Polynomial Kernelization for Removing Induced Claws and Diamonds

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Erik Jan van Leeuwen
  • Marcin Wrochna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

A graph is called {claw, diamond}-free if it contains neither a claw (a \(K_{1,3}\)) nor a diamond (a \(K_4\) with an edge removed) as an induced subgraph, or, equivalently, it is a line graph of a triangle-free graph. We consider the parameterized complexity of the {claw, diamond}-free Edge Deletion problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a {claw, diamond}-free graph. Our main result is that this problem admits a polynomial kernel. We also show that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time \(2^{o(k)}\cdot |V(G)|^{\mathcal {O}(1)}\), assuming the Exponential Time Hypothesis.

References

  1. 1.
    Beineke, L.W.: Characterizations of derived graphs. J. Comb. Theor. 9(2), 129–135 (1970)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bliznets, I., Fomin, F.V., Pilipczuk, M., Pilipczuk, M.: A subexponential parameterized algorithm for Interval Completion (2014). CoRR, abs/1402.3473Google Scholar
  3. 3.
    Bliznets, I., Fomin, F.V., Pilipczuk, M., Pilipczuk, M.: A subexponential parameterized algorithm for proper interval completion. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 173–184. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cai, L., Cai, Y.: Incompressibility of H-free edge modification. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 84–96. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Cai, Y.: Polynomial kernelisation of \(H\)-free edge modification problems. Master’s thesis. The Chinese University of Hong Kong, Hong Kong (2012)Google Scholar
  7. 7.
    Chudnovsky, M., Seymour, P.D.: Claw-free graphs. IV. Decomposition theorem. J. Comb. Theor. Ser. B 98(5), 839–938 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chudnovsky, M., Seymour, P.D.: Claw-free graphs. V. Global structure. J. Comb. Theor. Ser. B 98(6), 1373–1410 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cygan, M., Kowalik, L., Pilipczuk, M.: Open problems from workshop on kernels (2013). http://worker2013.mimuw.edu.pl/slides/worker-opl.pdf
  10. 10.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., van Leeuwen, E.J., Wrochna, M.: Polynomial kernelization for removing induced claws and diamonds (2015). CoRR, abs/1503.00704Google Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Drange, P.G., Fomin, F.V., Pilipczuk, M., Villanger, Y.: Exploring subexponential parameterized complexity of completion problems. In: STACS 2014, LIPIcs, vol. 25, pp. 288–299. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2014)Google Scholar
  13. 13.
    Drange, P.G., Pilipczuk, M.: A polynomial kernel for Trivially Perfect Editing. CoRR, abs/1412.7558 (2014)Google Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006)MATHGoogle Scholar
  15. 15.
    Fomin, F.V., Saurabh, S., Villanger, Y.: A polynomial kernel for proper interval vertex deletion. SIAM J. Discrete Math. 27(4), 1964–1976 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput. 42(6), 2197–2216 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ghosh, E., Kolay, S., Kumar, M., Misra, P., Panolan, F., Rai, A., Ramanujan, M.S.: Faster parameterized algorithms for deletion to split graphs. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 107–118. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Guillemot, S., Havet, F., Paul, C., Perez, A.: On the (non-)existence of polynomial kernels for \(P_l\)-free edge modification problems. Algorithmica 65(4), 900–926 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hermelin, D., Mnich, M., van Leeuwen, E.J.: Parameterized complexity of induced graph matching on claw-free graphs. Algorithmica 70(3), 513–560 (2014)MathSciNetMATHGoogle Scholar
  20. 20.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kloks, T., Kratsch, D., Müller, H.: Dominoes. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds.) WG 1994. LNCS, vol. 903, pp. 106–120. Springer, Heidelberg (1994)Google Scholar
  22. 22.
    Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Appl. Math. 160(15), 2259–2270 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 264–275. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. 24.
    Metelsky, Y., Tyshkevich, R.: Line graphs of Helly hypergraphs. SIAM J. Discrete Math. 16(3), 438–448 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 2
  • Michał Pilipczuk
    • 1
  • Erik Jan van Leeuwen
    • 3
  • Marcin Wrochna
    • 1
  1. 1.Institute of InformaticsUniversity of WarsawWarsawPoland
  2. 2.Department of Computer ScienceUniversity of WarwickWarwickUK
  3. 3.Max-Planck Institut Für InformatikSaarbrückenGermany

Personalised recommendations