Three-Dimensional Graph Drawing

A three-dimensional straight-line grid drawing of a graph, henceforth called a 3D drawing, represents the vertices by distinct grid-points in Z and represents each edge by the line segment between its end vertices, such that no two edges cross. In contrast to the case in the plane, it is folklore that every graph has a 3D drawing. For example, the “moment curve” algorithm places the i th vertex at .i; i ; i /. It is easily seen that no four vertices are coplanar, and thus no two edges cross. Since every graph has a 3D drawing, we are interested in optimizing certain measures of their aesthetic quality. If a 3D drawing is contained in an axis-aligned box with side lengths X 1, Y 1, and Z 1, then we speak of an X Y Z drawing with volume X Y Z. This entry considers the problem of producing a 3D drawing of a given graph with small volume.