Browsing by Author "Pilz, Alexander"
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3colorability of pseudotriangulations
Aichholzer, Oswin; Aurenhammer, Franz; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit (2015)
Article
Open AccessDeciding 3colorability for general plane graphs is known to be an NPcomplete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudotri ... 
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 policyWe use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(41.77n) crossingfree 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 (20191101)
Article
Open AccessWe use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(42.11n ) crossingfree geometric graphs. This improves the previously best known bound of ¿(41.18n ) by Aichholzer ... 
Cellpaths 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 AccessWe 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 (20170801)
Article
Open AccessAn 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 (20200605)
Article
Open AccessWe 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 NPhard: 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 AccessA 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 (20140901)
Article
Restricted access  publisher's policyThe 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 kholes
Aichholzer, Oswin; Fabila Monroy, Ruy; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit (20140701)
Article
Open AccessLet 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 Erdostype question on the least number h(k)(n) of convex kholes 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 policyWe 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 (20150101)
Article
Open AccessWe 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 AccessWe present production matrices for noncrossing 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 ...