On Making a Distinguished Vertex Minimum Degree by Vertex Deletion

  • Nadja Betzler
  • Robert Bredereck
  • Rolf Niedermeier
  • Johannes Uhlmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6543)


For directed and undirected graphs, we study the problem to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness, including a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding nonexistence results when replacing vertex cover number by treewidth or feedback vertex set number.


Social Network Analysis Vertex Cover Polynomial Kernel Tractability Result Vertex Deletion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: The maximum k-plex problem. Oper. Res. (2010) (to appear)Google Scholar
  2. 2.
    Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 5425–5442 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H., Downey, R., Fellows, M., Hermelin, D.: On problems without polynomial kernels. J. Comput. System Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels. Technical Report UU-CS-2008-030, Department of Information and Computing Sciences, Utrecht University (2008)Google Scholar
  6. 6.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Handbook of Theoretical Computer Science. Formal Models and Sematics (B), vol. B, pp. 193–242 (1990)Google Scholar
  7. 7.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Llull and Copeland voting computationally resist bribery and constructive control. J. Artifical Intelligence Res. 35(1), 275–341 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fellows, M.R., Guo, J., Moser, H., Niedermeier, R.: A generalization of Nemhauser and Trotter’s local optimization theorem. In: Proc. 26th STACS, pp. 409–420. IBFI Dagstuhl, Germany (2009)Google Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  12. 12.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  13. 13.
    Huberman, B.A., Romero, D.M., Wu, F.: Social networks that matter: Twitter under the microscope. First Monday 14(1) (2009)Google Scholar
  14. 14.
    Moser, H., Niedermeier, R., Sorge, M.: Algorithms and experiments for clique relaxations—finding maximum s-plexes. In: Proc. 8th SEA. LNCS, vol. 5526, pp. 233–244. Springer, Heidelberg (2009)Google Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Potterat, J., Phillips-Plummer, L., Muth, S., Rothenberg, R., Woodhouse, D., Maldonado-Long, T., Zimmerman, H., Muth, J.: Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs. Sexually Transmitted Infections 78(suppl. 1), 159–163 (2002)CrossRefGoogle Scholar
  17. 17.
    Romm-Livermore, C., Setzekorn, K.: Social Networking Communities and E-Dating Services: Concepts and Implications. Information Science Reference (2008)Google Scholar
  18. 18.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Nadja Betzler
    • 1
  • Robert Bredereck
    • 2
  • Rolf Niedermeier
    • 2
  • Johannes Uhlmann
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTechnische Universität BerlinBerlinGermany

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