Abstract order type extension and new results on the rectilinear crossing number

We extend the order type data base of all realizable order types in the plane to point sets of cardinality 11. More precisely, we provide a complete data base of all combinatorial different sets of up to 11 points in general position in the plane. In addition, we develop a novel and efficient method for a complete extension to order types of size 12 and more in an abstract sense, that is, without the need to store or realize the sets. The presented method is well suited for independent computations. Thus, time intensive investigations benefit from the possibility of distributed computing.Our approach has various applications to combinatorial problems which are based on sets of points in the plane. This includes classic problems like searching for (empty) convex k-gons ('happy end problem'), decomposing sets into convex regions, counting structures like triangulations or pseudo-triangulations, minimal crossing numbers, and more. We present some improved results to all these problems. As an outstanding result we have been able to determine the exact rectilinear crossing number of the complete graph Kn for up to n = 17, the largest previous range being n = 12, and slightly improved the asymptotic upper bound.

[1]  Stephane Durocher,et al.  The Rectilinear Crossing Number of K10 is 62 , 2000, Electron. J. Comb..

[2]  P. Erdös,et al.  Crossing Number Problems , 1973 .

[3]  Oswin Aichholzer.  The path of a triangulation , 1999, SCG '99.

[4]  Richard Pollack,et al.  Multidimensional Sorting , 1983, SIAM J. Comput..

[5]  J. Pach,et al.  Thirteen problems on crossing numbers , 2000 .

[6]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[7]  Jürgen Richter,et al.  On the Finding of Final Polynomials , 1990, Eur. J. Comb..

[8]  Hannes Krasser.  Order types of point sets in the plane , 2003 .

[9]  Komei Fukuda,et al.  Generation of Oriented Matroids—A Graph Theoretical Approach , 2002, Discret. Comput. Geom..

[10]  Franz Aurenhammer,et al.  On the Crossing Number of Complete Graphs , 2002, SCG '02.

[11]  József Balogh,et al.  Improved Bounds for the Number of (<=k)-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn , 2004, Graph Drawing.

[12]  Oswin Aichholzer,et al.  The point set order type data base: A collection of applications and results , 2001, CCCG.

[13]  Günter M. Ziegler,et al.  Oriented Matroids , 2017, Handbook of Discrete and Computational Geometry, 2nd Ed..

[14]  Bettina Speckmann,et al.  On pseudo-convex decompositions, partitions, and coverings , 2005, EuroCG.

[15]  W. T. Tutte.  Toward a theory of crossing numbers , 1970 .

[16]  Franz Aurenhammer,et al.  Enumerating Order Types for Small Point Sets with Applications , 2002, Order.

[17]  Géza Tóth,et al.  The Erdo"s-Szekeres Theorem: upper Bounds and Related Results , 2007 .

[18]  Oswin Aichholzer,et al.  On the structure of sets attaining the rectilinear crossing number , 2006 .

[19]  Bernd Sturmfels,et al.  Coordinate representation of order types requires exponential storage , 1989, STOC '89.

[20]  J. Pach.  Towards a Theory of Geometric Graphs , 2004 .

[21]  F. Harary,et al.  On the Number of Crossings in a Complete Graph , 1963, Proceedings of the Edinburgh Mathematical Society.

[22]  Jürgen Richter-Gebert,et al.  Two interesting oriented matroids , 1996, Documenta Mathematica.

[23]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[24]  Raimund Seidel,et al.  A simple and less slow method for counting triangulations and for related problems , 2004 .

[25]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[26]  Stefan Felsner,et al.  On the Number of Arrangements of Pseudolines , 1996, SCG '96.

[27]  József Balogh,et al.  k-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn , 2006, Discret. Comput. Geom..

[28]  E. Welzl,et al.  Convex Quadrilaterals and k-Sets , 2003 .

[29]  Marc Noy,et al.  A lower bound on the number of triangulations of planar point sets , 2004, Comput. Geom..

[30]  B. Sturmfels.  Oriented Matroids , 1993 .