# 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.