With the rise of social media, misinformation has significant negative effects on the decision-making of individuals, organizations and communities within society. Identifying and mitigating the spread of fake information is a challenging issue. We consider a generalization of the Domination problem that can be used to detect a set of individuals who, through an awareness process, can prevent the spreading of fake narratives. The considered problem, named (d ,  r)-Domination generalizes both distance and multiple domination. We study the parameterized complexity of the problem according to standard and structural parameters. We give fixed-parameter algorithms as well as polynomial compressions/kernelizations for some variants of the problem and parameter combinations.