# Universal 3-Dimensional Visibility Representations for Graphs

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
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 {\em 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 $O(n^2)$ algorithm. We also study the universality of classes of simple objects (translates of a single, not necessarily polygonal object) relative to cliques $K_n$ and similarly relative to complete bipartite graphs $K_{n,m}$.
Keywords :
Document type :
Reports
Domain :

https://hal.inria.fr/inria-00074064
Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Wednesday, May 24, 2006 - 2:24:53 PM
Last modification on : Friday, February 4, 2022 - 3:15:26 AM
Long-term archiving on: : Monday, April 5, 2010 - 12:04:14 AM

### Identifiers

• HAL Id : inria-00074064, version 1

### Citation

Helmut Alt, Michael Godau, Sue Whitesides. Universal 3-Dimensional Visibility Representations for Graphs. RR-2622, INRIA. 1995. ⟨inria-00074064⟩

Record views