Parameterized complexity in multiple-interval graphs: Domination, partition, separation, irredundancy

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 multipleinterval 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.