Upward Three-Dimensional Grid Drawings of Graphs

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a three-dimensional grid drawing with O ( n3/2 ) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward threedimensional grid drawing with O ( n3 ) volume, which is tight for the complete dag. The previous best upper bound was O ( n4 ) . Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O ( n2 ) volume. This result matches the bound in the undirected case, and improves the best known bound from O ( n3 ) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges. Research of Vida Dujmovic̀ is supported by NSERC. Research of David Wood is supported by the Government of Spain grant MEC SB2003-0270 and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224. V. Dujmović (B) School of Computer Science, Carleton University, Ottawa, Canada e-mail: [email protected] D. R. Wood Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain e-mail: [email protected] 2 Order (2006) 23: 1–20 Mathematics Subject Classification (1991) 05C62 (graph representations).