Results 1  10
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14
Tight degree bounds for pseudotriangulations of points
, 2003
"... We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this p ..."
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Cited by 33 (11 self)
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We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudotriangulation has at most four vertices). Both degree bounds are tight. Minimum pseudotriangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Convexity Minimizes PseudoTriangulations
 Computational Geometry: Theory and Applications
, 2002
"... The number of minimum pseudotriangulations is minimized for point sets in convex position. ..."
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Cited by 15 (2 self)
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The number of minimum pseudotriangulations is minimized for point sets in convex position.
A Lower Bound on the Number of Triangulations of Planar Point Sets
"... We show that the number of straightedge triangulations exhibited by any set of n points in general position in the plane is bounded from below by 4 :33 ). ..."
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Cited by 14 (3 self)
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We show that the number of straightedge triangulations exhibited by any set of n points in general position in the plane is bounded from below by 4 :33 ).
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 13 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
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Cited by 11 (1 self)
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We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is a O(1).)
On the number of plane graphs
 PROC. 17TH ANN. ACMSIAM SYMP. ON DISCRETE ALGORITHMS
, 2006
"... We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extre ..."
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Cited by 8 (1 self)
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We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the socalled double zigzag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.
On the number of pseudotriangulations of certain point sets
 J. Combin. Theory Ser. A
, 2007
"... We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significant ..."
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Cited by 8 (2 self)
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We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de
Enumerating pseudotriangulations in the plane
 COMPUT. GEOM. THEORY APPL
, 2005
"... A pseudotriangle is a simple polygon with exactly three convex vertices. A pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the le ..."
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Cited by 7 (0 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices. A pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the least number of pseudotriangles. We study the graph G whose vertices represent the pointed pseudotriangulations and whose edges represent flips. We present an algorithm for