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Extending Partial Representations of Trapezoid Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10520))

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Abstract

A trapezoid graph is an intersection graph of trapezoids spanned between two horizontal lines. The partial representation extension problem for trapezoid graphs is a generalization of the recognition problem: given a graph G and an assignment \(\xi \) of trapezoids to some vertices of G, can \(\xi \) be extended to a trapezoid intersection model of the entire graph G? We show that this can be decided in polynomial time. Thus, we determine the complexity of partial representation extension for one of the two major remaining classes of geometric intersection graphs for which it has been unknown (circular-arc graphs being the other).

The authors were partially supported by National Science Center of Poland grant 2015/17/B/ST6/01873.

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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), Article 32 (2015)

    Google Scholar 

  2. Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Trans. Algorithms 12(2), Article 16 (2016)

    Google Scholar 

  3. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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, Cham (2013). doi:10.1007/978-3-319-03841-4_12

    Chapter  Google Scholar 

  5. Cogis, O.: On the Ferrers dimension of a digraph. Discrete Math. 38(1), 47–52 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in \(O(E\log D)\) time. Combinatorica 21(1), 5–12 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cournier, A., Habib, M.: A new linear algorithm for modular decomposition. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 68–84. Springer, Heidelberg (1994). doi:10.1007/BFb0017474

    Chapter  Google Scholar 

  8. Dagan, I., Golumbic, M.C., Pinter, R.Y.: Trapezoid graphs and their coloring. Discrete Appl. Math. 21(1), 35–46 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63(3), 600–610 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  11. Even, S., Itai, A., Shamir, A.: On the complexity of time table and multi-commodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Felsner, S., Habib, M., Möhring, R.H.: On the interplay between interval dimension and dimension. SIAM J. Discrete Math. 7(1), 22–40 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fiala, J.: NP-completeness of the edge precoloring extension problem on bipartite graphs. J. Graph Theory 43(2), 156–160 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hung. 18(1–2), 25–66 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghouila-Houri, A.: Caractérisation des graphes non orientés dont on peut orienter les arrêtes de manière à obtenir le graphe d’une relation d’ordre. C. R. Acad. Sci. 254, 1370–1371 (1962)

    MathSciNet  MATH  Google Scholar 

  16. Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Canad. J. Math. 16, 539–548 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  17. Golumbic, M.C.: The complexity of comparability graph recognition and coloring. Computing 18(3), 199–208 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  19. Habib, M., Kelly, D., Möhring, R.H.: Interval dimension is a comparability invariant. Discrete Math. 88(2–3), 211–229 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. James, L.O., Stanton, R.G., Cowan, D.D.: Graph decomposition for undirected graphs. In: Hoffman, F., Levow, R.B., Thomas, R.S.D. (eds.) 3rd Southeastern Conference on Combinatorics, Graph Theory, and Computing (CGTC 1972). Congressus Numerantium, vol. 6, pp. 281–290. Utilitas Mathematica, Winnipeg (1972)

    Google Scholar 

  21. Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discrete Comput. Geom. 47(3), 548–568 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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). doi:10.1007/978-3-642-33090-2_58

    Chapter  Google Scholar 

  23. Klavík, P., Kratochvíl, J., Otachi, Y., Rutter, I., Saitoh, T., Saumell, M., Vyskočil, T.: Extending partial representations of proper and unit interval graphs. Algorithmica 77(4), 1071–1104 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskočil, T.: Extending partial representations of interval graphs. Algorithmica 78(3), 945–967 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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). doi:10.1007/978-3-642-20877-5_28

    Chapter  Google Scholar 

  27. Kratochvíl, J.: String graphs. II. Recognizing string graphs is NP-hard. J. Combin. Theory Ser. B 52(1), 67–78 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52(3), 233–252 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory Ser. B 62(2), 289–315 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ma, T.H., Spinrad, J.P.: On the 2-chain subgraph cover and related problems. J. Algorithms 17(2), 251–268 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Marx, D.: NP-completeness of list coloring and precoloring extension on the edges of planar graphs. J. Graph Theory 49(4), 313–324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Math. 201(1–3), 189–241 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mertzios, G.B., Corneil, D.G.: Vertex splitting and the recognition of trapezoid graphs. Discrete Appl. Math. 159(11), 1131–1147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006)

    Article  MathSciNet  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 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67(2), 365–380 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Schnyder, W.: Embedding planar graphs on the grid. In: Johnson, D. (ed.) 1st Annual ACM-SIAM Symposium Discrete Algorithms (SODA 1990), pp. 138–148. SIAM, Philadelphia (1990)

    Google Scholar 

  39. Spinrad, J.P.: Recognition of circle graphs. J. Algorithms 16(2), 264–282 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Spinrad, J.P.: Efficient Graph Representations, Field Institute Monographs, vol. 19. AMS, Providence (2003)

    Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

    Tomasz Krawczyk & Bartosz Walczak

Authors
  1. Tomasz Krawczyk
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  2. Bartosz Walczak
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Correspondence to Tomasz Krawczyk .

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Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany

    Gerhard J. Woeginger

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Krawczyk, T., Walczak, B. (2017). Extending Partial Representations of Trapezoid Graphs. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_27

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  • DOI: https://doi.org/10.1007/978-3-319-68705-6_27

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