Fixed-parameter tractability and completeness II: on completeness for W[1]
Theoretical Computer Science
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Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
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FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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We consider the parameterized complexity of the following problem under the framework introduced by Downey and Fellows: Given a graph G, an integer parameter k and a non-trivial hereditary property Π, are there k vertices of G that induce a subgraph with property Π? This problem has been proved NP-hard by Lewis and Yannakakis. We show that if Π includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class W[1] and is fixed parameter tractable otherwise. In the former case, if the forbidden set of the property is finite, we show, in fact, that the problem is W[1]-complete. Our proofs, both of the tractability as well as the hardness ones, involve clever use of Ramsey numbers.