# Can this special case of Node Weighted Steiner Tree be solved in polynomial time?

Consider the node-weighted steiner problem:

Input: a graph $$G=(V,E)$$, a set $$T\subseteq V$$ of terminals, a weight function $$w: V\setminus T \to \mathbb{R}_+$$.

Output: a minimum weight subset $$S \subseteq V\setminus T$$ of vertices such that $$G[T\cup S]$$ is connected.

This problem is known to be at least as hard as Set Cover. However, I am interested in a more special case of node-weighted steiner tree, in which:

1. Each steiner node is incident to exactly 2 terminals of $$T$$
2. $$T$$ is an independent set.

If I added a third condition that every pair of steiner nodes are adjacent, then this would be equivalent to an edge cover problem, which can be solved in polynomial time.

Hence, my special case lies somewhere in the middle between very hard to approximate (node weighted steiner tree), and exactly solvable (edge cover).

Since I don't know whether this is in P or NP-hard or to what extent it is approximable, any ideas about algorithms or hardness results on this problem would be welcome. Thanks!