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 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

Abstract. 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 n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n 3/2) 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 three-dimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) 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. 1.

3.1.2. Text algorithms and word combinatorics 2

The GLuskap system for interactive three-dimensional graph drawing applies techniques of scientific visualization and interactive systems to the construction, display, and analysis of graph drawings. Important features of the system include support for large-screen stereo-graphic 3D display with immersive head-tracking and motion-tracked interactive 3D wand control. A distributed rendering architecture contributes to the portability of the system, with user control performed on a laptop computer without specialized graphics hardware. An interface for implementing graph drawing layout and analysis algorithms in the Python programming language is also provided. This thesis describes comprehensively the work on the system by the author—this work includes the design and implementation of the ma-jor features described above. Further directions for continued development and research in cognitive tools for graph drawing research are also suggested.

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