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What would be the name of the problem of a vertex cover that covers all the vertices without the requirement of covering all the edges?

Vertex cover has to answer whether or not there is a set of $k$ vertices that covers all the edges of the graph, covering therefore, all the vertices. What is the name of the problem that, similarly to vertex cover, select a subset of $k$ vertices that has direct edges to all the vertices, but without necessarily covering all the edges of the graph?

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If I understand your description correctly, you are looking for a dominating set.

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