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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 24 (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
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 21 (7 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 13 (5 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.
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 10 (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.
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.
Bibliography
"... Automatic clustering of languages. Computational Linguistics, 18(3):339352. Berge, C. (1993). Graphs. North Holland, Amsterdam, 3rd edition. Berger, B., and Shor, P. (1990). Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the 1st ACMSIAM Symposium on Discrete ..."
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Automatic clustering of languages. Computational Linguistics, 18(3):339352. Berge, C. (1993). Graphs. North Holland, Amsterdam, 3rd edition. Berger, B., and Shor, P. (1990). Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the 1st ACMSIAM Symposium on Discrete Algorithms (SODA'90), pages 236243. Bertolazzi, P., Cohen, R. F., Di Battista, G., Tamassia, R., and Tollis, I. G. (1994a). How to draw a seriesparallel digraph. International Journal of Computational Geometry and Applications, 4:385402. Bertolazzi, P., Di Battista, G., and Didimo, W. (1997). Computing orthogonal drawings with the minimum number of bends. In Proceedings of the 5th Workshop on Algorithms and Data Structures (WADS'97), Spinger LNCS 1272, pages 331344. Bertolazzi, P., Di Battista, G., Liotta, G., and Mannino, C. (1994b). Upward drawings of triconnected digraphs. Algorithmica, 6(12):476