The Maximum Weighted Graph Set Packing (MaxGP) problem is defined as follows: given a graph G, a set \(\mathcal {G}\) of subgraphs of G, and a weight function \(\omega :\mathcal {G}\rightarrow \mathbb {Q}^+\) , the objective is to find a subset \(S \subseteq \mathcal {G}\) that maximizes \(\sum _{H \in S} \omega (H)\) and ensures that each pair of elements in S are edge-disjoint. This work introduces MaxGP and three specific cases of the problem, where \(\mathcal {G}\) contains only subgraphs from a particular class: stars (MaxSP), paths (MaxPP), or triangles (MaxTP). For the unweighted versions of these problems (where \(\omega (H) = 1\) for all \(H \in \mathcal {G}\) ), we derived several intractability and inapproximability results. We proved that both unweighted MaxSP and unweighted MaxPP are W[1]-hard and APX-hard, even when the maximum degree of G is restricted to 3. Additionally, we demonstrated that these two problems are not fixed-parameter tractable (FPT) when parameterized by the treewidth or the maximum degree of G, but they do admit fixed-parameter algorithms when parameterized by both treewidth and maximum degree. For MaxTP, we showed the existence of a fixed-parameter algorithm when parameterized solely by the treewidth of G. Moreover, we proved that the unweighted versions of MaxSP, MaxPP, and MaxTP are NP-hard on planar graphs. Using our FPT results, we proposed polynomial-time approximation schemes (PTASs) for these versions on planar graphs.