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 k2 + 2 k vertices in linear time. We first describe a very simple linear-time kernelization whose output has at most 4 k2 + 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 \(\max\{\frac{1}{2}k^2+\frac{7}{2}k,6 k\}\).


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  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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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