On the complexity of recognizing intersection and touching graphs of disks

  • Heinz Breu
  • David G. Kirkpatrick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)


Disk intersection (respectively, touching) graphs are the intersection graphs of closed disks in the plane whose interiors may (respectively, may not) overlap. In a previous paper [BK93], we showed that the recognition problem for unit disk intersection graphs (i.e. intersection graphs of unit disks) is NP-hard. That proof is easily modified to apply to unit disk touching graphs as well. In this paper, we show how to generalize our earlier construction to accomodate disks whose size may differ. In particular, we prove that the recognition problems for both bounded-ratio disk intersection graphs and bounded-ratio disk touching graphs are also NP-hard. (By bounded-ratio we refer to the natural generalization of the unit constraint in which the radius ratio of the largest to smallest permissible disk is bounded by some fixed constant.) The latter result contrasts with the fact that the disk touching graphs (of unconstrained ratio) are precisely the planar graphs, and are hence polynomial time recognizable. The recognition problem for disk intersection graphs (of unconstrained ratio) has recently been shown to be NP-hard as well [Kra95].


  1. [Bre95]
    Heinz Breu. Algorithmic Aspects of Constrained Unit Disk Graphs. Ph.D. thesis, University of British Columbia, Vancouver, Canada. In preparation.Google Scholar
  2. [BC87]
    Sandeep N. Bhatt and Stavros S. Cosmadakis. The complexity of minimizing wire lengths in VLSI layouts. Information Processing Letters, 25(4):263–267, 1987.CrossRefGoogle Scholar
  3. [BK93]
    Heinz Breu and David G. Kirkpatrick. Unit Disk Graph Recognition is NP-Hard. Technical Report 93-27, Department of Computer Science, University of British Columbia, August, 1993. (To appear in Computational Geometry: Theory and Applications.)Google Scholar
  4. [BL76]
    K.S. Booth and G.S. Luecker. Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms. J. Comput. System Sci., 13:335–379, 1976.Google Scholar
  5. [Can88]
    John Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual Symposium on the Theory of Computing, pages 460–467, Chicago, Illinois, 2–4 May 1988.Google Scholar
  6. [CCJ90]
    Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1/3):165–177, 1990.CrossRefGoogle Scholar
  7. [FG65]
    D.R. Fulkerson and O.A. Gross. Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15(3):835–355, 1965.Google Scholar
  8. [Fis83]
    Peter C. Fishburn. On the sphericity and cubicity of graphs. Journal of Combinatorial Theory, Series B, 35:309–318, 1983.Google Scholar
  9. [GJ79]
    Michael R. Garey and David S. Johnson. Computers and Intractability: a Guide to the Theory of NP-completeness. W.H. Freeman and Company, 1979.Google Scholar
  10. [Grä95]
    Albert Gräf. Coloring and Recognizing Special Graph Classes. PhD thesis, Musikwissenschaftliches Institut, Abteilung Musikinformatik, Johannes Gutenberg-Universität Mainz, February 1995. Available as technical report Bericht Nr. 20.Google Scholar
  11. [Hal80]
    William K. Hale. Frequency assignment: theory and applications. Proceedings of the IEEE, 68(12):1497–1514, 1980.Google Scholar
  12. [Hav82a]
    Timothy Franklin Havel. The Combinatorial Distance Geometry Approach to the Calculation of Molecular Conformation. PhD thesis, University of California, Berkeley, 1982. Cited by [Fis83].Google Scholar
  13. [JAMS91]
    David S. Johnson, Cecilia A. Aragon, Lyle.A. McGeogh, and Catherine Schevon. Optimization by simulated annealing: an experimental evaluation; part II, graph coloring and number partioning. Operations Research, 39(3):378–406, May–June 1991.Google Scholar
  14. [Kra91]
    J. Kratochvíl. String graphs II. Recognizing string graphs is NP-hard. J. Combin. Theory Ser. B, 52: 67–78, 1991.CrossRefGoogle Scholar
  15. [Kra94]
    J. Kratochvíl. A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Applied Mathematics, 52: 233–252, 1994.CrossRefGoogle Scholar
  16. [Kra95]
    J. Kratochvíl. Personal communication, February, 1995.Google Scholar
  17. [MHR92]
    M.V. Marathe, H.B. Hunt III, and S.S. Ravi. Geometry based approximations for intersection graphs. In Fourth Canadian Conference on Computational Geometry, pages 244–249, 10–14 August 1992.Google Scholar
  18. [MP92]
    S. Malitz and A. Papakostas. On the angular resolution of planar graphs. In Proceedings of the 24th Annual Symposium on the Theory of Computing, pages 527–538, May 1992.Google Scholar
  19. [Rob68]
    Fred S. Roberts. Indifference graphs. In Proof Techniques in Graph Theory, pages 139–146, Academic Press, New York and London, February 1968. Proceedings of the Second Ann Arbor Graph Theory Conference.Google Scholar
  20. [Rob91]
    Fred S. Roberts. Quo Vadis, Graph Theory. Technical Report 91-33, DIMACS, May 1991.Google Scholar
  21. [Sac94]
    Horst Sachs. Coin graphs, polyhedra, and conformal mapping. Discrete Mathematics, 134: 133–138, 1994.CrossRefGoogle Scholar
  22. [Spi85]
    Jeremy Spinrad. On comparability and permutation graphs. SIAM Journal on Computing, 14(3):658–670, August 1985.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Heinz Breu
    • 1
  • David G. Kirkpatrick
    • 1
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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