Bounded Representations of Interval and Proper Interval Graphs

  • Martin Balko
  • Pavel Klavík
  • Yota Otachi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals \(\mathfrak{L}_v\) and \(\mathfrak{R}_v\) called bounds. We ask whether there exists a bounded representation in which each interval \(\mathfrak{I}_v\) has its left endpoint in \(\mathfrak{L}_v\) and its right endpoint in \(\mathfrak{R}_v\). We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs.

Robert’s Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently.

The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benzer, S.: On the topology of the genetic fine structure. Proc. Nat. Acad. Sci. U.S.A. 45, 1607–1620 (1959)CrossRefGoogle Scholar
  2. 2.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA 2013 (2013)Google Scholar
  3. 3.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. System Sci. 13, 335–379 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. In: Wolff, A. (ed.) GD 2013. LNCS, vol. 8242, pp. 131–142. Springer, Heidelberg (accepted 2013)Google Scholar
  5. 5.
    Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Inf. Process. Lett. 55(2), 99–104 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deng, X., Hell, P., Huang, J.: Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM J. Comput. 25(2), 390–403 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dorbec, P., Kratochvíl, J., Montassier, M.: Contact representations of planar graph: Rebuilding is hard (submitted, 2013)Google Scholar
  8. 8.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hajós, G.: Über eine Art von Graphen. Internationale Mathematische Nachrichten 11, 65 (1957)Google Scholar
  11. 11.
    Jampani, K., Lubiw, A.: Simultaneous interval graphs. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 206–217. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B.: Extending partial representations of function graphs and permutation graphs. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 671–682. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Klavík, P., Kratochvíl, J., Otachi, Y., Rutter, I., Saitoh, T., Saumell, M., Vyskočil, T.: Extending partial representations of proper and unit interval graphs. In: Preparation (2012)Google Scholar
  14. 14.
    Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 444–454. Springer, Heidelberg (2012)Google Scholar
  15. 15.
    Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskočil, T.: Linear-time algorithm for partial representation extension of interval graphs. CoRR abs/1306.2182 (2013)Google Scholar
  16. 16.
    Klavík, P., Kratochvíl, J., Vyskočil, T.: Extending partial representations of interval graphs. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 276–285. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Köbler, J., Kuhnert, S., Watanabe, O.: Interval graph representation with given interval and intersection lengths. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 517–526. Springer, Heidelberg (2012)Google Scholar
  18. 18.
    Pe’er, I., Shamir, R.: Realizing interval graphs with size and distance constraints. SIAM J. Discrete Math. 175, 349–372 (1997)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Roberts, F.S.: Indifference graphs. Proof Techniques in Graph Theory, 139–146 (1969)Google Scholar
  20. 20.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SICOMP 5(2), 266–283 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin Balko
    • 1
  • Pavel Klavík
    • 2
  • Yota Otachi
    • 3
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  3. 3.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan

Personalised recommendations