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32
Drawing on Physical Analogies
, 2001
"... in Sections 4.2 and 4.3. An asset of physical modeling that is often overlooked is its inherent exibility. For this reason, we conclude this chapter by listing examples of model speci cations tailored to speci c layout objectives. 4.1 The Springs Given a connected undirected graph with no partic ..."
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Cited by 30 (2 self)
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in Sections 4.2 and 4.3. An asset of physical modeling that is often overlooked is its inherent exibility. For this reason, we conclude this chapter by listing examples of model speci cations tailored to speci c layout objectives. 4.1 The Springs Given a connected undirected graph with no particular background information, the following two criteria of readable layout seem to be generally agreed upon for the conventional twodimensional straightline representation. 1. Vertices should spread well on the page. 2. Adjacent vertices should be close. Only intuitive explanations can be oered. While uniform vertex distribution reduces clutter, the implied uniform edge lengths leave an undistorted impression of the graph. Since \clutter" and \distortion" already have physical connotations, it seems fairly natural to start thinking of a more speci c physical analogy. We are used to observing even spacing between repelling objects. This makes it natural to imagine vertices behaving l
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.
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 (9 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.
Drawing Clusters and Hierarchies
, 2001
"... with respect to edges can be of interest as well. A method to do this can be found in Paulish (1993, Chapter 5). Clustering of graphs means grouping of vertices into components called clusters. Thus, clustering is related to partitioning the vertex set. Denition 8.1 (Partition). A (kway) partitio ..."
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Cited by 15 (0 self)
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with respect to edges can be of interest as well. A method to do this can be found in Paulish (1993, Chapter 5). Clustering of graphs means grouping of vertices into components called clusters. Thus, clustering is related to partitioning the vertex set. Denition 8.1 (Partition). A (kway) partition of a set C is a family of subsets (C 1 ; : : : ; C k ) with { S k i=1 C i = C and { C i \ C j = ; for i 6= j. The C i are called parts. We refer to a 2way partition as a bipartition. Now, we can dene one of the most basic denitions of clustered graphs. 8. Drawing Clusters and Hierarchies 195<F14.
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.
Graph Layout Problems Parameterized by Vertex Cover
"... In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are ..."
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Cited by 11 (5 self)
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In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to reemphasize the importance and utility of this algorithm.
Bounded degree book embeddings and threedimensional orthogonal graph drawing
 Proc. 9th International Symp. on Graph Drawing (GD ’01), volume 2265 of Lecture Notes in Computer Science
, 2002
"... Abstract. A book embedding of a graph consists of a linear ordering 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 grap ..."
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Cited by 8 (3 self)
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Abstract. A book embedding of a graph consists of a linear ordering 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 → N be a function such that 1 ≤ f(v) ≤ deg(v). We present a Las Vegas algorithm which produces a book embedding of G with O( √E  ·maxvdeg(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 O(V 3/2E) volume, and singlerow drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some Zplane; such drawings are without socalled crosscuts, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with O(V E) volume but with crosscuts. These results establish improved bounds for the volume of 3D orthogonal graph drawings. 1
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...
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 7 (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
On the complexity of the balanced vertex ordering problem
 in Proc. COCOON2005, LNCS 3595
, 2005
"... Abstract. We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, w ..."
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Cited by 5 (1 self)
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Abstract. We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [1]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NPhard for graphs with maximum degree six. One of our main results is closing the gap in these results, by proving NPhardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NPhard for planar graphs with maximum degree six and for 5regular graphs. On the other hand we present a polynomial time algorithm that determines whether there is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an ‘almost balanced ’ ordering. 1