Three-dimensional graph drawing

AbstractGraph 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.