Finding a vertex coverage that is also an independent set
This problem is called the independent vertex cover problem and it is solvable in polynomial time. Note that the set $S$ must be a vertex cover means that each edge has an endpoint in the set. On the other side, being an independent set means that each edge has at most one endpoint in the set. Hence, each edge has exactly one endpoint in the set. That means we are looking for the smallest possible subset of vertices $S$ such that $(S, V\setminus S)$ is a bipartition of the graph. This can be done in linear time since each connected component has either non or exactly one bipartition.

Here is also another solution of the problem based on a reduction to the 2-sat problem (also using the fact that each edge has exactly one endpoint in the set).

https://cstheory.stackexchange.com/questions/36707/smallest-vertex-cover-which-is-also-an-independent-set