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Algorithmica

, Volume 33, Issue 4, pp 461–493 | Cite as

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

  •  Alber
  •  Bodlaender
  •  Fernau
  •  Kloks
  •  Niedermeier

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c 1 ^ \sqrt k n) time, where c 1 =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.

Key words. NP-complete problems, Fixed parameter tractability, Planar graphs, planar dominating set, face cover, Outerplanarity, Treewidth. 

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Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  •  Alber
    • 1
  •  Bodlaender
    • 2
  •  Fernau
    • 1
  •  Kloks
    • 3
  •  Niedermeier
    • 1
  1. 1.Universität Tübingen, Wilhelm-Schickard-Institut für Informatik, Sand 13, D-72076 Tübingen, Germany. {alber,fernau,niedermr}@informatik.uni-tuebingen.de.DE
  2. 2.Department of Computer Science, Utrecht University, Padualaan 14, De Uithof, NL-3584 CH Utrecht, The Netherlands. hansb@cs.uu.nl.NL
  3. 3.ton@win.tue.nl.NL

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