Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Selected papers from the second Krakow conference on Graph theory
Journal of the ACM (JACM)
Deciding first-order properties of locally tree-decomposable structures
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Fixed-Parameter Tractability, Definability, and Model-Checking
SIAM Journal on Computing
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
The dominating set problem is fixed parameter tractable for graphs of bounded genus
Journal of Algorithms
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up
SIAM Journal on Computing
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Claw-free graphs. I. Orientable prismatic graphs
Journal of Combinatorial Theory Series B
Claw-free graphs. II. Non-orientable prismatic graphs
Journal of Combinatorial Theory Series B
Claw-free graphs. III. Circular interval graphs
Journal of Combinatorial Theory Series B
Claw-free graphs. IV. Decomposition theorem
Journal of Combinatorial Theory Series B
Claw-free graphs. V. Global structure
Journal of Combinatorial Theory Series B
Optimization problems in multiple-interval graphs
ACM Transactions on Algorithms (TALG)
Claw-free graphs VI. Colouring
Journal of Combinatorial Theory Series B
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Extremal Combinatorics: With Applications in Computer Science
Extremal Combinatorics: With Applications in Computer Science
Domination when the stars are out
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Parameterized Complexity
Parameterized complexity of induced h-matching on claw-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Parameterized domination in circle graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Parameterized Domination in Circle Graphs
Theory of Computing Systems
Hi-index | 5.23 |
We show that the Dominating Set problem parameterized by solution size is fixed-parameter 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 claw-free 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@?N, the complete bipartite graph K"i","j as a subgraph. For i,j=2, the class of claw-free 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 fixed-parameter 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 t-claw 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@?N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free 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.