We show that the problem k-Dominating Set and its several variants including k-Connected Dominating Set, k-Independent Dominating Set, and k-Dominating Clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-Distance k-Dominating Set for d >= 2 is W[2]-complete in multiple-interval graphs and their complements. We also show that k-Perfect Code and d-Distance k-Perfect Code for d >= 2 are W[1]-complete even in unit 2-track interval graphs. In addition, we present various new results on the parameterized complexities of k-Vertex Clique Partition and k-Separating Vertices in multiple-interval graphs and their complements, and present a very simple alternative proof of the W[1]-hardness of k-Irredundant Set in general graphs.
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