What is the optimal way to explore the planets in Master of Orion? (Maximize $\sum_{v\in T}r(v)$ where $|T|=n$ and $T\supset S$.)

Question

The general problem is, given

1. a graph $G=(V,E)$ with vertices $V,$ and edges in $E$,
2. a path connected subset $S$ of $V$ (the previously explored planets),
3. a mapping $r:V\rightarrow (0, m)$ where $m\in\mathbb R$ ($r$ is the reward for exploring each planet), and
4. an integer $n$, where $|V|>n>|S|$ ($n$ is the number of planets to be explored),

which path connected superset $T$ of $S$ of size $n$ maximizes $$\sum_{v\in T}r(v)\ \ ?$$

Mainly, I would like to know if a polynomial time algorithm exists to solve this problem.

Also, in the actual game $V\subset\mathbb R^2$ and there exists an positive integer $d$ such that $e\in E$ if and only if the Euclidean distance between the vertices of $e$ is less than $d$. Would that condition allow for a faster or simpler algorithm?

There is one more condition that is satisfied in the actual game, the incidence count for each vertex is less than 20. (i.e. $\max_{v\in V} |\{e| v\in e\}| < 20$). It's possible that adding that condition would allow for a polynomial time algorithm.