Bounded degree book embeddings and three-dimensional orthogonal graph drawing

Abstract

A book embedding of a graph consists of a linear ordering of the vertices along a line in 3-space (the spine), and an assignment of edges to half-planes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V, E), let f: V → ℕ be a function such that 1 ≤ f(v) ≤ deg(v). We present a Las Vegas algorithm which produces a book embedding of G with O(√|E| · maxv[deg(v)/f(v)]) pages, such that at most f(v) edges incident to a vertex v are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3-D orthogonal drawings with one bend per edge and O(|V|3/2|E|) volume, and single-row drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some Z-plane; such drawings are without so-called cross-cuts, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with O(|V||E|) volume but with cross-cuts. These results establish improved bounds for the volume of 3-D orthogonal graph drawings. © Springer-Verlag Berlin Heidelberg 2002.