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15
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
Abstract

Cited by 27 (10 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
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 7 (4 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.
1Bend 3D Orthogonal BoxDrawings: Two Open Problems Solved
, 2001
"... This paper studies threedimensional orthogonal boxdrawings where edgeroutes have at most one bend. Two open problems for such drawings are: (1) Does every drawing of Kn have volume# n 3 )? (2) Is there a drawing of Kn for which additionally the vertices are represented by cubes with surface O( ..."
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Cited by 3 (2 self)
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This paper studies threedimensional orthogonal boxdrawings where edgeroutes have at most one bend. Two open problems for such drawings are: (1) Does every drawing of Kn have volume# n 3 )? (2) Is there a drawing of Kn for which additionally the vertices are represented by cubes with surface O(n)? This paper answers both questions in the negative, and provides related results concerning volume bounds as well. Communicated by G. Liotta: submitted May 2000; revised November 2000 and March 2001. Research partially supported by NSERC. The results in this paper were presented at the 12th Canadian Conference on Computational Geometry, August 2000. T. Biedl, 1Bend 3D Orthogonal BoxDrawings , JGAA, 5(3) 115 (2001) 2 1 Background A 3D orthogonal boxdrawing of a graph is a drawing of the graph where vertices are represented by disjoint axisparallel boxes and edges are represented by disjoint routes along an underlying threedimensional rectangular grid. (Since no other type of drawings will be studied here, the term drawing is used to mean a 3D orthogonal boxdrawing from now on.) The route of each edge thus consists of a sequence of contiguous grid segments, i.e., axisparallel line segments for which the fixed coordinates are integers. The transition from one grid segment to another is called a bend. A drawing is called a kbend drawing if all edge routes have at most k bends. Every vertex is represented by an axisparallel box with integral boundaries; such a box is called a grid box. An Xplane is a plane that is perpendicular to the Xaxis. It is called an Xgrid plane if its fixed coordinate is integral. Y planes and Zplanes are defined similarly. For any vertex v, let X(v) be the number of Xgrid planes that intersect the box of v; Y (v) and Z(v) are de...
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 3 (1 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 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 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.
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 3 (2 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.
Private communication
, 2001
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; w ..."
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Cited by 3 (0 self)
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We show that planar xyz graphs can be recognized in linear time, but that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given nvertex graph is an xyz graph. Submitted:
The Topology of Bendless ThreeDimensional Orthogonal Graph Drawing
, 2007
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; ..."
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Cited by 2 (1 self)
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We describe a correspondence between xyz graphs and facecolored embeddings of the graph onto twodimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and threeconnected, and that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given graph is an xyz graph.
Minimising the Number of Bends and Volume in 3Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2004
"... A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3d ..."
Abstract

Cited by 1 (0 self)
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A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, socalled diagonal drawings. This vertexlayout strategy was introduced in the 3BENDS algorithm of Eades et al. [Discrete Applied Math. 103:5587, 2000]. 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 in linear time. Using two heuristics for determining this vertexordering 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 o(n ) volume, which is the best known upper bound for the volume of 3dimensional orthogonal graph drawings with at most three bends per edge.
BoundedDegree Book Embedings and ThreeDimensional Orthogonal Graph Drawing
 Proc. 9th International Symp. on Graph Drawing (GD '01), volume 2265 of Lecture Notes in Comput. Sci
, 2002
"... A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$ ..."
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A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$, let $f:V\rightarrow\mathbb{N}$ be a function such that $1\leq f(v)\leq\deg(v)$. We present a Las Vegas algorithm which produces a book embedding of $G$ with \Oh{\sqrt{E\cdot\max_v\ceil{\deg(v)/f(v)}}} pages, such that at most $f(v)$ edges incident to a vertex $v$ are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3D orthogonal drawings with one bend per edge and \Oh{V^{3/2}E} volume, and \emph{singlerow} drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some $Z$plane; such drawings are without socalled \emph{crosscuts}, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with \Oh{VE} volume but with crosscuts. These results establish improved bounds for the volume of 3D orthogonal graph drawings.