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Extending Partial Representations of Interval Graphs

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Abstract

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph G can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker (J Comput Syst Sci 13:335–379, 1976) based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph G with a partial representation \({{{\mathcal {R}}}}'\) fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation \({{{\mathcal {R}}}}\) of the entire graph G extending \({{{\mathcal {R}}}}'\). We generalize the characterization of interval graphs by Fulkerson and Gross (Pac J Math 15:835–855, 1965) to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.

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Notes

  1. For the purpose of Sect. 3, we allow empty intervals with \(\ell _v = r_v\).

  2. This can be done in constant time if we remember in each moment the positions of the two leftmost lower handles in the ordering, and update this information after removing one of them from .

  3. We also need to update here since it might happen that the interval is empty for some maximal clique a. This can happen only if some pre-drawn interval of P(a) is a singleton.

References

  1. Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32:1–32:42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balko, M., Klavík, P., Otachi, Y.: Bounded representations of interval and proper interval graphs. In: Algorithms and Computation, Lecture Notes in Computer Science, vol. 8283, pp. 535–546. Springer, Berlin (2013)

  3. Bang-Jensen, J., Huang, J., Zhu, X.: Completing orientations of partially oriented graphs. CoRR (2015). arXiv:1509.01301

  4. Benzer, S.: On the topology of the genetic fine structure. Proc. Natl. Acad. Sci. USA 45, 1607–1620 (1959)

    Article  Google Scholar 

  5. Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Trans. Algorithms 12(2), 16:1–16:46 (2015)

    Article  MathSciNet  Google Scholar 

  6. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J.: Contact representations of planar graphs: extending a partial representation is hard. In: Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, pp. 139–151 (2014)

  8. Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. In: Graph Drawing, Lecture Notes in Computer Science, vol. 8242, pp. 131–142. Springer, Berlin (2013)

  9. Chaplick, S., Guśpiel, G., Gutowski, G., Krawczyk, T., Liotta, G.: The partial visibility representation extension problem. CoRR (2015). arXiv:1512.00174

  10. Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discrete Math. 23(4), 1905–1953 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fishburn, P.: Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. Wiley, New York (1985)

    MATH  Google Scholar 

  12. Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. North-Holland Publishing Co, Amsterdam (2004)

    MATH  Google Scholar 

  14. Habib, M., McConnell, R., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comput. Sci. 234, 59–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hajós, G.: Über eine Art von Graphen. Int. Math. Nachr. 11, 65 (1957)

    Google Scholar 

  16. Hsu, W.: \(O(M \cdot N)\) algorithms for the recognition and isomorphism problems on circular-arc graphs. SIAM J. Comput. 24(3), 411–439 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jampani, K.R., Lubiw, A.: Simultaneous interval graphs. In: Algorithms and Computation, Lecture Notes in Computer Science, vol. 6506, pp. 206–217 (2010)

  18. Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. J. Graph Algortihms Appl. 16(2), 283–315 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B.: Extending partial representations of function graphs and permutation graphs. In: Algorithms—ESA 2012, Lecture Notes in Computer Science, vol. 7501, pp. 671–682 (2012)

  20. Klavík, P., Kratochvíl, J., Otachi, Y., Rutter, I., Saitoh, T., Saumell, M., Vyskočil, T.: Extending partial representations of proper and unit interval graphs. Algorithmica (2016). doi:10.1007/s00453-016-0133-z

  21. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. Theor. Comput. Sci. 576, 85–101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klavík, P., Kratochvíl, J., Vyskočil, T.: Extending partial representations of interval graphs. In: Theory and Applications of Models of Computation—8th Annual Conference, TAMC 2011, Lecture Notes in Computer Science, vol. 6648, pp. 276–285 (2011)

  23. Klavík, P., Otachi, Y., Šejnoha, J.: On the classes of interval graphs of limited nesting and count of lengths. CoRR (2015). arXiv:1510.03998

  24. Klavík, P., Saumell, M.: Minimal obstructions for partial representations of interval graphs. In: Ahn, H.-K., Shin, C.-S. (eds.) Algorithms and Computation, ISAAC 2014. Lecture Notes in Computer Science, vol. 8889, pp. 401–413. Springer International Publishing (2014). http://link.springer.com/chapter/10.1007%2F978-3-319-13075-0_32

  25. Kobler, J., Kuhnert, S., Watanabe, O.: Interval graph representation with given interval and intersection lengths. In: Algorithms and Computation, Lecture Notes in Computer Science, vol. 7676, pp. 517–526 (2012)

  26. Korte, N., Möhring, R.: An incremental linear-time algorithm for recognizing interval graphs. SIAM J. Comput. 18(1), 68–81 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lekkerkerker, C., Boland, D.: Representation of finite graphs by a set of intervals on the real line. Fundam. Math. 51, 45–64 (1962)

    MathSciNet  MATH  Google Scholar 

  28. Lindzey, N., McConnell, R.M.: On finding tucker submatrices and Lekkerkerker–Boland subgraphs. In: Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 8165, pp. 345–357 (2013)

  29. Marczewski, E.S.: Sur deux propriétés des classes d’ensembles. Fundam. Math. 33, 303–307 (1945)

    MATH  Google Scholar 

  30. McConnell, R.M.: Linear-time recognition of circular-arc graphs. Algorithmica 37(2), 93–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)

  32. Patrignani, M.: On extending a partial straight-line drawing. In: GD’2006, Lecture Notes in Computer Science, vol. 3843, pp. 380–385 (2006)

  33. Pe’er, I., Shamir, R.: Realizing interval graphs with size and distance constraints. SIAM J. Discrete Math. 10(4), 662–687 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Roberts, F.S.: Indifference graphs. Proof Tech. Graph Theory 139, 146 (1969)

    MATH  Google Scholar 

  35. Roberts, F.S.: Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems. Prentice-Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  36. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. Soulignac, F.J.: Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. CoRR (2014). arXiv:1408.3443

  38. Spinrad, J.P.: Efficient Graph Representations. Field Institute Monographs. American Mathematical Society (2003)

  39. Stoffers, K.E.: Scheduling of traffic lights—a new approach. Transp. Res. 2, 199–234 (1968)

    Article  Google Scholar 

  40. Trotter, W.T.: New perspectives on interval orders and interval graphs. In: Bailey, R.A. (ed.) In Surveys in Combinatorics, pp. 237–286. Cambridge University Press, Cambridge (1997)

  41. Zeman, P.: Extending partial representations of subclasses of circular-arc graphs (2016, in preparation)

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Acknowledgments

We are very thankful to Pavol Hell for suggesting the PQ-trees approach, and to Martin Balko and Jiří Fiala for comments concerning writing. The first two authors are supported by CE-ITI (P202/12/G061 of GAČR) and Charles University as GAUK 196213.

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Authors and Affiliations

  1. Computer Science Institute, Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Pavel Klavík

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Jan Kratochvíl

  3. School of Information Science, Japan Advanced Institute of Science and Technology, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan

    Yota Otachi

  4. Graduate School of Engineering, Kobe University, Rokkodai 1-1, Nada, Kobe, 657-8501, Japan

    Toshiki Saitoh

  5. Department of Computer Science, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Tomáš Vyskočil

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  1. Pavel Klavík
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  2. Jan Kratochvíl
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Correspondence to Pavel Klavík.

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The conference version of this paper appeared in TAMC 2011 [22].

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Klavík, P., Kratochvíl, J., Otachi, Y. et al. Extending Partial Representations of Interval Graphs. Algorithmica 78, 945–967 (2017). https://doi.org/10.1007/s00453-016-0186-z

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