why does the poly-time reduction from dominating set to vertex cover require adding a vertex to every edge?
The key reason we need $w_{uv}$ is because not every dominating set is necessarily a vertex cover, but $w_{uv}$ &quot;patches&quot; that up.
In particular, assume isolated vertices can be ignored. Then every vertex cover is a dominating set, but not every dominating set is a vertex cover. Hence, if we tried to do the reduction without adding $w_{uv}$, then only one direction would work: we can show that if $G$ has a vertex cover of size $k$, then $G$ also has a dominating set of the same size. However, the converse is not true; only by adding the $w_{uv}$ does every dominating set of $G^\prime$ necessarily dominate the $w_{uv}$ as well, and thus cover all edges in $G$.