We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves, XNL-complete when a maximum length ℓ for the sequence is given in binary in the input, and XNLP-complete when ℓ is given in unary. The problems were known to be W[1]- and W[2]-hard respectively when ℓ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.

Let XNLP be the class of parameterized prob-lems such that an instance of size n with parameter k can be solved nondeterministically in time f (k) nO(1) and space f (k) log(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloringand Precoloring Extensionwith pathwidth as parameter, Scheduling Of Jobs With Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidthand variants of Weighted Cnf-satisfiability and reconfiguration problems. In particular, this implies that all these problems are W[t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem.

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves and XNL-complete when a maximum length ℓ for the sequence is given in binary in the input. The problems are known to be XNLP-complete when ℓ is given in unary instead, and W[1]- and W[2]-hard respectively when ℓ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.