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"... Lower bounds on the maximum number of noncrossing acyclic graphs Clemens Huemer∗, Anna de Mier† This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of noncrossing spanning trees and forests. We show that the socal ..."
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Lower bounds on the maximum number of noncrossing acyclic graphs Clemens Huemer∗, Anna de Mier† This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of noncrossing spanning trees and forests. We show that the so
Binary labelings for bipartite graphs
, 2007
"... Part of the authors introduced in [C. Huemer, S. Kappes, A binary labelling for plane Laman graphs and quadrangulations, in Proceedings of the 22nd European Workshop on Computational Geometry 83–86, 2006] a binary labeling for the angles of plane quadrangulations, similar to Schnyder labelings of th ..."
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Part of the authors introduced in [C. Huemer, S. Kappes, A binary labelling for plane Laman graphs and quadrangulations, in Proceedings of the 22nd European Workshop on Computational Geometry 83–86, 2006] a binary labeling for the angles of plane quadrangulations, similar to Schnyder labelings
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 13 (3 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.
Compatible Geometric Matchings
, 2007
"... This paper studies noncrossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also noncrossing. Our first result states that for any two perfect matchings M and M ′ of the same set of n points, for some k ∈ O(log n), there ..."
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Cited by 15 (9 self)
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This paper studies noncrossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also noncrossing. Our first result states that for any two perfect matchings M and M ′ of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M0, M1,..., Mk = M ′ , such that each Mi is compatible with Mi+1. This improves the previous best bound of k ≤ n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edgedisjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edgedisjoint compatible matching with approximately
Large bichromatic point sets admit empty monochromatic 4gons
 in Proceedings of the 25th European Workshop on Computational Geometry
, 2009
"... Large bichromatic point sets admit empty monochromatic 4gons ..."
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Cited by 5 (2 self)
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Large bichromatic point sets admit empty monochromatic 4gons
Empty monochromatic triangles
 COMPUTATIONAL GEOMETRY, THEORY AND APPLICATIONS
, 2008
"... We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex kgons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As ..."
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Cited by 7 (4 self)
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We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex kgons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bicolored set of n points in R 2 in general position determines a superlinear number of empty monochromatic triangles, namely Ω(n 5/4).
Binary labelings for plane quadrangulations and their relatives
, 2008
"... Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for tr ..."
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Cited by 13 (8 self)
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Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2book. Furthermore, as Schnyder labelings have been extended to 3connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2connected bipartite graphs. Finally, we propose a binary labeling for Laman graphs.
Transforming spanning trees and pseudotriangulations
 Inf. Process. Lett
, 2006
"... Let TS be the set of all crossingfree straight line spanning trees of a planar npoint set S. Consider the graph TS where two members T and T ′ of TS are adjacent if T intersects T ′ only in points of S or in common edges. We prove that the diameter of TS is O(log k), where k denotes the number of ..."
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Cited by 5 (1 self)
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Let TS be the set of all crossingfree straight line spanning trees of a planar npoint set S. Consider the graph TS where two members T and T ′ of TS are adjacent if T intersects T ′ only in points of S or in common edges. We prove that the diameter of TS is O(log k), where k denotes the number of convex layers of S. Based on this result, we show that the flip graph PS of pseudotriangulations of S (where two pseudotriangulations are adjacent if they differ in exactly one edge – either by replacement or by removal) has a diameter of O(n log k). This sharpens a known O(n log n) bound. Let � PS be the induced subgraph of pointed pseudotriangulations of PS. We present an example showing that the distance between two nodes in � PS is strictly larger than the distance between the corresponding nodes in PS. 1
On kGons and kHoles in Point Sets
"... We consider a variation of the classical ErdősSzekeres problems on the existence and number of convex kgons and kholes (empty kgons) in a set of n points in the plane. Allowing the kgons to be nonconvex, we show bounds and structural results on maximizing and minimizing their numbers. Most not ..."
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Cited by 2 (2 self)
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We consider a variation of the classical ErdősSzekeres problems on the existence and number of convex kgons and kholes (empty kgons) in a set of n points in the plane. Allowing the kgons to be nonconvex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of kholes, and show that this number is maximized by sets in convex position. We also provide an improved lower bound for the number of convex 6holes. 1
Results 1  10
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