Touching graphs of unit balls

  • Petr Hliněný
Packing, Compressing, and Touching
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)


The touching graph of balls is a graph that admits a representation by non-intersecting balls in the space (of prescribed dimension), so that its edges correspond to touching pairs of balls. By a classical result of Koebe [5], the disc touching graphs are exactly the planar graphs. This paper deals with a recognition of unit-ball touching graphs. The 2-dimensional case was proved to be NP-hard by Breu and Kirkpatrick [1]. We show in this paper that also unit-ball touching graphs in dimensions 3 and 4 are NP-hard to recognize. By a recent result of Kirkpatrick and Rote, these results may be transferred in ball-touching graphs in one dimension higher.


  1. 1.
    H. Breu, D.G. Kirkpatrick, On the complexity of recognizing intersection and touching graphs of discs, In: Graph Drawing (F.J. Brandenburg ed.), Proceedings Graph Drawing '95, Passau, September 1995, Lecture Notes in Computer Science 1027, Springer Verlag 1996, 88–98.Google Scholar
  2. 2.
    H. de Fraysseix, P.O. de Mendez, P. Rosenstiehl, On triangle contact graphs, Combinatorics, Probability and Computing 3 (1994), 233–246.Google Scholar
  3. 3.
    M.R. Garey, D.S. Johnson, Computers and Intractability, W.H. Freeman and Company, New York 1978.Google Scholar
  4. 4.
    P. Hliněný, Contact graphs of curves (extended abstract), In: Graph Drawing (F. J. Brandenburg ed.), Proceedings Graph Drawing '95, Passau, September 1995; Lecture Notes in Computer Science 1027, Springer Verlag 1996, 312–323.Google Scholar
  5. 5.
    P. Koebe, Kontaktprobleme der konformen Abbildung, Berichte über die Verhandlungen der Sächsischen, Akad. d. Wiss., Math.-Physische Klasse 88 (1936), 141–164.Google Scholar
  6. 6.
    J. Kratochvíl, Intersection graphs of noncrossing arc-connected sets in the plane, Proceedings Graph Drawing '96, Berkeley, September 1996; Lecture Notes in Computer Science 1190, Springer Verlag, Berlin Heidelberg 1997.Google Scholar
  7. 7.
    C.B. Lekkerkerker, J.C. Boland, Representation of finite graphs by a set of intervals on the real line, Fund. Math. 51 (1962), 45–64.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Petr Hliněný
    • 1
  1. 1.Dept. of Applied MathematicsCharles UniversityPraha 1

Personalised recommendations