Vertex cover problem modification such that every vertex is connected to the set, NP-Hard? 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.
This is called the Dominating Set problem, and it is indeed NP-hard. In fact, it's in some sense harder than Vertex Cover, since it's not fixed parameter tractable (FPT) with respect to the solution size $k$.