Minimal Obstructions for Partial Representations of Interval Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

Abstract

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation.

In this paper, we characterize the minimal obstructions which make a partial representation non-extendible. This generalizes Lekkerkerker and Boland’s characterization of minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to the first polynomial-time certifying algorithm for partial representation extension of intersection graphs.

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References

  1. 1.
    Angelini, P., Battista, G.D., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: SODA 2010, pp. 202–221 (2010)Google Scholar
  2. 2.
    Balko, M., Klavík, P., Otachi, Y.: Bounded representations of interval and proper interval graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 535–546. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Benzer, S.: On the topology of the genetic fine structure. Proc. Nat. Acad. Sci. U.S.A. 45, 1607–1620 (1959)CrossRefGoogle Scholar
  4. 4.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA 2013, pp. 1030–1043 (2013)Google Scholar
  5. 5.
    Booth, K., Lueker, G.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. System Sci. 13, 335–379 (1976)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J.: Contact representations of planar graph: Rebuilding is hard. In: WG 2014 (2014)Google Scholar
  7. 7.
    Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 131–142. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Colbourn, C.J., Booth, K.S.: Linear times automorphism algorithms for trees, interval graphs, and planar graphs. SIAM J. Comput. 10(1), 203–225 (1981)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hajós, G.: Über eine Art von Graphen. Internat. Math. News 11, 65 (1957)Google Scholar
  11. 11.
    Jelínek, V., Kratochvíl, J., Rutter, I.: A kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. 46(4), 466–492 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kendall, D.G.: Incidence matrices, interval graphs and seriation in archaeology. Pac. J. Math 28(3), 565–570 (1969)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    Klavík, P., Kratochvíl, J., Otachi, Y., Rutter, I., Saitoh, T., Saumell, M., Vyskočil, T.: Extending partial representations of proper and unit interval graphs. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 253–264. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  15. 15.
    Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 444–454. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskočil, T.: Extending partial representations of interval graphs. CoRR, abs/1306.2182 (2013)Google Scholar
  17. 17.
    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
  18. 18.
    Korte, N., Möhring, R.: An incremental linear-time algorithm for recognizing interval graphs. SIAM J. Comput. 18(1), 68–81 (1989)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Lekkerkerker, C., Boland, D.: Representation of finite graphs by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)MATHMathSciNetGoogle Scholar
  20. 20.
    Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Roberts, F.S.: Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems. Prentice-Hall, Englewood Cliffs (1976)MATHGoogle Scholar
  22. 22.
    Skrien, D.: Chronological orderings of interval graphs. Discrete Appl. Math. 8(1), 69–83 (1984)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Computer Science InstituteCharles University in PraguePragueCzech Republic
  2. 2.Department of Mathematics and European Centre of Excellence NTISUniversity of West BohemiaPilsenCzech Republic

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