Geometric Representation of Graphs in Low Dimension

  • L. Sunil Chandran
  • Naveen Sivadasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


An axis-parallel k–dimensional box is a Cartesian product R 1 ×R 2 ×⋯×R k where R i (for 1 ≤ik) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicitybox(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis–parallel) boxes in k–dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc.

A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem has logn approximation ratio for boxicity 2 graphs. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in 1.5 (Δ+ 2) ln n dimensions, where Δ is the maximum degree of G. We also show that box(G) ≤ (Δ + 2) ln n for any graph G. Our bound is tight up to a factor of ln n. The only previously known general upper bound for boxicity was given by Roberts, namely box(G) ≤ n/2. Our result gives an exponentially better upper bound for bounded degree graphs.

We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm.

Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, its boxicity is upper bound by c Open image in new window (d av + 1) ln n where d av is the average degree and c is a small constant. Also, we show that for any graph G, \(\ensuremath{\mathrm{box}}(G) \le \sqrt{8 n d_{av} \ln n}\), which is tight up to a factor of \(b \sqrt{\ln n}\) for a constant b.


Polynomial Time Maximum Degree Intersection Graph Interval Graph Unit Disk Graph 
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  1. 1.
    Afshani, P., Chan, T.: Approximation algorithms for maximum cliques in 3d unit-disk graphs. In: Proc. 17th Canadian Conference on Computational Geometry (CCCG), pp. 6–9 (2005)Google Scholar
  2. 2.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11, 209–218 (1998)MATHGoogle Scholar
  3. 3.
    Bellantoni, S., Hartman, I.B.-A., Przytycka, T., Whitesides, S.: Grid intersection graphs and boxicity. Discrete mathematics 114, 41–49 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Berman, P., DasGupta, B., Muthukrishnan, S., Ramaswami, S.: Efficient approximation algorithms for tiling and packing problems with rectangles. J. Algorithms 41, 443–470 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–21 (1993)MATHMathSciNetGoogle Scholar
  6. 6.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  7. 7.
    Chandran, L.S., Sivadasan, N.: Geometric representation of graphs in low dimension,
  8. 8.
    Chandran, L.S., Sivadasan, N.: Treewidth and boxicity (submitted), available at:
  9. 9.
    Chang, Y.W., West, D.B.: Rectangle number for hyper cubes and complete multipartite graphs. In: 29th SE conf. Comb., Graph Th. and Comp., Congr. Numer., vol. 132, pp. 19–28 (1998)Google Scholar
  10. 10.
    Chang, Y.W., West, D.B.: Interval number and boxicity of digraphs. In: Proceedings of the 8th International Graph Theory Conf. (1998)Google Scholar
  11. 11.
    Cozzens, M.B.: Higher and multidimensional analogues of interval graphs. Ph. D thesis, Rutgers University, New Brunswick, NJ (1981)Google Scholar
  12. 12.
    Cozzens, M.B., Roberts, F.S.: Computing the boxicity of a graph by covering its complement by cointerval graphs. Discrete Applied Mathematics 6, 217–228 (1983)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal of Computing (to appear)Google Scholar
  14. 14.
    Feinberg, R.B.: The circular dimension of a graph. Discrete mathematics 25, 27–31 (1979)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kloks, T.: Treewidth: Computations And Approximations. LNCS, vol. 842. Springer, Heidelberg (1994)MATHGoogle Scholar
  16. 16.
    Kratochvil, J.: A special planar satisfiability problem and a consequence of its NP–completeness. Discrete Applied Mathematics 52, 233–252 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Roberts, F.S.: Recent Progresses in Combinatorics. In: ch. On the boxicity and Cubicity of a graph, pp. 301–310. Academic Press, New York (1969)Google Scholar
  18. 18.
    Scheinerman, E.R.: Intersectin classes and multiple intersection parameters. Ph. D thesis, Princeton University (1984)Google Scholar
  19. 19.
    Shearer, J.B.: A note on circular dimension. Discrete mathematics 29, 103 (1980)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Thomassen, C.: Interval representations of planar graphs. Journal of combinatorial theory, Ser B 40, 9–20 (1986)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Trotter, W.T., West, J.D.B.: Poset boxicity of graphs. Discrete Mathematics 64, 105–107 (1987)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    van Leeuwen, E.J.: Approximation algorithms for unit disk graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 351–361. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM Journal on Algebraic Discrete Methods 3, 351–358 (1982)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • L. Sunil Chandran
    • 1
  • Naveen Sivadasan
    • 2
  1. 1.Dept. of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
  2. 2.Strand Life SciencesBangaloreIndia

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