A refined search tree technique for Dominating Set on planar graphs

We establish a refined search tree technique for the parameterized D OMINATING S ET 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 time...

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Bibliographic Details
Published inJournal of computer and system sciences Vol. 71; no. 4; pp. 385 - 405
Main Authors Alber, Jochen, Fan, Hongbing, Fellows, Michael R., Fernau, Henning, Niedermeier, Rolf, Rosamond, Fran, Stege, Ulrike
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.11.2005
Subjects
Dominating set
Exact algorithm
Fixed-parameter tractability
NP-complete problem
Planar graph
Search tree
Search tree
Dominating set
NP-complete problem
Exact algorithm
Fixed-parameter tractability
Planar graph
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Summary:We establish a refined search tree technique for the parameterized D OMINATING S ET 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 k n ) and O ( 8 k k + 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.
ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2004.03.007