The dominating set problem is :

Given an $ n$ vertex graph $ G=(V,E)$ , find a set $ S(\subseteq V)$ such that $ |N[S]|$ is exactly $ n$ , where $ $ N[S] := \{x~ | \text{ either $ x$ or a neighbor of $ x$ lies in $ S$ }\}$ $

My question is if the following (new problem) has a definite name in literature, and if not what should be the most appropriate name.

New Problem:Given an $ n$ vertex graph $ G=(V,E)$ and an integer $ k$ , find a set $ S(\subseteq V)$ of size $ k$ such that $ |N[S]|$ is maximized.

For the second problem, some of the names I have seen in the literature are maximum-graph-coverage; partial-coverage; k-dominating-set, (however, the exact same names are also used in other contexts).