Skip to main content
SpringerLink
Log in

Cluster Deletion on Interval Graphs and Split Related Graphs

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (\(P_5\)-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of a polynomial-time algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of \(P_5\)-free chordal graphs. Although the later result arises from the already-known reduction for \(P_5\)-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split 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
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Note that the class of split graphs is closed under the addition of true twins in the clique.

  2. The term laminar comes from the notion of laminar family of sets: a family of sets is called laminar if any two of its sets are either disjoint or one includes the other.

References

  1. Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56, 89–113 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bonomo, F., Durán, G., Napoli, A., Valencia-Pabon, M.: A one-to-one correspondence between potential solutions of the cluster deletion problem and the minimum sum coloring problem, and its application to \(P_4\)-sparse graphs. Inf. Proc. Lett. 115, 600–603 (2015)

    Article  Google Scholar 

  3. Bonomo, F., Durán, G., Valencia-Pabon, M.: Complexity of the cluster deletion problem on subclasses of chordal graphs. Theor. Comput. Science 600, 59–69 (2015)

    Article  MathSciNet  Google Scholar 

  4. Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  5. Bui-Xuan, B., Telle, J.A., Vatshelle, M.: Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theor. Comput. Sci. 511, 66–76 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cerioli, M.R., Szwarcfiter, J.L.: Characterizing intersection graphs of substars of a star. ARS Comb. 79, 21–31 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. Proc. FOCS 2003, 524–533 (2003)

    MATH  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  MathSciNet  Google Scholar 

  9. Dessmark, A., Jansson, J., Lingas, A., Lundell, E., Persson, M.: On the approximability of maximum and minimum edge clique partition problems. Int. J. Found. Comput. Sci. 18, 217–226 (2007)

    Article  MathSciNet  Google Scholar 

  10. Földes, S., Hammer, P.L.: Split graphs. Congr. Numer. 19, 311–315 (1977)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Gao, Y., Hare, D.R., Nastos, J.: The cluster deletion problem for cographs. Discrete Math. 313, 2763–2771 (2013)

    Article  MathSciNet  Google Scholar 

  13. Gerber, M.U., Kobler, D.: Algorithms for vertex-partitioning problems on graphs with fixed clique-width. Theor. Comput. Sci. 299, 719–734 (2003)

    Article  MathSciNet  Google Scholar 

  14. Golovach, P.A., Heggernes, P., Konstantinidis, A.L., Lima, P.T., Papadopoulos, C.: Parameterized aspects of strong subgraph closure. Algorithmica 82(7), 2006–2038 (2020)

    Article  MathSciNet  Google Scholar 

  15. Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11, 423–443 (2000)

    Article  MathSciNet  Google Scholar 

  16. Grüttemeier, N., Komusiewicz, C.: On the relation of strong triadic closure and cluster deletion. Proc. WG 2018, 239–251 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1, 275–284 (1981)

    Article  MathSciNet  Google Scholar 

  18. Hansen, P., Jaumard, B.: Cluster analysis and mathematical programming. Math. Program. 79, 191–215 (1997)

    MathSciNet  MATH  Google Scholar 

  19. Hartigan, J.: Clustering Algorithms. Wiley, New York (1975)

    MATH  Google Scholar 

  20. Heggernes, P., Lokshtanov, D., Nederlof, J., Paul, C., Telle, J.A.: Generalized graph clustering: recognizing \((p, q)\)-cluster graphs. Proc. WG 2010, 171–183 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Kanj, I.A., Komusiewicz, C., Sorge, M., van Leeuwen, E.J.: Solving partition problems almost always requires pushing many vertices around. In: Proceedings of ESA 2018, pp. 51:1–51:14 (2018)

  22. Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Appl. Math. 160, 2259–2270 (2012)

    Article  MathSciNet  Google Scholar 

  23. Konstantinidis, A.L., Nikolopoulos, S.D., Papadopoulos, C.: Strong triadic closure in cographs and graphs of low maximum degree. Theor. Comput. Sci. 740, 76–84 (2018)

    Article  MathSciNet  Google Scholar 

  24. Konstantinidis, A.L., Papadopoulos, C.: Cluster deletion on interval graphs and split related graphs. In: Proceedings of MFCS 2019, pp. 12:1–12:14 (2019)

  25. Konstantinidis, A.L., Papadopoulos, C.: Maximizing the strong triadic closure in split graphs and proper interval graphs. Discrete Appl. Math. 285, 79–95 (2020)

    Article  MathSciNet  Google Scholar 

  26. Schaeffer, S.E.: Graph clustering. Comput. Sci. Rev. 1(1), 27–64 (2007)

    Article  Google Scholar 

  27. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144, 173–182 (2004)

    Article  MathSciNet  Google Scholar 

  28. Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math. 10, 529–550 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank Paloma T. Lima for bringing to our attention the class of starlike graphs introduced in [6]. A preliminary version of this paper appeared as an extended abstract in the proceedings of MFCS 2019 [24]. This long version contains full proofs of all statements, a new characterization of stable-like graphs, and an extended polynomial case of starlike graphs (laminar-like graphs) not appeared in [24]. The research work done by A. L. Konstantinidis is co-financed by Greece and the European Union (European Social Fund—ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (IKY). The research work done by C. Papadopoulos was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant”, Project FANTA (eFficient Algorithms for NeTwork Analysis), number HFRI-FM17-431.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Athanasios L. Konstantinidis & Charis Papadopoulos

Authors
  1. Athanasios L. Konstantinidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charis Papadopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charis Papadopoulos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konstantinidis, A.L., Papadopoulos, C. Cluster Deletion on Interval Graphs and Split Related Graphs. Algorithmica 83, 2018–2046 (2021). https://doi.org/10.1007/s00453-021-00817-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-021-00817-8

Keywords

Search

Navigation