Three-dimensional graph drawing

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.

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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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Cohen, R.F., Eades, P., Lin, T. et al. Three-dimensional graph drawing. Algorithmica 17, 199–208 (1997). https://doi.org/10.1007/BF02522826

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Key Words

  • Graph drawing
  • Algorithms
  • Three-dimensional