Given a weighted graph $G(V,E,w)$ where $w$ is the weight function on edges and a subset of vertices $S\subseteq Q$ called terminals, a Steiner Tree is a connected subgraph which connects all vertices in $S$. Finding minimum weight Steiner Tree is called Minimum Steiner Tree Problem. This problem and many of its variants are known to be NP-Complete.
I thought about this variant where we dont need to connect all vertices in $S$. That is, along with the input, we are also given a natural number $k$, such that $k<|S|$. Now, find a minimum weight connected subgraph which connects at least $|S|-k$ vertices in $S$.
Notice that when $k$=0, this is the usual Steiner Tree Problem and when $k=|S|-2$, it is Shortest path problem. Is this problem NP-Complete for any arbitrary $k$?
I do not know whether this question already exists (quite possible considering the number of variants of Steiner Tree Problem). I did a brief search and I couldnt find anything related to this problem.