A refined search tree technique for Dominating Set on planar graphs

  • Authors:

  • Jochen Alber;Hongbing Fan;Michael R. Fellows;Henning Fernau;Rolf Niedermeier;Fran Rosamond;Ulrike Stege

  • Affiliations:

  • Universität Tübingen, Wilhelm-Schickard-Institut für Informatik, Sand 13, D-72076 Tübingen, Federal Republic of Germany;Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada T1K 3M4;School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, NSW 2308 Callaghan, Australia;Universität Tübingen, Wilhelm-Schickard-Institut für Informatik, Sand 13, D-72076 Tübingen, Federal Republic of Germany;Universität Tübingen, Wilhelm-Schickard-Institut für Informatik, Sand 13, D-72076 Tübingen, Federal Republic of Germany;School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, NSW 2308 Callaghan, Australia;Department of Computer Science, University of Victoria, Victoria B.C., Canada V8W 3P6

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2005

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Abstract

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8^kn) and O(8^kk+n^3), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general ''annotated'' problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.