Skip to main content
SpringerLink
Log in

The Sandwich Problem for Decompositions and Almost Monotone Properties

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in \({\mathcal {C}}\) as an induced subgraph is almost monotone if \({\mathcal {C}}\) is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length \(\equiv j \mod n\) is almost monotone if and only if \(j \equiv 2 \mod n\) or \(n \le 2\). Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.

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
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Berge, C.: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10(114), 114 (1961)

    Google Scholar 

  2. Berry, A., Golumbic, M.C., Lipshteyn, M.: Recognizing chordal probe graphs and cycle-bicolorable graphs. SIAM J. Discrete Math. 21(3), 573–591 (2007)

    Article  MathSciNet  Google Scholar 

  3. Brandstädt, A., Dragan, F.F., Szymczak, T.: On stable cutsets in graphs. Discrete Appl. Math. 105(1), 39–50 (2000)

    Article  MathSciNet  Google Scholar 

  4. Cerioli, M.R., Everett, H., de Figueiredo, C.M.H., Klein, S.: The homogeneous set sandwich problem. Inf. Process. Lett. 67(1), 31–35 (1998)

    Article  MathSciNet  Google Scholar 

  5. Chudnovsky, M.: Berge trigraphs and their applications. PhD thesis, Princeton University (2003)

  6. Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P., Vušković, K.: Recognizing Berge graphs. Combinatorica 25(2), 143–186 (2005)

    Article  MathSciNet  Google Scholar 

  7. Chudnovsky, M.: Berge trigraphs. J. Graph Theory 53(1), 1–55 (2006)

    Article  MathSciNet  Google Scholar 

  8. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164(1), 51–229 (2006)

    Article  MathSciNet  Google Scholar 

  9. Chudnovsky, M., Kapadia, R.: Detecting a theta or a prism. SIAM J. Discrete Math. 22(3), 1164–1186 (2008)

    Article  MathSciNet  Google Scholar 

  10. Chudnovsky, M., Seymour, P.: The three-in-a-tree problem. Combinatorica 30(4), 387–417 (2010)

    Article  MathSciNet  Google Scholar 

  11. Chvátal, V.: Recognizing decomposable graphs. J. Graph Theory 8(1), 51–53 (1984)

    Article  MathSciNet  Google Scholar 

  12. Chvátal, V.: Star-cutsets and perfect graphs. J. Comb. Theory Ser. B 39(3), 189–199 (1985)

    Article  MathSciNet  Google Scholar 

  13. Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: Even-hole-free graphs part II: recognition algorithm. J. Graph Theory 40(4), 238–266 (2002)

    Article  MathSciNet  Google Scholar 

  14. Couto. F, Faria, L., Gravier, S., Klein, S.: On the forbidden induced subgraph probe and sandwich problems. Discrete Appl. Math. 234, 56–66 (2016)

  15. Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebr. Discrete Methods 3(2), 214–228 (1982)

    Article  MathSciNet  Google Scholar 

  16. Dantas, S., de Figueiredo, C.M.H., da Silva, M.V.G., Teixeira, R.B.: On the forbidden induced subgraph sandwich problem. Discrete Appl. Math. 159(16), 1717–1725 (2011)

    Article  MathSciNet  Google Scholar 

  17. de Figueiredo, C.M.H., Klein, S., Kohayakawa, Y., Reed, B.A.: Finding skew partitions efficiently. J. Algorithms 2(37), 505–521 (2000)

    Article  MathSciNet  Google Scholar 

  18. de Figueiredo, C.M.H., Klein, S., Vušković, K.: The graph sandwich problem for 1-join composition is NP-complete. Discrete Appl. Math. 121(1), 73–82 (2002)

    Article  MathSciNet  Google Scholar 

  19. Dirac, G.A.: On rigid circuit graphs. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 25, no. 1, pp. 71–76. Springer, Berlin (1961)

    Article  MathSciNet  Google Scholar 

  20. Edwards, K.: The complexity of colouring problems on dense graphs. Theor. Comput. Sci. 43, 337–343 (1986)

    Article  MathSciNet  Google Scholar 

  21. Everett, H., Klein, S., Reed, B.: An algorithm for finding homogeneous pairs. Discrete Appl. Math. 72(3), 209–218 (1997)

