Is this a graph theory problem?

My basic problem includes a graph where each node $ i$ is associated with a weight $ c_i$ , and the problem is to find a minimum (or maximum) weighted independent set with a fixed cardinality $ p$ . This is I believe a well-known problem in graph theory that is well-studied for different types of graphs.

Now, suppose I am dealing with a generalized form of the problem as following. The weight of each node can take $ p$ different values, that is each node is associated with $ p$ different weights. The aim is again to find a minimum (or maximum) weighted independent set with a fixed cardinality $ p$ , however, each type of weight can be selected only once. Precisely, if the weight type $ j$ is selected for the node $ i$ , i.e., we select the weight $ c_{ij}$ , then the other selected nodes cannot take a weight of type $ j$ .

My question is that, is this still a graph theory problem? Is it a known generalization in the graph theory problems?

Any help and/or reference is appreciated.