Independent Sets with Domination Constraints

A ρ-independent set S in a graph is parameterized by a set ρ of non-negative integers that constrains how the independent set S can dominate the remaining vertices (fV υ ∋ S: ¦N(υ) ∩ S¦ ∃ ρ.) For all values of ρ, we classify as either NP-complete or polynomial-time solvable the problems of deciding if a given graph has a ρ-independent set. We complement this with approximation algorithms and inapproximability results, for all the corresponding optimization problems.