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 arbitrary n-vertex graph with all vertices located at integer grid points, in an n×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 a H × V integer grid to a three-dimensional drawing with [√H] × [√H] × V volume. Using this technique we show, for example, that three-dimensional drawings of binary trees can be computed with volume [√n] × [√n] × [log n]. We give an algorithm for producing drawings of rooted trees in which the z coordinate of a node represents the depth of the node in the tree; our algorithm minimizes the footprint of the drawing, that is, the size of the projection in the xy plane. Finally, we list significant unsolved problems in algorithms for three-dimensional graph drawing.