12 research outputs found
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
On 4-connected geometric graphs
Given a set S of n points in the plane, in this paper we give a necessary and sometimes sufficient condition to build a 4-connected non-crossing geometric graph on S.Gobierno de AragĂłnMinisterio de EconomĂa y CompetitividadEuropean Science FoundationMinisterio de Ciencia e InnovaciĂł
Four-connected triangulations of planar point sets
In this paper, we consider the problem of determining in polynomial time
whether a given planar point set of points admits 4-connected
triangulation. We propose a necessary and sufficient condition for recognizing
, and present an algorithm of constructing a 4-connected
triangulation of . Thus, our algorithm solves a longstanding open problem in
computational geometry and geometric graph theory. We also provide a simple
method for constructing a noncomplex triangulation of which requires
steps. This method provides a new insight to the structure of
4-connected triangulation of point sets.Comment: 35 pages, 35 figures, 5 reference
Geometric biplane graphs I: maximal graphs
Postprint (author’s final draft
Geometric Biplane Graphs II: Graph Augmentation
We study biplane graphs drawn on a nite point set
S
in the plane in general position.
This is the family of geometric graphs whose vertex set is
S
and which can be decomposed
into two plane graphs. We show that every su ciently large point set admits a 5-connected
biplane graph and that there are arbitrarily large point sets that do not admit any 6-
connected biplane graph. Furthermore, we show that every plane graph (other than a
wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are
arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph
by adding pairwise noncrossing edges.Peer ReviewedPostprint (author’s final draft
Geometric biplane graphs II: graph augmentation
We study biplane graphs drawn on a finite point set in the plane in general position. This is the family of geometric graphs whose vertex set is and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author's final draft
Geração de malhas para domĂnios 2,5 dimensionais usando triangulação de delaunay restrita
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro TecnolĂłgico, Programa de PĂłs-graduação em Engenharia Mecânica, FlorianĂłpolis, 2001.Gerar uma malha consiste em discretizar um domĂnio geomĂ©trico em pequenos elementos de forma geomĂ©trica simplificada, como triângulos e/ou quadriláteros, em duas dimensões, e tetraedros e/ou hexaedros em trĂŞs dimensões. Malhas sĂŁo utilizadas em diversas áreas, como geologia, geografia e cartografia, onde elas fornecem uma representação compacta dos dados do terreno; em computação gráfica, a grande maioria dos objetos sĂŁo mapeados e imagens; e, em matemática aplicada e computação cientĂfica, sĂŁo essenciais na solução numĂ©rica de equações diferenciais parciais, resultantes do modelamento de problemas fĂsicos. Este trabalho concentra-se no desenvolvimento de um gerador de malhas voltadas para esta Ăşltima aplicação, mas que podem, tambĂ©m, ser empregadas nas outras áreas. Mais especificamente, o interesse está na geração de malhas de triângulos nĂŁo-estruturadas, atravĂ©s do processo de triangulação de Delaunay, para aplicações na solução de problemas de transferĂŞncia de calor em superfĂcies planas tridimensionais. Devido Ă utilização do mĂ©todo CVFEM (Control Volume based Finite Element Method) para a modelagem numĂ©rica, um paralelo entre a Triangulação de Delaunay e Diagramas de Voronoi Ă© delineado, apresentando suas propriedades e aplicaçõe .SĂŁo estudados os mĂ©todos de geração de triangulações de Delaunay para superfĂcies planas de inversĂŁo de aresta, divide-and-conquer e incremental. A estrutura de dados utilizada Ă© a triangular, e o mĂ©todo de refino para garantia de qualidade de malha Ă© baseado no algoritmo de Ruppert. Restrições geomĂ©tricas sĂŁo tratadas de forma que a malha gerada obedeça as intersecções e conexões entre diversas superfĂcies. A contribuição fundamental do presente trabalho está na extensĂŁo de mĂ©todos de riangulação de Delaunay e de refino de malha bidimensionais para domĂnios 2,5 dimensionais compostos, isto Ă© mĂşltiplos planos interconectados no espaço tridimensional tratados simultaneamente. Otimização de ângulos internos, tamanho e forma dos elementos atravĂ©s da especificação de parâmetros, conferem ao gerador desenvolvido versatilidade e generalidade
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Triangulating with high connectivity
We consider the problem of triangulating a given point set, using straight-line edges, so that the resulting graph is "highly connected." Since the resulting graph is planar, it can be at most 5-connected. Under the nondegeneracy assumption that no three points are collinear, we characterize the point sets with three vertices on the convex hull that admit 4-connected triangulations. More generally, we characterize the planar point sets that admit triangulations having neither chords nor noncomplex (i.e., nonfacial) triangles. We also show that any planar point set can be augmented with at most 2 extra points to admit a 4-connected triangulation. All our proofs are constructive, and the resulting triangulations can be constructed in 0(n log n) time. We conclude by stating several open problems