Polynomial reductions between the Limited Packing and Tuple Domination problems in graphs

The Limited Packing and Tuple Domination problems in graphs have closely-related definitions and the same computational complexity on several graph classes. In this work we present two polynomial reductions between these problems. Thus, by considering graph classes which are closed under these transformations, computational complexity results that are valid for one of the problems give rise to results for the other. The question concerning the existence of a class where one of the problems is polynomial and the other NP-complete is still open.