    Article  MathSciNet  Google Scholar 

  22. Feder, T., Hell, P., Klein, S., Motwani, R.: List partitions. SIAM J. Discrete Math. 16(3), 449–478 (2003)

    Article  MathSciNet  Google Scholar 

  23. Golumbic, M.C., Shamir, R.: Complexity and algorithms for reasoning about time: a graph-theoretic approach. J. ACM: JACM 40(5), 1108–1133 (1993)

    Article  MathSciNet  Google Scholar 

  24. Golumbic, M.C., Kaplan, H.: Graph sandwich problems. J. Algorithms 19(3), 449–473 (1995)

    Article  MathSciNet  Google Scholar 

  25. Golumbic, M.C., Lipshteyn, M.: Chordal probe graphs. Discrete Appl. Math. 143(1), 221–237 (2004)

    Article  MathSciNet  Google Scholar 

  26. Kennedy, W.S., Reed, B.: Fast skew partition recognition. In: Ito, H., Kano, M., Katoh, N., Uno, Y. (eds.) Computational Geometry and Graph Theory, pp. 101–107. Springer, Berlin (2008)

    Chapter  Google Scholar 

  27. Maffray, F., Trotignon, N.: Algorithms for perfectly contractile graphs. SIAM J. Discrete Math. 19(3), 553–574 (2005)

    Article  MathSciNet  Google Scholar 

  28. McMorris, F.R., Wang, C., Zhang, P.: On probe interval graphs. Discrete Appl. Math. 88(1), 315–324 (1998)

    Article  MathSciNet  Google Scholar 

  29. Moshi, A.M.: Matching cutsets in graphs. J. Graph Theory 13(5), 527–536 (1989)

    Article  MathSciNet  Google Scholar 

  30. Ramsey, F. P.: On a problem of formal logic. In: Proceedings of the London Mathematical Society, 2nd Series, vol. 30, pp. 264–344 (1930)

    Article  MathSciNet  Google Scholar 

  31. Spinrad, J.: \(P_4\)-trees and substitution decomposition. Discrete Appl. Math. 39(3), 263–291 (1992)

    Article  MathSciNet  Google Scholar 

  32. Teixeira, R.B., de Figueiredo, C.M.H.: The sandwich problem for cutsets: clique cutset, \(k\)-star cutset. Discrete Appl. Math. 154(13), 1791–1798 (2006)

    Article  MathSciNet  Google Scholar 

  33. Teixeira, R.B., Dantas, S.: Skew partition sandwich problem is NP-complete. Electron. Notes Discrete Math. 35, 9–14 (2009)

    Article  MathSciNet  Google Scholar 

  34. Truemper, K.: Alpha-balanced graphs and matrices and \(GF(3)\)-representability of matroids. J. Comb. Theory Ser. B 32(2), 112–139 (1982)

    Article  MathSciNet  Google Scholar 

  35. Whitesides, S.H.: An algorithm for finding clique cut-sets. Inf. Process. Lett. 12(1), 31–32 (1981)

    Article  MathSciNet  Google Scholar 

  36. Zhang, P., Schon, E.A., Fischer, S.G., Cayanis, E., Weiss, J., Kistler, S., Bourne, P.E.: An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. Comput. Appl. Biosci.: CABIOS 10(3), 309–317 (1994)

    Google Scholar 

Download references

Acknowledgements

We are thankful to Paul Seymour for many helpful discussions. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant No. W911NF-16-1-0404.

Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Maria Chudnovsky & Sophie Spirkl

  2. COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Celina M. H. de Figueiredo

Authors
  1. Maria Chudnovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Celina M. H. de Figueiredo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sophie Spirkl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sophie Spirkl.

Additional information

Maria Chudnovsky was supported by National Science Foundation Grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Celina M. H. de Figueiredo was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chudnovsky, M., de Figueiredo, C.M.H. & Spirkl, S. The Sandwich Problem for Decompositions and Almost Monotone Properties. Algorithmica 80, 3618–3645 (2018). https://doi.org/10.1007/s00453-018-0409-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-018-0409-6

Keywords

Mathematics Subject Classification

Search

Navigation