Extending Partial Representations of Interval Graphs

  • Pavel Klavík
  • Jan Kratochvíl
  • Tomáš Vyskočil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


We initiate the study of the computational complexity of the question of extending partial representations of geometric intersection graphs. In this paper we consider classes of interval graphs – given a collection of real intervals that forms an intersection representation of an induced subgraph of an input graph, is it possible to add intervals to achieve an intersection representation of the entire graph? We present an \(\mathcal{O}(n^2)\) time algorithm that solves this problem and constructs a representation if one exists. Our algorithm can also be used to list all nonisomorphic extensions with \(\mathcal{O}(n^2)\) delay.

Although the classes of proper and unit interval graphs coincide, the partial representation extension problems differ on them. We present an \(\mathcal{O}(mn)\) time decision algorithm for partial representation extension of proper interval graphs, but for unit interval graphs the complexity remains open.

Finally we show how our methods can be used for solving the problem of simultaneous interval representations. We prove that this problem is fixed-paramater tractable with the size of the common intersection of the input graphs being the parameter.


Maximal Clique Intersection Representation Intersection Graph Interval Graph Input Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)CrossRefzbMATHGoogle Scholar
  2. 2.
    Angelini, P., Battista, G.D., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: SODA 2010: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (2010)Google Scholar
  3. 3.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using pq-tree algorithms. Journal of Computational Systems Science 13, 335–379 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Corneil, D.G.: A simple 3-sweep lbfs algorithm for the recognition of unit interval graphs. Discrete Appl. Math. 138(3), 371–379 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Corneil, D.G., Olariu, S., Stewart, L.: The lbfs structure and recognition of interval graphs. SIAM Journal on Discrete Mathematics 23(4), 1905–1953 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiala, J.: NP completeness of the edge precoloring extension problem on bipartite graphs. J. Graph Theory 43(2), 156–160 (2003)MathSciNetCrossRefzbMATHGoogle 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.
    Golumbic, M., Kaplan, H., Shamir, R.: Algorithms and complexity of sandwich problems in graphs (extended abstract). In: van Leeuwen, J. (ed.) WG 1993. LNCS, vol. 790, pp. 57–69. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. North-Holland Publishing Co., Amsterdam (2004)zbMATHGoogle Scholar
  11. 11.
    Golumbic, M.C., Shamir, R.: Complexity and algorithms for reasoning about time: a graph-theoretic approach. J. ACM 40(5), 1108–1133 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hajós, G.: Über eine Art von Graphen. Internationale Mathematische Nachrichten 11, 65 (1957)Google Scholar
  13. 13.
    Hell, P., Huang, J.: Lexicographic orientation and representation algorithms for comparability graphs, proper circular arc graphs, and proper interval graphs. Journal of Graph Theory 20(3), 361–374 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jampani, K., Lubiw, A.: Simultaneous interval graphs. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 206–217. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Krokhin, A., Jeavons, P., Jonsson, P.: Reasoning about temporal relations: The tractable subalgebras of allen’s interval algebra. J. ACM 50(5), 591–640 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Looges, P.J., Olariu, S.: Optimal greedy algorithms for indi erence graphs. Comput. Math. Appl. 25, 15–25 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marx, D.: NP-completeness of list coloring and precoloring extension on the edges of planar graphs. J. Graph Theory 49(4), 313–324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  19. 19.
    Patrignani, M.: On extending a partial straight-line drawing. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 380–385. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press, London (1969)Google Scholar
  21. 21.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing 5(2), 266–283 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Spinrad, J.P.: Efficient Graph Representations. Field Institute Monographs (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Pavel Klavík
    • 1
  • Jan Kratochvíl
    • 1
  • Tomáš Vyskočil
    • 1
  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer ScienceCharles UniversityPragueCzech Republic

Personalised recommendations