Graphs and Combinatorics

, Volume 29, Issue 5, pp 1221–1234 | Cite as

Helly Numbers of Polyominoes

  • Jean Cardinal
  • Hiro Ito
  • Matias Korman
  • Stefan Langerman
Original Paper

Abstract

We define the Helly number of a polyomino P as the smallest number h such that the h-Helly property holds for the family of symmetric and translated copies of P on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyominoes of Helly number k for any k ≠ 1, 3.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bárány I., Matousek J.: A fractional Helly theorem for convex lattice sets. Adv. Math. 174(2), 227–235 (2003)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barbosa R.M., Coelho E.M.M., Dourado M.C., Szwarcfiter J.L.: The colorful helly theorem and general hypergraphs. Eur. J. Comb. 33(5), 743–749 (2012)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beck J.: Combinatorial games: Tic–Tac–Toe theory. Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  4. 4.
    Berge, C.: Graphs and hypergraphs. North-Holland, Amsterdam, New York ([rev. ed.] translated by edward minieka. edition) (1973)Google Scholar
  5. 5.
    Cardinal J., Collette S., Ito H., Sakaidani H., Korman M., Langerman Taslakian P.: Cannibal animal games: a new variant of tic-tac-toe. In: Proceedings of the 27th European Workshop on Computational Geometry, pp. 131–134 (2011)Google Scholar
  6. 6.
    Doignon J.P.: Convexity in crystallographic lattices. J. Geom. 3, 71–85 (1973)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dourado M.C., Protti F., Szwarcfiter J.L.: Computational aspects of the helly property: a survey. J. Brazil. Comput. Soc. 12(1), 7–33 (2006)MathSciNetGoogle Scholar
  8. 8.
    Golomb S.W.: Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd edn. Princeton University Press, New Jersey (1996)Google Scholar
  9. 9.
    Golumbic M.C., Jamison R.E.: The edge intersection graphs of paths in a tree. J. Comb. Theory Ser. B 38(1), 8–22 (1985)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Golumbic M.C., Lipshteyn M., Stern M.: Edge intersection graphs of single bend paths on a grid. Networks 54(3), 130–138 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2004)Google Scholar
  12. 12.
    Graham, R.L., Grötschel, M., Lovász, L. (eds.): Handbook of combinatorics, vol. 1. MIT Press, Cambridge (1995)Google Scholar
  13. 13.
    Hugh D., Hugh D.: Counting polyominoes: yet another attack. Discrete Math. 36(3), 191–203 (1981)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Morishita, T.: Constructing a program for enumerating the helly number of polyominoes. Unpublished manuscript (in japanese) (2011). Available at: http://llati.upc.es/mkorman/papers/cpehnp.pdf.

Copyright information

© Springer 2012

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Hiro Ito
    • 2
  • Matias Korman
    • 1
  • Stefan Langerman
    • 1
  1. 1.Computer Science DepartmentUniversité Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations