Faster Parameterized Algorithms for Deletion to Split Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We study the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split.We initiate a systematic study and give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely,

  1. 1

    for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an \({\cal O}^*(2^k)\) algorithm improving on the previous best bound of \({\cal O}^*({2.32^k})\). We also give an \({\cal O}(k^3)\)-sized kernel for the problem.

  2. 2

    For Split Edge Deletion, the problem of determining whether there are k edges whose deletion results in a split graph, we give an \({\cal O}^*( 2^{ O(\sqrt{k}\log k) } )\) algorithm. We also prove the existence of an \({\cal O}(k^2)\) kernel.


In addition, we note that our algorithm for Split Edge Deletion  adds to the small number of subexponential parameterized algorithms not obtained through bidimensionality, and on general graphs.


Vertex Cover Reduction Rule Graph Class Split Graph Colored Instance 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abu-Khzam, F.: A kernelization algorithm for d-hitting set. Journal of Computer and System Sciences 76(7), 524–531 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58(4), 171–176 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67, 789–807 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  7. 7.
    Foldes, S., Hammer, P.: Split graphs. Congressus Numerantium 19, 311–315 (1977)MathSciNetGoogle Scholar
  8. 8.
    Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. In: SODA, pp. 1737–1746 (2012)Google Scholar
  9. 9.
    Fujito, T.: A unified approximation algorithm for node-deletion problems. Discrete Appl. Math. 86, 213–231 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  11. 11.
    Guo, J.: Problem Kernels for NP-Complete Edge Deletion Problems: Split and Related Graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 915–926. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Heggernes, P., Mancini, F.: Minimal split completions. Discrete Applied Mathematics 157(12), 2659–2669 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Heggernes, P., van’t Hof, P., Jansen, B.M.P., Kratsch, S., Villanger, Y.: Parameterized Complexity of Vertex Deletion into Perfect Graph Classes. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 240–251. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. CoRR, abs/1203.0833 (2012)Google Scholar
  16. 16.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41, 960–981 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62(3-4), 807–822 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Lp can be a cure for parameterized problems. In: STACS, pp. 338–349 (2012)Google Scholar
  20. 20.
    Tyshkevich, R.I., Chernyak, A.A.: Yet another method of enumerating unmarked combinatorial objects. Mathematical Notes 48, 1239–1245 (1990)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Indian Institute of TechnologyMadrasIndia
  3. 3.Chennai Mathematical InstituteIndia

Personalised recommendations