Three-dimensional 1-bend graph drawings

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n/ log n) volume and one bend per edge. The best previous bound was O(n). Article Type Communicated by Submitted Revised concise paper G. Liotta April 2004 March 2005 Presented at the 16th Canadian Conference on Computational Geometry (CCCG ’04), Concordia University, Montréal, Canada, August 9–11, 2004. Research of Pat Morin supported by NSERC. Research of David Wood supported by the Government of Spain grant MEC SB20030270, and partially completed at Carleton University, Canada. Morin and Wood, 1-Bend Graph Drawings, JGAA, 8(3) 357–366 (2004) 358