Now showing items 1-12 of 12

    • 3-colorability of pseudo-triangulations 

      Aichholzer, Oswin; Aurenhammer, Franz; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit (2015)
      Article
      Open Access
      Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-tri ...
    • A new lower bound on the maximum number of plane graphs using production matrices 

      Huemer, Clemens; Pilz, Alexander; Silveira, Rodrigo Ignacio (2018)
      Conference lecture
      Restricted access - publisher's policy
      We use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(41.77n) crossing-free geometric graphs. This improves the previously best known bound of ¿(41.18n) by Aichholzer ...
    • A new lower bound on the maximum number of plane graphs using production matrices 

      Huemer, Clemens; Pilz, Alexander; Silveira, Rodrigo Ignacio (2019-11-01)
      Article
      Open Access
      We use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(42.11n ) crossing-free geometric graphs. This improves the previously best known bound of ¿(41.18n ) by Aichholzer ...
    • Cell-paths in mono- and bichromatic line arrangements in the plane 

      Aichholzer, Oswin; Cardinal, Jean; Hackl, Thomas; Hurtado Díaz, Fernando Alfredo; Korman Cozzetti, Matías; Pilz, Alexander; Silveira, Rodrigo Ignacio; Uehara, Ryuhei; Vogtenhuber, Birgit; Welzl, Emo (2014)
      Conference report
      Open Access
      We show that in every arrangement of n red and blue lines | in general position and not all of the same color | there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and no ...
    • Characteristic polynomials of production matrices for geometric graphs 

      Huemer, Clemens; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio (2017-08-01)
      Article
      Open Access
      An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production ...
    • Convex quadrangulations of bichromatic point sets 

      Pilz, Alexander; Seara Ojea, Carlos (2020-06-05)
      Article
      Open Access
      We consider quadrangulations of red and blue points in the plane where each face is convex and no edge connects two points of the same color. In particular, we show that the following problem is NP-hard: Given a finite set ...
    • Empty triangles in good drawings of the complete graph 

      Aichholzer, Oswin; Hackl, Thomas; Pilz, Alexander; Ramos Alonso, Pedro Antonio; Sacristán Adinolfi, Vera; Vogtenhuber, Birgit (2015)
      Article
      Open Access
      A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges ...
    • Geodesic order types 

      Aichholzer, Oswin; Korman Cozzetti, Matías; Pilz, Alexander; Vogtenhuber, Birgit (2014-09-01)
      Article
      Restricted access - publisher's policy
      The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained ...
    • Lower bounds for the number of small convex k-holes 

      Aichholzer, Oswin; Fabila Monroy, Ruy; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit (2014-07-01)
      Article
      Open Access
      Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds ...
    • New results on stabbing segments with a polygon 

      Díaz Bañez, José Miguel; Korman Cozzetti, Matías; Pérez Lantero, Pablo; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio (Springer, 2013)
      Conference report
      Restricted access - publisher's policy
      We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon P if at least one of its two endpoints is contained in P. A segment set S is stabbed by P ...
    • New results on stabbing segments with a polygon 

      Díaz Bañez, José Miguel; Korman Cozzetti, Matías; Pérez Lantero, Pablo; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio (2015-01-01)
      Article
      Open Access
      We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed ...
    • Production matrices for geometric graphs 

      Huemer, Clemens; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio (2016)
      Article
      Open Access
      We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot ...