ω-Searchlight Obedient Graph Drawings

  • Gill Barequet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1984)

Abstract

A drawing of a graph in the plane is ω-searchlight obedient if every vertex of the graph is located on the centerline of some strip of width ω, which does not contain any other vertex of the graph. We estimate the maximum possible value ω(n) of an ω-searchlight obedient drawing of a graph with n vertices, which is contained in the unit square. We show a lower bound and an upper bound on ω(n), namely, ω(n) = Ω(log n=n) and ω(n) = ω(n) O(1/n 4/7−), for an arbitrarily small ε > 0. Any improvement for either bound will also carry on to the famous Heilbronn’s triangle problem

Keywords

Geometric optimization Heilbronn’s triangle problem 

References

  1. 1.
    Cheng, C.C., Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Drawing planar graphs with circular arcs. Proc. 7th Graph Darwing Conference, Prague, Czech, 1999, 117–126. Lecture Notes in Computer Science 1731, Springer-VerlagGoogle Scholar
  2. 2.
    Goodman, J.E., Lutwak, E., Malkevitch, J., Pollack, R. (eds.): Discrete Geometry and Convexity. Ann. New York Acad. Sci. (440), 1985Google Scholar
  3. 3.
    Hadwiger, H., Debrunner, H. (translation by V. Klee): Combinatorial Geometry in the Plane. Holt, Rinehart, and Winston, New York, 1964Google Scholar
  4. 4.
    Komlós, J., Pintz, J., Szemerédi, E.: On Heilbronn’s triangle problem. J. London Mathematical Society (2) 24 (1981) 385–396MATHCrossRefGoogle Scholar
  5. 5.
    Komlós, J., Pintz, J., Szemerédi, E.: A lower bound for Heilbronn’s problem. J. London Mathematical Society (2) 25 (1982) 13–24MATHCrossRefGoogle Scholar
  6. 6.
    Moser W., Pach, J.: Research problems in Discrete Geometry. Mineographed Notes, 1985Google Scholar
  7. 7.
    Roth, K.F.: On a problem of Heilbronn. Proc. London Mathematical Society 26(1951) 198–204Google Scholar
  8. 8.
    Roth, K.F.: On a problem of Heilbronn, III. Proc. London Mathematical Society (3) 25 (1972) 543–549Google Scholar
  9. 9.
    Roth, K.F.: Developments in Heilbronn’s triangle problem. Advances in Mathematics 22 (1976) 364–385MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Gill Barequet
    • 1
  1. 1.The Technion—Israel Institute of TechnologyHaifaIsrael

Personalised recommendations