# 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