I am trying to reduce the vertex cover (decision) problem to the dominating set (decision) problem in order to prove that the latter is NP-hard. After some research online, I found that many articles use a reduction that transforms the input for the vertex cover problem to an input for the dominating set problem by creating a triangle for each edge. Here is one of such articles (See question 7 in the link).
The question that I would like to ask is, if we drop isolated vertices in the input to the dominating set problem, then, we could easily find a counterexample to the reduction - Let the input to the vertex cover problem be a graph containing $N$ isolated nodes and parameter $k=N$. Now, the input to the dominating set problem will clearly be an empty graph with the parameter $k=N$. Now, there is a vertex cover of size $N$. But it is not a dominating set of the transformed graph (i.e. the answer to the vertex cover problem is yes but the answer to the dominating set cover problem is no).
How could I fix the reduction? Could someone please advise me?