, Volume 17, Issue 2, pp 199-208

Three-dimensional graph drawing

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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.

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.