Being new to complexity problems, I've met a question that is quite similar to the Vertex Cover Problem and I am not sure if this one is NP-Hard. We know that the vertex cover problem is the following: given a graph $(V, E)$, selecting a set of vertices $S$ such that every edge $e \in E$ in the graph is connected to some vertices in $S$. The problem of minimizing the vertex cover problem is a well-known NP-hard problem.
My question is the following: given a graph $(V, E)$, selecting a set of vertices $S$ such that every node $v \in V$ is either in $S$ or connected to the nodes in $S$. Is this problem of minimizing the set of $S$ NP-Hard? This problem seems to be quite intuitive after learning the vertex cover problem, but I didn't find a similar question after searching. I apologize if I asked duplicated questions.