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Introduction Counting Triangulations of a Convex Polygon
"... problem of counting the number of triangulations of a convex polygon. Euler, one of the most ..."
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problem of counting the number of triangulations of a convex polygon. Euler, one of the most
Minimum weight triangulation is NPhard
 IN PROC. 22ND ANNU. ACM SYMPOS. COMPUT. GEOM
, 2006
"... A triangulation of a planar point set S is a maximal plane straightline graph with vertex set S. In the minimum weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem ..."
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Cited by 42 (0 self)
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is NPhard. We use a reduction from PLANAR1IN3SAT. The correct working of the gadgets is established with computer assistance, using geometric inclusion and exclusion criteria for MWT edges, such as the diamond test and the LMTSkeleton heuristic, as well as dynamic programming on polygonal faces.
Counting polygon dissections in the projective plane
 Advances in Applied Mathematics
, 2008
"... For each value of k ≥ 2, we determine the number pn of ways of dissecting a polygon in the projective plane into n subpolygons with k + 1 sides each. In particular, if k = 2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the Möbius band having n labelled points o ..."
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Cited by 2 (0 self)
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For each value of k ≥ 2, we determine the number pn of ways of dissecting a polygon in the projective plane into n subpolygons with k + 1 sides each. In particular, if k = 2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the Möbius band having n labelled points
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 25 (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
Flip Distance Between Triangulations of a Simple Polygon is NPComplete
 EXTENDED ABSTRACT IN PROC. 29 TH EUROCG
, 2013
"... Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangul ..."
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Cited by 5 (1 self)
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show that computing the flip distance between two triangulations of a simple polygon is NPcomplete. This complements a recent result that shows APXhardness of determining the flip distance between two triangulations of a planar point set.
AN O(n log log n)TIME ALGORITHM FOR TRIANGULATING A SIMPLE POLYGON
, 1988
"... Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that tria ..."
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Cited by 38 (3 self)
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that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.
Expansive motions and the polytope of pointed pseudotriangulations
 Discrete and Computational Geometry  The GoodmanPollack Festschrift, Algorithms and Combinatorics
, 2003
"... We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriang ..."
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Cited by 52 (14 self)
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We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriangulations
Counting Crossing Free Structures
"... Let P be a set of n points in the plane. A crossingfree structure on P is a straightedge planar graph with vertex set in P. Examples of crossingfree structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount ..."
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Let P be a set of n points in the plane. A crossingfree structure on P is a straightedge planar graph with vertex set in P. Examples of crossingfree structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount
Counting Carambolas
, 2014
"... We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane. Configurations of interest include convex polygons, starshaped polygons and monotone paths. We also consider related prob ..."
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We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane. Configurations of interest include convex polygons, starshaped polygons and monotone paths. We also consider related
On kConvex Polygons ∗
, 2011
"... We introduce a notion of kconvexity and explore polygons in the plane that have this property. Polygons which are kconvex can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUMhard problem. We give a characterization of 2convex polygons, a particularl ..."
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Cited by 3 (2 self)
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We introduce a notion of kconvexity and explore polygons in the plane that have this property. Polygons which are kconvex can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUMhard problem. We give a characterization of 2convex polygons, a
Results 1  10
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