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In Graph theory, a minimum vertex cover is the smallest set of vertices that covers all edges.

Minimum Vertex Cover

In this diagram, the minimum Vertex cover would require both 3 and either 1 or 2 (to cover the edge between 1 and 2).

Is there an equivalent set of minimum vertex required to connect at least once to all other vertices. Limited Minimum Vertex

In this diagram, only 3 would be required, as it connects to edges that connect to all other Vertices.

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The set of vertices you are looking for is called a dominating set and the size of the smallest dominating set is the dominating number of the graph. These have been extensively researched and a quick Google search will yield plenty of hits.

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