We describe new results in parameterized complexity theory. In particular, we prove a number of concrete hardness results for W[P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parameterized complexity of analogues of P SP ACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W[P] and related classes. For instance, we show that W[P] can be characterized in terms of the DTIME(2^o(n)) and NP
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