I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for. The story/idea is we have ally locations and enemy locations. We want to visit all allies while encountering few enemies. I thought of this when reading some papers in combinatorial optimization, but this is not my area of expertise. So, I don't know if I am searching the correct words.
The graph theoretic formulation is that we have an edge weighted graph $G = (V,E)$ which is bicolored so $V = A \cup B$ (note the coloring need not be proper, i.e. $G$ doesn't have to be bipartite). We are given an integer $k$ and want to find a minimum weight cycle (or alternatively path) which contains all vertices in $A$ and at most $k$ vertices in $B$.
Has this problem been studied somewhere? Does it have a name or fit inside a known generalized TSP? Or maybe there is some reduction to known version of TSP?
When $k=0$ we can delete vertices in $B$ along with all edges incident to them and we have the usual TSP.