Dominating Set is Fixed Parameter Tractable in Clawfree Graphs
Abstract
We show that the dominating set problem parameterized by solution size is fixedparameter tractable (FPT) in graphs that do not contain the claw (K(1,3)), the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k^2)n^O(1) time and polynomial space to decide whether a clawfree graph on n vertices has a dominating set of size at most k. Note that this parameterization of dominating set is W[2]hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of dominating set is the class of K(i,j)free graphs, which exclude, for some fixed i,j, the complete bipartite graph K(i,j) as a subgraph. For i,i >= 2, the class of clawfree graphs and any class of K(i,j)free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of dominating set is known to be fixedparameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the dominating set problem hard. More precisely, we show that for any t ?>= 4, the dominating set problem parameterized by the solution size is W[2]hard in graphs that exclude the tclaw K(1,t) as an induced subgraph. Our arguments also imply that the related connected dominating set and dominating clique problems are W[2]hard in these graph classes. Finally, we show that for any t, the clique problem parameterized by solution size, which is W[1]hard on general graphs, is FPT in tclawfree graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 DOI:
 10.48550/arXiv.1011.6239
 arXiv:
 arXiv:1011.6239
 Bibcode:
 2010arXiv1011.6239C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 F.2.2