Let $G=(V,E)$ be a connected graph, and let $S\subset V$. How to find a minimum subset $H\supset S$,and $G(H)$ is a connected graph.
It seems different from Steiner tree problem. Since in Steiner tree problem, the weight is added in edges and the goal is to find a tree. While in our problem, the goal is to find a induced subgraph.