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inria-00074064, version 1

## Universal 3-Dimensional Visibility Representations for Graphs

Helmut Alt 1, Michael Godau, Sue Whitesides

N° RR-2622 (1995)

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 $K_{n,m}$.

• 1:  PRISME (INRIA Sophia Antipolis)
• INRIA
• Domain : Computer Science/Other
• Keywords : COMPUTATIONAL GEOMETRY / VISIBILITY / GRAPHS
• Internal note : RR-2622

• inria-00074064, version 1
• oai:hal.inria.fr:inria-00074064
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• Submitted on: Wednesday, 24 May 2006 14:24:53
• Updated on: Wednesday, 31 May 2006 14:24:28