Three-dimensional graph drawing R. F. Cohen P. Eades Tao Lin F. Ruskey Article Received: 12 December 1994 Revised: 07 August 1995 DOI :
10.1007/BF02522826

Cite this article as: Cohen, R.F., Eades, P., Lin, T. et al. Algorithmica (1997) 17: 199. doi:10.1007/BF02522826
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Abstract Graph drawing research has been mostly oriented toward two-dimensional drawings. This paper describes an investigation of fundamental aspects of three-dimensional graph drawing. In particular we give three results concerning the space required for three-dimensional drawings.

We show how to produce a grid drawing of an arbitraryn -vertex graph with all vertices located at integer grid points, in ann ×2n ×2n grid, such that no pair of edges cross. This grid size is optimal to within a constant. We also show how to convert an orthogonal two-dimensional drawing in anH ×V integer grid to a three-dimensional drawing with\(\left\lceil {\sqrt H } \right\rceil \times \left\lceil {\sqrt H } \right\rceil \times V\) volume. Using this technique we show, for example, that three-dimensional drawings of binary trees can be computed with volume\(\left\lceil {\sqrt n } \right\rceil \times \left\lceil {\sqrt n } \right\rceil \times \left\lceil {\log n} \right\rceil \) . We give an algorithm for producing drawings of rooted trees in which thez -coordinate of a node represents the depth of the node in the tree; our algorithm minimizes thefootprint of the drawing, that is, the size of the projection in thexy plane.

Finally, we list significant unsolved problems in algorithms for three-dimensional graph drawing.

Key Words Graph drawing Algorithms Three-dimensional This work was performed as part of the Information Visualization Group(IVG) at the University of Newcastle. The IVG is supported in part by IBM Toronto Laboratory.

Communicated by T. Nishizeki.

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Authors and Affiliations R. F. Cohen P. Eades Tao Lin F. Ruskey 1. Department of Computer Science University of Newcastle Callaghan Australia 2. CSIRO DIT Camberra Australia 3. Department of Computer Science University of Victoria Victoria Canada