Skip to main content
Springer Nature Link
Log in

Strong Triadic Closure in Cographs and Graphs of Low Maximum Degree

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10392))

Included in the following conference series:

  • 1553 Accesses

Abstract

The MaxSTC problem is an assignment of the edges with strong or weak labels having the maximum number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent. The Cluster Deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques. Both problems are known to be NP-hard and an optimal solution for the Cluster Deletion problem provides a solution for the MaxSTC problem, however not necessarily an optimal one. In this work we give the first systematic study that reveals graph families for which the optimal solutions for MaxSTC and Cluster Deletion coincide. We first show that MaxSTC coincides with Cluster Deletion on cographs and, thus, MaxSTC is solvable in quadratic time on cographs. As a side result, we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs. Furthermore we study how low degree bounds influence the complexity of the MaxSTC problem. We show that this problem is polynomial-time solvable on graphs of maximum degree three, whereas MaxSTC becomes NP-complete on graphs of maximum degree four. The latter implies that there is no subexponential-time algorithm for MaxSTC unless the Exponential-Time Hypothesis fails.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The \(O^*\) notation suppresses polynomial factors of n.

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. Cochefert, M., Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: Parameterized algorithms for finding square roots. Algorithmica 74, 602–629 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Corneil, D.G., Lerchs, H., Stewart, L.K.: Complement reducible graphs. Discret. Appl. Math. 3, 163–174 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  8. Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  9. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Granovetter, M.: The strength of weak ties. Am. J. Sociol. 78, 1360–1380 (1973)

    Article  Google Scholar 

  12. Hon, W.-K., Kloks, T., Liu, H.-H., Poon, S.-H., Wang, Y.-L.: On independence domination. In: Gąsieniec, L., Wolter, F. (eds.) FCT 2013. LNCS, vol. 8070, pp. 183–194. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40164-0_19

    Chapter  Google Scholar 

  13. Imrich, W., Klavzar, S., Rall, D.F.: Topics in Graph Theory: Graphs and Their Cartesian Product. AK Peters Ltd., Wellesley (2008)

    MATH  Google Scholar 

  14. Jha, P.K., Slutzki, G.: Independence numbers of product graphs. Appl. Math. Lett. 7, 91–94 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Konstantinids, A., Papadopoulos, C.: Maximizing the strong triadic closure in split graphs and proper interval graphs. CoRR, abs/1609.09433 (2016)

    Google Scholar 

  17. Le, V.B.: Gallai graphs and anti-gallai graphs. SIAM J. Discret. Math. 159, 179–189 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shamir, R., Sharan, R.: A fully dynamic algorithm for modular decomposition and recognition of cographs. Discret. Appl. Math. 136, 329–340 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Sintos, S., Tsaparas, P.: Using strong triadic closure to characterize ties in social networks. In: Proceedings of KDD 2014, pp. 1466–1475 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Athanasios L. Konstantinidis & Charis Papadopoulos

  2. Department of Computer Science and Engineering, University of Ioannina, Ioannina, Greece

    Stavros D. Nikolopoulos

Authors
  1. Athanasios L. Konstantinidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stavros D. Nikolopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Charis Papadopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charis Papadopoulos .

Editor information

Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Konstantinidis, A.L., Nikolopoulos, S.D., Papadopoulos, C. (2017). Strong Triadic Closure in Cographs and Graphs of Low Maximum Degree. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62389-4_29

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62388-7

  • Online ISBN: 978-3-319-62389-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics