Extending Partial Representations of Circle Graphs

  • Steven Chaplick
  • Radoslav Fulek
  • Pavel Klavík
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation \(\mathcal{R'}\) giving some pre-drawn chords that represent an induced subgraph of G. The question is whether one can extend \(\mathcal{R'}\) to a representation \(\mathcal{R}\) of the entire G, i.e., whether one can draw the remaining chords into a partially pre-drawn representation.

Our main result is a polynomial-time algorithm for partial representation extension of circle graphs. To show this, we describe the structure of all representation a circle graph based on split decomposition. This can be of an independent interest.


  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: ISAAC (to appear, 2013)Google Scholar
  3. 3.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA 2013, pp. 1030–1043 (2013)Google Scholar
  4. 4.
    Bouchet, A.: Reducing prime graphs and recognizing circle graphs. Combinatorica 7(3), 243–254 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bouchet, A.: Unimodularity and circle graphs. Discrete Mathematics 66(1-2), 203–208 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Annals of Mathematics 164, 51–229 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Courcelle, B.: Circle graphs and monadic second-order logic. J. Applied Logic 6(3), 416–442 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cunningham, W.: Decomposition of directed graphs. SIAM J. Alg. and Disc. Methods 3, 214–228 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dahlhaus, E.: Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition. Journal of Algorithms 36(2), 205–240 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    de Fraysseix, H.: Local complementation and interlacement graphs. Discrete Mathematics 33(1), 29–35 (1981)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    de Fraysseix, H., de Mendez, P.O.: On a characterization of gauss codes. Discrete & Computational Geometry 22(2), 287–295 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Even, S., Itai, A.: Queues, stacks, and graphs. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computation, pp. 71–76 (1971)Google Scholar
  13. 13.
    Gabor, C.P., Supowit, K.J., Hsu, W.: Recognizing circle graphs in polynomial time. J. ACM 36(3), 435–473 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient circle graph recognition. Algorithmica, 1–30 (2013)Google Scholar
  15. 15.
    Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. Journal of Graph Algortihms and Applications 16(2), 283–315 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    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, 2013)Google Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Kostochka, A., Kratochvíl, J.: Covering and coloring polygon-circle graphs. Discrete Mathematics 163(1-3), 299–305 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Naji, W.: Graphes de Cordes: Une Caracterisation et ses Applications. PhD thesis, l’Université Scientifique et Médicale de Grenoble (1985)Google Scholar
  22. 22.
    Oum, S.: Rank-width and vertex-minors. J. Comb. Theory, Ser. B 95(1), 79–100 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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
  24. 24.
    Spinrad, J.P.: Recognition of circle graphs. J. of Algorithms 16(2), 264–282 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Spinrad, J.P.: Efficient Graph Representations. Field Institute Monographs (2003)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Radoslav Fulek
    • 1
  • Pavel Klavík
    • 2
  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

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