In this paper, we introduce new concepts of domination and packing functions in graphs, which generalize, respectively, the labeled dominating and packing functions defined by Lee and Chang in 2008, and Hinrichsen et al. in 2019. These generalized functions offer a unified and simpler framework for addressing many of the variations of domination and packing concepts in graphs explored in the literature. Interestingly, their associated optimization problems turn out to be equivalent, providing insight to explain the observed coincidences in computational complexity results for graph classes where both problems, the domination one and its corresponding packing variation, have been analyzed. This equivalence also allows us to solve some computational complexity open questions, for some graph classes. Furthermore, we prove that the generalized problems remain solvable in polynomial time for graphs with bounded clique-width and strongly chordal graphs.

In this paper, we present the Generalized Packing Problem (GPF) in graphs in such a way that all other packing problems in graphs presented in the literature correspond to particular instances of it. We find the first class of graphs where one of these previously defined packing problems is tractable and another is NP-hard. We show that GPF remains linear-time solvable in strongly chordal graphs and, when reduced to instances with bounded neighborhood capacities (M-GPF), remains polynomial-time solvable in bounded clique-width graphs. To design specific algorithms that take advantage of the structural properties of certain subclasses of bounded clique-width graphs, we obtain technical results related to the behavior of the new packing parameter under several graph operations. In particular, we present a linear-time algorithm to solve GPF in block graphs and a polynomial-time algorithm to solve M-GPF in P4-tidy graphs.