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ThreeDimensional 1Bend Graph Drawings
 Concordia University
, 2004
"... We consider threedimensional griddrawings 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 ..."
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Cited by 5 (1 self)
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We consider threedimensional griddrawings 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 threedimensional griddrawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).
Upward threedimensional grid drawings of graphs
, 2005
"... A threedimensional 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 threedimensional grid drawings with small bounding box volume. Our first main resu ..."
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Cited by 4 (3 self)
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A threedimensional 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 threedimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a threedimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A threedimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the zdirection. We prove that every dag has an upward threedimensional 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 ccolourable dag (c constant) has an upward threedimensional grid drawing with O(n²) 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 threedimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward threedimensional grid drawings with bends in the edges.
VISUALIZING THREEDIMENSIONAL GRAPH DRAWINGS
, 2006
"... The GLuskap system for interactive threedimensional 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 largescreen stereographic 3D display with imm ..."
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The GLuskap system for interactive threedimensional 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 largescreen stereographic 3D display with immersive headtracking and motiontracked 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 major features described above. Further directions for continued development and research in cognitive tools for graph drawing research are also suggested.
Laboratoire Lorrain de Recherche en Informatique et ses Applications  2004 Activity Report
, 2004
"... The research, both theoretical and applied, is centered around five major themes: • highperformance computing, networks and visualization, • teleoperations and intelligent assistants, • language engineering, document engineering, scientific and technical information engineering, • sofware and compu ..."
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The research, both theoretical and applied, is centered around five major themes: • highperformance computing, networks and visualization, • teleoperations and intelligent assistants, • language engineering, document engineering, scientific and technical information engineering, • sofware and computer system quality and safety, • bioinformatics and applications in genomics. Research activity is performed by 25 research teams including 15 INRIA project teams, with the assistance of 8 research support services. In 2003, three new research teams have been created: Design on objectoriented langages and systems, Madynes on management of dynamic networks and services, and Algorille on algorithms for the Grid. The research teams Cassis et Modbio became INRIA project teams. In autumn 2003, 4 INRIA and 3 CNRS permanent researchers, 5 assistant professors joined the research teams. More than half of them are coming from outside Lorraine. Detailled activities and results are described in this annual report. Without attempting any exhaustivity, some of them mentioned below illustrate applicative potential and scientific relevance
PointSet Embedding in Three Dimensions
, 2012
"... Given a graph G with n vertices and m edges, and a set P of n points on a threedimensional integer grid, the 3D PointSet Embeddability problem is to determine a (threedimensional) crossingfree drawing of G with vertices located at P and with edges drawn as polylines with bendpoints at integer ..."
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Given a graph G with n vertices and m edges, and a set P of n points on a threedimensional integer grid, the 3D PointSet Embeddability problem is to determine a (threedimensional) crossingfree drawing of G with vertices located at P and with edges drawn as polylines with bendpoints at integer grid points. We solve a variant of the problem in which the points of P lie on a plane. The resulting drawing lies in a bounding box of reasonable volume and uses at most O(logm) bends per edge. If a particular pointset P is not specified, we show that the graph G can be drawn crossingfree with at most O(logm) bends per edge in a volume bounded by O((n + m) logm). Our construction is asymptotically similar to previously known drawings, however avoids a possibly nonpolynomial preprocessing step.