Skip to main content
SpringerLink
Log in

On some combinatorial problems in cographs

  • Published:
International Journal of Advances in Engineering Sciences and Applied Mathematics Aims and scope Submit manuscript

Abstract

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and finding some constrained vertex separators are linear-time solvable in cographs. We propose polynomial-time algorithms for some connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs and our framework yields polynomial-time algorithms for the three problems.

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

References

  1. Brandstädt, A., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  2. Bretscher, A., Corneil, D., Habib, M., Paul, C.: A simple linear time LexBFS cograph recognition algorithm. In: International Workshop on Graph-Theoretic Concepts in Computer Science, pp. 119-130 (2003)

  3. Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23(1), 11–24 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Végh, L.A.: Augmenting undirected node-connectivity by one. SIAM J. Discrete Math. 25(2), 695–718 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Frank, A., Jordän, T.: Minimal edge-coverings of pairs of sets. J. Comb. Theory Ser. B 65(1), 73–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Narayanaswamy, N.S., Sadagopan, N.: A unified framework for Bi (Tri) connectivity and chordal augmentation. Int. J. Found. Comput. Sci. 21(1), 67–93 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math. 5(1), 25–53 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. J. Comput. Syst. Sci. 35(1), 96–144 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. West, D.B.: Introduction to Graph Theory, vol. 2. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  12. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  13. Kirkpatrick, D.G., Przytycka, T.: Parallel recognition of complement reducible graphs and cotree construction. Discrete Appl. Math. 29(1), 79–96 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discrete Appl. Math. 3(3), 163–174 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Narayanaswamy, N.S., Sadagopan, N.: Connected \((s, t)\)-vertex separator parameterized by chordality. J. Graph Algorithms Appl. 19, 549–565 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dhanalakshmi, S., Sadagopan, N., Manogna, V.: On \(2K_2\)-free graphs. Int. J. Pure Appl. Math. 109(7), 167–173 (2016)

    Google Scholar 

  17. Brandstadt, A., Dragan, F.F., Le, V.B., Szymczak, T.: On stable cutsets in graphs. Discrete Appl. Math. 105, 39–50 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yu, M.-S., Yang, C.-H.: An \(O(n)\) time algorithm for maximum matching on cographs. Inf. Process. Lett. 47(2), 89–93 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gabow, H.N.: Data structures for weighted matching and extensions to \(b\)-matching and \(f\)-factors. arxiv:1611.07541

  20. Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The longest path problem has a polynomial solution on interval graphs. Algorithmica 61(2), 320–341 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mertzios, G.B., Corneil, D.G.: A simple polynomial algorithm for the longest path problem on cocomparability graphs. SIAM J. Discrete Math. 26(3), 940–963 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett. 103(2), 71–77 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is partially supported by DST-ECRA project - ECR/2017/001442.

Author information

Authors and Affiliations

  1. Indian Institute of Information Technology Design and Manufacturing, Kancheepuram, Chennai, India

    Harshita Kona & N. Sadagopan

Authors
  1. Harshita Kona
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Sadagopan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harshita Kona.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kona, H., Sadagopan, N. On some combinatorial problems in cographs. Int J Adv Eng Sci Appl Math 11, 25–39 (2019). https://doi.org/10.1007/s12572-019-00244-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12572-019-00244-7

Keywords

Search

Navigation