Kernels for Edge Dominating Set: Simpler or Smaller

  • Torben Hagerup
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


A kernelization for a parameterized computational problem is a polynomial-time procedure that transforms every instance of the problem into an equivalent instance (the so-called kernel) whose size is bounded by a function of the value of the chosen parameter. We present new kernelizations for the NP-complete Edge Dominating Set problem which asks, given an undirected graph G = (V,E) and an integer k, whether there exists a subset D ⊆ E with |D| ≤ k such that every edge in E shares at least one endpoint with some edge in D. The best previous kernelization for Edge Dominating Set, due to Xiao, Kloks and Poon, yields a kernel with at most 2 k 2 + 2 k vertices in linear time. We first describe a very simple linear-time kernelization whose output has at most 4 k 2 + 4 k vertices and is either a trivial “no” instance or a vertex-induced subgraph of the input graph in which every edge dominating set of size ≤ k is also an edge dominating set of the input graph. We then show that a refinement of the algorithm of Xiao, Kloks and Poon and a different analysis can lower the bound on the number of vertices in the kernel by a factor of about 4, namely to \(\max\{\frac{1}{2}k^2+\frac{7}{2}k,6 k\}\).


Undirected Graph Incident Edge Input Graph Kernel Size Reduction Rule 
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.
    Fernau, H.: edge dominating set: Efficient Enumeration-Based Exact Algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 142–153. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  3. 3.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)Google Scholar
  4. 4.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  5. 5.
    Prieto, E.: Systematic Kernelization in FPT Algorithm Design. Ph.D. thesis, The University of Newcastle, Callaghan, N.S.W., Australia (2005)Google Scholar
  6. 6.
    Weston, M.: A fixed-parameter tractable algorithm for matrix domination. Inform. Process. Lett. 90, 267–272 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Xiao, M., Kloks, T., Poon, S.H.: New Parameterized Algorithms for the Edge Dominating Set Problem. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 604–615. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Torben Hagerup
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

Personalised recommendations