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Drawing 3polytopes with good vertex resolution
 In GD’09, Proc. 17th International Symposium on Graph Drawing, 2009, Lecture Notes in Computer Science
, 2010
"... Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an ..."
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Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained in a 2(n−2)×2×1 box. The constructed embedding can be scaled to a grid embedding whose xcoordinates are contained in [0, 2(n − 2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant. 1
3D straightline drawings of ktrees
"... This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. ..."
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This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. We extend the study to the problem of drawing general ktrees on a set of parallel grid lines. Lower bounds and upper bounds on the number of such grid lines are presented. The results in this paper extend and improve similar ones already described in the literature.
Small grid embeddings of 3polytopes
, 2009
"... We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadri ..."
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Cited by 2 (0 self)
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We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2 5.54n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.
Resolving Loads with Positive Interior Stresses
, 2009
"... We consider the pair (pi, fi) as a force with twodimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a noncrossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, ..."
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We consider the pair (pi, fi) as a force with twodimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a noncrossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, such that for every pi the sum of the weighted edges (seen as vectors) around pi yields −fi. As additional constraint we restrict ourselves to weights that are nonnegative on every edge that is not on the convex hull of the point set. We show that (under a generic assumption) for any reasonable set of forces there is exactly one pointed pseudotriangulation that fulfils the desired properties. Our results will be obtained by linear programming duality over the PPTpolytope. For the case where the forces appear only at convex hull vertices we show that the pseudotriangulation that resolves the load can be computed as weighted Delaunay triangulation. Our observations lead to a new characterization of pointed pseudotriangulations, structures that have been proven to be extremely useful in the design and analysis of efficient geometric algorithms. As an application, we discuss how to compute the maximal locally convex function for a polygon whose corners lie on its convex hull.
Embedding Stacked Polytopes on a PolynomialSize Grid
, 2014
"... A stacking operation adds a dsimplex on top of a facet of a simplicial dpolytope while maintaining the convexity of the polytope. A stacked dpolytope is a polytope that is obtained from a dsimplex and a series of stacking operations. We show that for a fixed d every stacked dpolytope with n ver ..."
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A stacking operation adds a dsimplex on top of a facet of a simplicial dpolytope while maintaining the convexity of the polytope. A stacked dpolytope is a polytope that is obtained from a dsimplex and a series of stacking operations. We show that for a fixed d every stacked dpolytope with n vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n2+2 log(d−1)), except for one axis, where the coordinates are bounded by O(n3+3 log(d−1)). The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding. 1
Small grid embeddings of 3polytopes
, 2010
"... We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(27.55n) = O(188n). If the graph contains a triangle we can bound the integer coordinates by O(24.82n). If the graph contains a quadrilat ..."
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We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(27.55n) = O(188n). If the graph contains a triangle we can bound the integer coordinates by O(24.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(25.46n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face. 1