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25
ThreeDimensional Orthogonal Graph Drawing with Optimal Volume
"... An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axisaligned boxes, and edges represented by paths in the grid which only possibly intersect at common endpoints. In this paper, we study threedimensional orthogonal drawings and prov ..."
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Cited by 23 (8 self)
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An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axisaligned boxes, and edges represented by paths in the grid which only possibly intersect at common endpoints. In this paper, we study threedimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where vertices have bounded aspect ratio, (2) drawings where the surface of vertices is proportional to their degree, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that match the lower bounds in all scenarios within an order of magnitude.
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.
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.
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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Cited by 9 (4 self)
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we
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...
On the orthogonal drawing of outerplanar graphs
 in Proceedings of COCOON ’04, Lect. Notes Comput. Sci. 3106
, 2004
"... Abstract. In this paper we show that an outerplanar graph G with maximum degree at most 3 has a 2D orthogonal drawing with no bends if and only if G contains no triangles. We also show that an outerplanar graph G with maximum degree at most 6 has a 3D orthogonal drawing with no bends if and only i ..."
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Cited by 5 (1 self)
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Abstract. In this paper we show that an outerplanar graph G with maximum degree at most 3 has a 2D orthogonal drawing with no bends if and only if G contains no triangles. We also show that an outerplanar graph G with maximum degree at most 6 has a 3D orthogonal drawing with no bends if and only if G contains no triangles. 1
Bidimensionality and Geometric Graphs
"... Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problem ..."
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Cited by 5 (3 self)
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Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for FEEDBACK VERTEX SET, VERTEX COVER, CONNECTED VERTEX COVER, DIAMOND HITTING SET, on map graphs and unit disk graphs, and for CYCLE PACKING and MINIMUMVERTEX FEEDBACK EDGE SET on unit disk graphs. To the best of our knowledge, these results were previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for VERTEX COVER, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. Our results are based on the recent decomposition theorems proved by Fomin et al. in [SODA
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.
private communication
, 2005
"... We show that every graph of maximum degree three can be drawn without crossings in three dimensions with at most two bends per edge, and with 120 ◦ angles between all pairs of edge segments that meet at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensi ..."
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Cited by 2 (0 self)
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We show that every graph of maximum degree three can be drawn without crossings in three dimensions with at most two bends per edge, and with 120 ◦ angles between all pairs of edge segments that meet at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5 ◦ angles, i. e., the angular resolution of the diamond lattice, between all pairs of edge segments that meet at a vertex or a bend. The angles in these drawings are the best possible given the degrees of the vertices. Submitted: