily polygonal object) relative to cliques Kn and similarly relative to complete bipartite graphs Kn;m .

### Citations

86 | A unified approach to visibility representations of planar graphs - Tamassia, Tollis - 1986 |

36 | Characterizing bar line-of-sight graphs - Wismath - 1985 |

26 | A rational rotation method for robustgeometric algorithms
- Canny, Donald, et al.
- 1992
(Show Context)
Citation Context ... easily and with the same time complexity to produce convex polygonal (or polyhedral) pieces. Furthermore, these pieces can be made to have all vertex angles of at least =6. By using the technique of =-=[CDR]-=-, it is also possible to implement the algorithm in O(n 2 ) time with respect to a Turing machine model of computation. 4.1 The Basic Pieces Let W denote a regular, convex 2n-gon centered at the origi... |

22 | The number of polytopes, configurations and real matroids - ALON - 1986 |

19 | Rectanglevisibility representations of bipartite graphs - Dean, Hutchinson - 1997 |

18 | A visibility representation for graphs in three dimensions
- Bose, Everett, et al.
- 1998
(Show Context)
Citation Context ...sses of objects are universal, i.e., can give visibility representations for all graphs, or all graphs of a given kind? The visibility representation considered in this paper has also been studied in =-=[SS]-=- (an abstract was presented at GD'92), in [Rom], and in [FHW]. In these papers, the objects representing vertices are axis-aligned rectangles, or disks, and the properties of graphs that can be repres... |

14 | Tamassia and I. Tollis, "Algorithms for Drawing Graphs: An Annotated Bibliography - Battista, Eades, et al. - 1994 |

7 | New results on a visibility representation of graphs in 3D - Fekete, Houle, et al. - 1996 |

7 | Über Treffanzahlen bei translationsgleichen Eikörpern - Hadwiger - 1957 |

6 | Directed VR-representable graphs have unbounded dimension
- Romanik
- 1995
(Show Context)
Citation Context ...visibility representations for all graphs, or all graphs of a given kind? The visibility representation considered in this paper has also been studied in [SS] (an abstract was presented at GD'92), in =-=[Rom]-=-, and in [FHW]. In these papers, the objects representing vertices are axis-aligned rectangles, or disks, and the properties of graphs that can be represented by these objects are studied. By contrast... |

6 |
A uni approach to visibility representations of planar graphs
- Tamassia, Tollis
- 1986
(Show Context)
Citation Context ..., VLSI wire routing, algorithm animation, CASE tools and circuit board layout. See [DETT] for a survey on graph drawing in general; for 2-dimensional visibility representations, see for example [DH], =-=[TT]-=-, [KKU], [W]. Exploration of 3-dimensional visibility is still in the early stages. From the point of view of geometric graph theory, it is natural to consider visibility representations of graphs in ... |

4 | On the number of directions in visibility representations of graphs
- Kranakis, Krizanc, et al.
- 1995
(Show Context)
Citation Context ... wire routing, algorithm animation, CASE tools and circuit board layout. See [DETT] for a survey on graph drawing in general; for 2-dimensional visibility representations, see for example [DH], [TT], =-=[KKU]-=-, [W]. Exploration of 3-dimensional visibility is still in the early stages. From the point of view of geometric graph theory, it is natural to consider visibility representations of graphs in dimensi... |

3 | personal communication - Felsner - 1995 |

1 |
iThe number of polytopes, congurations and real matroids", Mathematika 33
- Alon
- 1986
(Show Context)
Citation Context ...hich are not neccessarily distinct). A family of m lines can be turned into a family of n points by dualization and the arrangement of the lines corresponds to the order type of the dual points. Alon =-=[A]-=- has shown that the number of dioeerent order types of m points is at most 2 flm log m for some constant fl ? 0. (The points do not neccessarily have to be in general position, not even to be distinct... |

1 |
iRectangle-visibility representations of bipartite graphs,j
- Dean, Hutchison
- 1994
(Show Context)
Citation Context ...rawing, VLSI wire routing, algorithm animation, CASE tools and circuit board layout. See [DETT] for a survey on graph drawing in general; for 2-dimensional visibility representations, see for example =-=[DH]-=-, [TT], [KKU], [W]. Exploration of 3-dimensional visibility is still in the early stages. From the point of view of geometric graph theory, it is natural to consider visibility representations of grap... |

1 |
iAlgorithms for drawing graphs: an annotated bibliographyj
- Battista, Eades, et al.
- 1994
(Show Context)
Citation Context ...notion of 2-dimensional visibility has received wide attention due to its applications to such areas as graph drawing, VLSI wire routing, algorithm animation, CASE tools and circuit board layout. See =-=[DETT]-=- for a survey on graph drawing in general; for 2-dimensional visibility representations, see for example [DH], [TT], [KKU], [W]. Exploration of 3-dimensional visibility is still in the early stages. F... |

1 |
iOn a Conjecture of H
- Grnbaum
(Show Context)
Citation Context ...1;n ns6 is not possible since one square cannot intersect 5 or more disjoint squares of the same size. b) (Sketch) Here we use some results from convexity theory obtained by Hadwiger [H] and Gr#nbaum =-=[G]-=-. In fact, they showed that at most 8 translates of a convex body B in two dimensions can touch B without intersecting it or each other. The number 8 is only achieved by parallelograms, otherwise it i... |

1 |
i#ber Treoeanzahlen bei translationsgleichen Eik#rpernj
- Hadwiger
- 1957
(Show Context)
Citation Context ...raightforward. K 1;n ns6 is not possible since one square cannot intersect 5 or more disjoint squares of the same size. b) (Sketch) Here we use some results from convexity theory obtained by Hadwiger =-=[H]-=- and Gr#nbaum [G]. In fact, they showed that at most 8 translates of a convex body B in two dimensions can touch B without intersecting it or each other. The number 8 is only achieved by parallelogram... |

1 |
iCharacterizing bar line-of-sight graphs,j
- Wismath
- 1985
(Show Context)
Citation Context ...outing, algorithm animation, CASE tools and circuit board layout. See [DETT] for a survey on graph drawing in general; for 2-dimensional visibility representations, see for example [DH], [TT], [KKU], =-=[W]-=-. Exploration of 3-dimensional visibility is still in the early stages. From the point of view of geometric graph theory, it is natural to consider visibility representations of graphs in dimensions h... |

1 | Szekeres, "A Combinatorial Problem in Geometry - Erdos, Gy - 1935 |

1 | On a Conjecture of H. Hadwiger - Grunbaum - 1961 |