Advertisement

Refined Search Tree Technique for Dominating Set on Planar Graphs

  • Jochen Alber
  • Hongbing Fan
  • Michael R. Fellows
  • Henning Fernau
  • Rolf Niedermeier
  • Fran Rosamond
  • Ulrike Stege
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2136)

Abstract

We establish refined search tree techniques for the parameterized DOMINATING SET problem on planar graphs. We derive a fixed parameter algorithm with running time O(8 k n), where k is the size of the dominating set and n is the number of vertices in the graph. For our search tree, we firstly provide a set of reduction rules. Secondly, we prove an intricate branching theorem based on the Euler formula. In addition, we give an example graph showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final algorithm is very easy (to implement); its analysis, however, is involved.

Keywords

dominating set planar graph fixed parameter algorithm search tree 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Alber, H. L. Bodlaender, H. Fernau, and R. Niedermeier. Fixed parameter algorithms for planar dominating set and related problems. In 7th Scandinavian Workshop on Algorithm Theory SWAT, volume 1851 of LNCS, pages 97–110, Springer-Verlag, 2000. A lng version has been accepted for publication in Algorithmica.CrossRefGoogle Scholar
  2. 2.
    J. Alber, H. Fernau, and R. Niedermeier. Graph separators: a parameterized view. Technical Report WSI-2001-8, Universität Tübingen (Fed. Rep. of Germany), Wilhelm-Schickard-Institut für Informatik, 2001. Extended abstract accepted at COCOON 2001, to appear in LNCS, Springer-Verlag, August 2001.Google Scholar
  3. 3.
    J. Alber, H. Fernau, and R. Niedermeier. Parameterized complexity: exponential speedup for planar graph problems. Technical Report TR01-023, ECCC Reports, Trier (Fed. Rep. of Germany), March 2001. Extended abstract accepted at ICALP 2001, to appear in LNCS, Springer-Verlag, July 2001.Google Scholar
  4. 4.
    L. Cai, M. Fellows, D. Juedes, and F. Rosamond. Efficient polynomial-time approximation schemes for problems on planar graph structures: upper and lower bounds. Manuscript, May 2001.Google Scholar
  5. 5.
    L. Cai and D. Juedes. Subexponential parameterized algorithms collapse the W-hierarchy. Extended abstract accepted at ICALP 2001, to appear in LNCS, Springer-Verlag, July 2001.Google Scholar
  6. 6.
    R. Diestel. Graph Theory. Springer-Verlag, 1997.Google Scholar
  7. 7.
    R. G. Downey and M. R. Fellows. Parameterized computational feasibility. In Feasible Mathematics II, pages 219–244. Birkhäuser, 1995.Google Scholar
  8. 8.
    R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.Google Scholar
  9. 9.
    R. G. Downey, M. R. Fellows, and U. Stege. Parameterized complexity: A framework for systematically confronting computational intractability. DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, 49:49–99, 1999.MathSciNetGoogle Scholar
  10. 10.
    R. Niedermeier and P. Rossmanith. A general method to speed up fixed-parameter-tractable algorithms. Information Processing Letters, 73:125–129, 2000.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jochen Alber
    • 1
  • Hongbing Fan
    • 2
  • Michael R. Fellows
    • 2
  • Henning Fernau
    • 1
  • Rolf Niedermeier
    • 1
  • Fran Rosamond
    • 2
  • Ulrike Stege
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany
  2. 2.Department of Computer ScienceUniversity of VictoriaVictoria B.C.Canada

Personalised recommendations