Reduction Vertex cover into Dominating Set I have a question to the reduction from Vertex Cover into Dominating Set.

So my lecture says if I have a undirected Graph $G = (V,E)$ where $S \subseteq V$ is a vertex cover. Then we construct a new graph $G'$ which has the same vertices of G, except the isolated ones. For each edge $\{u,v\}$ in $G$, we add a new vertex that is connected with $u,v$.

So my question is, why I have to form every edge into triangles? Why is it not enough to only remove the isolated vertices? 

Every vertex cover without isloated vertices is a dominating set at the same time
Remember that you also have to care for the reverse direction. In this case, you should see whether every dominating set is also a vertex cover (hint: no, but why?).