# maximum coverage version of dominating set

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).