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20
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 33 (13 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Refinement of ThreeDimensional Orthogonal Graph Drawings
, 2001
"... In this paper we introduce a number of techniques for the re nement of threedimensional orthogonal drawings of maximum degree six graphs. We have implemented several existing algorithms for threedimensional orthogonal graph drawing including a number of heuristics to improve their performance. The ..."
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Cited by 25 (3 self)
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In this paper we introduce a number of techniques for the re nement of threedimensional orthogonal drawings of maximum degree six graphs. We have implemented several existing algorithms for threedimensional orthogonal graph drawing including a number of heuristics to improve their performance. The performance of the re nements on the produced drawings is then evaluated in an extensive experimental study. We measure the aesthetic criteria of the bounding box volume, the average and maximum number of bends per edge, and the average and maximum edge length. On the same set of graphs used in Di Battista et al. [3], our main re nement algorithm improves the above aesthetic criteria by 80%, 38%, 10%, 54% and 49%, respectively.
GIOTTO3D: A System for Visualizing Hierarchical Structures in 3D
 Proceedings of Graph Drawing ’96), Lecture Notes in Computer Science 1190
, 1997
"... Hierarchical structures represented by directed acyclic graphs are widely used in visualization applications (e.g., class inheritance diagrams and scheduling diagrams). 3D information visualization has received increasing attention in the last few years, motivated by the advances in hardware and ..."
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Cited by 18 (1 self)
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Hierarchical structures represented by directed acyclic graphs are widely used in visualization applications (e.g., class inheritance diagrams and scheduling diagrams). 3D information visualization has received increasing attention in the last few years, motivated by the advances in hardware and software technology for 3D computer graphics.
MultiDimensional Orthogonal Graph Drawing with Small Boxes
 Proc. 7th International Symp. on Graph Drawing (GD '99
, 1999
"... In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane. ..."
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Cited by 14 (6 self)
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In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.
3DCube: a Tool for Three Dimensional Graph Drawing
 Graph Drawing (Proc. GD '97), volume 1353 of Lecture Notes Comput. Sci
, 1997
"... . In this paper we describe a tool that is a general frame for the threedimensional representation of graphs, especially devoted to the algorithms evaluation, refinement and development. 3DCube (3D Diagram Drawer) offers innovative features in the user interaction and contains a set of threedi ..."
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Cited by 11 (1 self)
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. In this paper we describe a tool that is a general frame for the threedimensional representation of graphs, especially devoted to the algorithms evaluation, refinement and development. 3DCube (3D Diagram Drawer) offers innovative features in the user interaction and contains a set of threedimensional algorithms both taken from the literature and proposed by the authors. 1 Introduction Three dimensional graph drawing is an emerging field in the graph drawing area. Several tools are already available for the representation of graphs in the plane, and the most sophisticated of them also allow some threedimensional representation, either as an additional presentation feature of basically 2D results (see GMB [13] and PLUM [17]), or as the result of an actual 3Ddrawing algorithm (see 3DSA [4], COMAIDE [6], the ffGraph library [9], GEM3D [2], GIOTTO3D [11], GOLD [10], GOVE [18]), and PARSA [14]). The tool we describe in this paper offers to the user a general frame for the repr...
Minimising the Number of Bends and Volume in ThreeDimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2000
"... A 3D orthogonal drawing of graph with maximum degree at most six positions the vertices at gridpoints in the 3D orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. In this paper we present two algorithms for producing 3D orthogonal grap ..."
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Cited by 9 (6 self)
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A 3D orthogonal drawing of graph with maximum degree at most six positions the vertices at gridpoints in the 3D orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. In this paper we present two algorithms for producing 3D orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so called diagonal drawings. This vertexlayout strategy was introduced in the 3Bends algorithm of Eades et al. [11]. We show that minimising the number of bends in a diagonal drawing of a given graph is NPhard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal. Using two heuristics for determining this vertex ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the abovementioned 3Bends algorithm, produces 3bend drawings with n^3 + o(n^3) volume, which is the best known upper bound for the volume of 3D orthogonal graph drawings with at most 3 bends per edge.
A New Algorithm and Open Problems in ThreeDimensional Orthogonal Graph Drawing
 Curtin University of Technology
, 1999
"... . In this paper we present an algorithm for 3D orthogonal drawing of arbitrary degree nvertex medge multigraphs with O(m 2 = p n) bounding box volume and 6 bends per edge route. This is the smallest known bound on the bounding box volume of 3D orthogonal multigraph drawings. We continue ..."
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Cited by 7 (3 self)
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. In this paper we present an algorithm for 3D orthogonal drawing of arbitrary degree nvertex medge multigraphs with O(m 2 = p n) bounding box volume and 6 bends per edge route. This is the smallest known bound on the bounding box volume of 3D orthogonal multigraph drawings. We continue the study of the tradeoff between bounding box volume and the number of bends in orthogonal graph drawings through a refined algorithm with O(m 2 ) bounding box volume and 5 bends per edge route. Many open problems in 3D orthogonal graph drawing are presented and potential avenues for their solution are discussed. 1 Introduction With applications including VLSI circuit design [4, 18, 20] and software engineering [14, 19, 23], there has been recent interest in 3D graph visualization. Proposed models include straightline drawings [6, 13, 16] and of interest in this paper orthogonal drawings [1, 2, 5, 8, 9, 10, 11, 15, 17, 25, 26, 27, 28]. The 3D orthogonal grid consists of grid po...
Orthogonal drawings with few layers
 PROC. 9TH INTERNATIONAL SYMP. ON GRAPH DRAWING (GD '01
, 2002
"... In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very smal ..."
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Cited by 4 (3 self)
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In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain pointdrawings with 3layers and 4 bends per edge, and algorithms to obtain boxdrawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
Lower Bounds for the Number of Bends in ThreeDimensional Orthogonal Graph Drawings
, 2003
"... This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tigh ..."
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Cited by 4 (2 self)
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This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of cconnected simple graphs, multigraphs, and pseudographs (2 &le; c &le; 6) of maximum degree &Delta; (3 &le; &Delta; &le; 6), with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3D orthogonal graph drawings. These results have significant ramifications for the `2bends problem', which is one of the most important open problems in the field.
Advances in the Theory and Practice of Graph Drawing
 Theor. Comp. Sci
, 1996
"... The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph d ..."
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Cited by 4 (0 self)
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The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing. Specific topics include bounds and tradeoffs for drawing properties, threedimensional representations, methods for constraint satisfaction, and experimental studies. 1 Introduction In this paper, we survey selected research trends in graph drawing, and overview some recent results of the author and his collaborators. Graph drawing addresses the problem of constructing geometric representations of graphs, a key component of support tools for complex applications in science and engineering. Graph drawing is a young research field that has growth very rapidly in the last decade. One of its distinctive characteristics is to have furthered collaborative efforts between computer scien...