Skip to main content
SpringerLink
Log in

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The Contractibility problem takes as input two graphs G and H, and the task is to decide whether H can be obtained from G by a sequence of edge contractions. The Induced Minor and Induced Topological Minor problems are similar, but the first allows both edge contractions and vertex deletions, whereas the latter allows only vertex deletions and vertex dissolutions. All three problems are NP-complete, even for certain fixed graphs H. We show that these problems can be solved in polynomial time for every fixed H when the input graph G is chordal. Our results can be considered tight, since these problems are known to be W[1]-hard on chordal graphs when parameterized by the size of H. To solve Contractibility and Induced Minor, we define and use a generalization of the classic Disjoint Paths problem, where we require the vertices of each of the k paths to be chosen from a specified set. We prove that this variant is NP-complete even when k=2, but that it is polynomial-time solvable on chordal graphs for every fixed k. Our algorithm for Induced Topological Minor is based on another generalization of Disjoint Paths called Induced Disjoint Paths, where the vertices from different paths may no longer be adjacent. We show that this problem, which is known to be NP-complete when k=2, can be solved in polynomial time on chordal graphs even when k is part of the input. Our results fit into the general framework of graph containment problems, where the aim is to decide whether a graph can be modified into another graph by a sequence of specified graph operations. Allowing combinations of the four well-known operations edge deletion, edge contraction, vertex deletion, and vertex dissolution results in the following ten containment relations: (induced) minor, (induced) topological minor, (induced) subgraph, (induced) spanning subgraph, dissolution, and contraction. Our results, combined with existing results, settle the complexity of each of the ten corresponding containment problems on chordal graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Belmonte, R., Golovach, P.A., Heggernes, P., van ’t Hof, P., Kamiński, M., Paulusma, D.: Finding contractions and induced minors in chordal graphs via disjoint paths. In: Proceedings of ISAAC 2011. LNCS, vol. 7074, pp. 110–119. Springer, Berlin (2011)

    Google Scholar 

  2. Belmonte, R., Heggernes, P., van ’t Hof, P.: Edge contractions in subclasses of chordal graphs. In: Proceedings of TAMC 2011. LNCS, vol. 6648, pp. 528–539. Springer, Berlin (2011)

    Google Scholar 

  3. Bienstock, D.: On the complexity of testing for odd holes and induced odd paths. Discrete Math. 90, 85–92 (1991). See also Corrigendum. Discrete Math. 102, 109 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blair, J.R.S., Peyton, B.W.: An introduction to chordal graphs and clique trees. In: Graph Theory and Sparse Matrix Computations. IMA Volumes in Mathematics and Its Applications, vol. 56, pp. 1–29. Springer, Berlin (1993)

    Chapter  Google Scholar 

  5. 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, 335–379 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  7. Brouwer, A.E., Veldman, H.J.: Contractibility and NP-completeness. J. Graph Theory 11, 71–79 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dirac, G.A.: On rigid circuit graphs. Abh. Math. Semin. Univ. Hamb. 25, 71–76 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  10. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theor. Comput. Sci. 141, 109–131 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fiala, J., Kamiński, M., Lidický, B., Paulusma, D.: The k-in-a-path problem for claw-free graphs. Algorithmica 62, 499–519 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fiala, J., Kamiński, M., Paulusma, D.: A note on contracting claw-free graphs. Manuscript (2011)

  13. Fellows, M.R., Kratochvíl, J., Middendorf, M., Pfeiffer, F.: The complexity of induced minors and related problems. Algorithmica 13, 266–282 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  16. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory, Ser. B 16, 47–56 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. George, A., Liu, J.W.: Computer Solutions of Large Sparse Positive Definite Systems. Prentice Hall, New York (1981)

    Google Scholar 

  18. Golovach, P.A., Kamiński, M., Paulusma, D.: Contracting a chordal graph to a split graph or a tree. In: Proceedings of MFCS 2011. LNCS, vol. 6907, pp. 339–350. Springer, Berlin (2011)

    Google Scholar 

  19. Golovach, P.A., Kamiński, M., Paulusma, D., Thilikos, D.M.: Containment relations in split graphs. Discrete Appl. Math. 160, 155–163 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  21. Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proceedings of STOC 2011, pp. 479–488 (2011)

    Google Scholar 

  22. van ’t Hof, P., Kamiński, M., Paulusma, D., Szeider, S., Thilikos, D.M.: On graph contractions and induced minors. Discrete Appl. Math. 160, 799–809 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. van ’t Hof, P., Paulusma, D., Woeginger, G.J.: Partitioning graphs in connected parts. Theor. Comput. Sci. 410, 4834–4843 (2009)

    Article  MATH  Google Scholar 

  24. Kamiński, M., Thilikos, D.M.: Contraction checking in graphs on surfaces. In: Proceedings of STACS, 2012, to appear

    Google Scholar 

  25. Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Berlin (1994)

    MATH  Google Scholar 

  26. Lévêque, B., Lin, D.Y., Maffray, F., Trotignon, N.: Detecting induced subgraphs. Discrete Appl. Math. 157, 3540–3551 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Levin, A., Paulusma, D., Woeginger, G.J.: The computational complexity of graph contractions I: polynomially solvable and NP-complete cases. Networks 51, 178–189 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Levin, A., Paulusma, D., Woeginger, G.J.: The computational complexity of graph contractions II: two tough polynomially solvable cases. Networks 52, 32–56 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Matoušek, J., Thomas, R.: On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Math. 108, 343–364 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. Natarajan, S., Sprague, A.P.: Disjoint paths in circular arc graphs. Nord. J. Comput. 3, 256–270 (1996)

    MathSciNet  Google Scholar 

  31. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  32. de Ridder, H.N., et al.: Information System on Graph Classes and their Inclusions (ISGCI). http://www.graphclasses.org, 2001–2012

  33. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  35. Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 66–579 (1984)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Rémy Belmonte, Pinar Heggernes & Pim van ’t Hof

  2. School of Engineering and Computing Sciences, Durham University, Durham, UK

    Petr A. Golovach & Daniël Paulusma

  3. Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Marcin Kamiński

Authors
  1. Rémy Belmonte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr A. Golovach
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pinar Heggernes
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pim van ’t Hof
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marcin Kamiński
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Daniël Paulusma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniël Paulusma.

Additional information

This work is supported by EPSRC (EP/G043434/1) and Royal Society (JP100692), and by the Research Council of Norway (197548/F20). Preliminary versions of different parts of the paper were presented at MFCS 2011 [18] and ISAAC 2011 [1].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belmonte, R., Golovach, P.A., Heggernes, P. et al. Detecting Fixed Patterns in Chordal Graphs in Polynomial Time. Algorithmica 69, 501–521 (2014). https://doi.org/10.1007/s00453-013-9748-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-013-9748-5

Keywords

Search

Navigation