Volume upper bounds for 3D graph drawing

Many systems, particularly those which present relational information, include a graph drawing function. Such systems have motivated a great deal of research on algorithms for drawing graphs; a recent survey lists over 250 references. Almost all this work has been oriented toward two-dimensional drawings. This paper describes an investigation of mathematically fundamental aspects of three-dimensional graph drawing. In particular we give three results concerning the space required for three-dimensional drawings.We showhow to produce a grid drawing of an arbitrary n-vertex graph with all vertices located at integer grid point, 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⌉ × ⌈logn⌉. 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.