# 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*(*υ*) ≤ deg(*υ*). We present a Las Vegas algorithm which produces a book embedding of *G* with \(
O(\sqrt {|E| \cdot \max _\upsilon \left\lceil {\deg (\upsilon )/f(\upsilon )} \right\rceil } )
\) 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.

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