From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

Authors Alexander Pilz, Emo Welzl, Manuel Wettstein

Thumbnail PDF


  • Filesize: 0.61 MB
  • 16 pages

Document Identifiers

Author Details

Alexander Pilz
Emo Welzl
Manuel Wettstein

Cite AsGet BibTex

Alexander Pilz, Emo Welzl, and Manuel Wettstein. From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n). Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.
  • Geometric Graph
  • Wheel Set
  • Simplicial Depth
  • Gale Transform
  • Polytope


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Bernardo M. Ábrego and Silvia Fernández-Merchant. A lower bound for the rectilinear crossing number. Graphs Combin., 21(3):293-300, 2005. URL:
  2. Peyman Afshani, Donald R. Sheehy, and Yannik Stein. Approximating the simplicial depth. CoRR, abs/1512.04856, 2015. Google Scholar
  3. Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, Hannes Krasser, and Birgit Vogtenhuber. On the number of plane geometric graphs. Graphs Combin., 23:67-84, 2007. URL:
  4. Esther M. Arkin, Samir Khuller, and Joseph S. B. Mitchell. Geometric knapsack problems. Algorithmica, 10(5):399-427, 1993. URL:
  5. Andries E. Brouwer. The enumeration of locally transitive tournaments. Technical Report ZW 138/80, Mathematisch Centrum, Amsterdam, 1980. Google Scholar
  6. Andrew Y. Cheng and Ming Ouyang. On algorithms for simplicial depth. In Proc. 13superscriptth Canadian Conference on Computational Geometry, pages 53-56, 2001. Google Scholar
  7. Serge Dulucq and Jean-Guy Penaud. Cordes, arbres et permutations. Discr. Math., 117(1):89-105, 1993. URL:
  8. Herbert Edelsbrunner and Ernst P. Mücke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph., 9(1):66-104, 1990. URL:
  9. Herbert Edelsbrunner, Joseph O'Rourke, and Raimund Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15(2):341-363, 1986. URL:
  10. David Eppstein, Mark H. Overmars, Günter Rote, and Gerhard J. Woeginger. Finding minimum area k-gons. Discr. Comput. Geom., 7:45-58, 1992. URL:
  11. Philippe Flajolet and Marc Noy. Analytic combinatorics of non-crossing configurations. Discr. Math., 204(1-3):203-229, 1999. URL:
  12. Joseph Gil, William L. Steiger, and Avi Wigderson. Geometric medians. Discr. Math., 108(1-3):37-51, 1992. URL:
  13. Jacob E. Goodman and Richard Pollack. Multidimensional sorting. SIAM J. Comput., 12(3):484-507, 1983. Google Scholar
  14. Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. J. Combin. Theory Ser. A, 37(3):257-293, 1984. Google Scholar
  15. Branko Grünbaum. Convex Polytopes. Springer, 2nd edition, 2003. Google Scholar
  16. Samir Khuller and Joseph S. B. Mitchell. On a triangle counting problem. Inf. Process. Lett., 33(6):319-321, 1990. URL:
  17. Svante Linusson. The number of M-sequences and f-vectors. Combinatorica, 19(2):255-266, 1999. URL:
  18. R. Y. Liu. On a notion of data depth based on random simplices. Annals of Statistics, 18:405-414, 1990. Google Scholar
  19. László Lovász, Katalin Vesztergombi, Uli Wagner, and Emo Welzl. Convex quadrilaterals and k-sets. In Towards a Theory of Geometric Graphs, pages 139-148. AMS, Providence, 2004. Google Scholar
  20. Jiří Matoušek. Lectures on Discrete Geometry. Springer, 2002. Google Scholar
  21. Juan José Montellano-Ballesteros and Ricardo Strausz. Counting polytopes via the Radon complex. J. Comb. Theory, Ser. A, 106(1):109-121, 2004. URL:
  22. Theodore S. Motzkin. Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products. Bull. Amer. Math. Soc., 54(4):352-360, 1948. Google Scholar
  23. Edgar M. Palmer and Robert W. Robinson. Enumeration of self-dual configurations. Pacific J. Math., 110(1):203-221, 1984. Google Scholar
  24. Dana Randall, Günter Rote, Francisco Santos, and Jack Snoeyink. Counting triangulations and pseudo-triangulations of wheels. In Proc. 13superscriptth Canadian Conference on Computational Geometry, pages 149-152, 2001. Google Scholar
  25. Peter J. Rousseeuw and Ida Ruts. Bivariate location depth. J. Royal Stat. Soc. Ser. C, 45(4):516-526, 1996. Google Scholar
  26. Andres J. Ruiz-Vargas and Emo Welzl. Crossing-free perfect matchings in wheel point sets. Unpublished manuscript, September 2015. Google Scholar
  27. Micha Sharir and Adam Sheffer. Counting triangulations of planar point sets. Electr. J. Combin., 18(1), 2011. Google Scholar
  28. Micha Sharir, Adam Sheffer, and Emo Welzl. Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn’s technique. J. Comb. Theory Ser. A, 120(4):777-794, 2013. URL:
  29. Micha Sharir and Emo Welzl. On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput., 36(3):695-720, 2006. URL:
  30. Uli Wagner. On the rectilinear crossing number of complete graphs. In Proc. 14superscriptth Annual Symposium on Discrete Algorithms, pages 583-588. ACM/SIAM, 2003. Google Scholar
  31. Uli Wagner and Emo Welzl. A continuous analogue of the Upper Bound Theorem. Discr. Comput. Geom., 26(2):205-219, 2001. URL:
  32. Emo Welzl. Entering and leaving j-facets. Discr. Comput. Geom., 25(3):351-364, 2001. URL:
  33. Günter M. Ziegler. Lectures on Polytopes. Springer, 1995. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail