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I’m trying to understand the difference between vertex cover and dominating set.
From what understand, in dominating set, the set D contains vertices that adjacent to other vertices that are not in D (for every v in V, either v is in D or it is adjacent to one in D).
In vertex cover all the vertices in D cover all the edges, but by doing that they are adjacent to other vertices they are not in D – So why is it not a dominating set?
I found some graphs on Wikipedia’s Dominating Set article that illustrate this difference.
These examples show dominating sets (in red) that aren’t vertex covers, the opposite of what you asked later on in your question. The edges in (V-D) prevent them from being vertex covers.