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Graph Drawing by HighDimensional Embedding
 In GD02, LNCS
, 2002
"... We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seco ..."
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Cited by 70 (10 self)
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We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 13 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
GRAPH DRAWINGS WITH FEW SLOPES
, 2006
"... The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a ..."
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Cited by 12 (3 self)
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The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a graph with bounded degree. We prove upper and lower bounds on the slopenumber of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slopenumber of interval graphs, cocomparability graphs, and ATfree graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slopenumber at most O(log n). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
The Complexity of Computing Nice Viewpoints of Objects in Space
 IN VISION GEOMETRY IX, PROC. SPIE INTERNATIONAL SYMPOSIUM ON OPTICAL SCIENCE AND TECHNOLOGY
, 2000
"... A polyhedral object in 3dimensional space is often well represented by a set of points and line segments that act as its features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of the object, relevant for some task, are clearly visible. Such a vie ..."
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Cited by 4 (0 self)
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A polyhedral object in 3dimensional space is often well represented by a set of points and line segments that act as its features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of the object, relevant for some task, are clearly visible. Such a view is often called a nondegenerate view or projection. In this paper we are concerned with computing nondegenerate orthogonal and perspective projections of sets of points and line segments (objects) in 3dimensional space. We outline the areas in which such problems arize, discuss recent research on the computational complexity of these problems, illustrate the fundamental ideas used in the design of algorithms for computing nondegenerate projections, and provide pointers to the literature where the results can be found.
Really straight drawings II: Nonplanar graphs
, 2005
"... We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings ..."
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Cited by 1 (1 self)
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We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.