On the Complexity Landscape of the Domination Chain

In this paper, we survey and supplement the complexity landscape of the domination chain parameters as a whole, including classifications according to approximability and parameterised complexity. Moreover, we provide clear pointers to yet open questions. As this posed the majority of hitherto unsettled problems, we focus on Upper IRREDUNDANCE and LOWER IRREDUNDANCE that correspond to finding the largest irredundant set and resp. the smallest maximal irredundant set. The problems are proved NP-hard even for planar cubic graphs. While LOWER IRREDUNDANCE is proved not c log(n)-approximable in polynomial time unless NP subset of DTIME(n(log log n)), no such result is known for Upper Irredundance. Their complementary versions are constantfactor approximable in polynomial time. All these four versions are APXhard even on cubic graphs.