# Dominating Set Gap Problem Reduction

In the reduction from Vertex Cover problem to Dominating Set a new vertex is added for each edge in the original graph. Specifically, Given graph $$G=(V,E)$$ where $$|V|=n$$ with $$VC$$ with size $$k$$ we construct a new graph $$G'=(V',E')$$ where $$n\le |V'| \le n^2$$ which has a $$DS$$ of size $$k$$ as well. The bounds are for the edge cases where $$G$$ has no edges and when it's a "full graph".

Consider the Vertex Cover & Dominating Set gap problem. I'm interested in proving the following: $$Gap[\alpha n,\beta n]-VC \le Gap[\alpha' n,\beta' n]-DS$$

Note that $$n$$ in the two problems might be different as stated above.

The same reduction works, but I'm not sure regarding the choice of $$\alpha'$$ and $$\beta'$$. Consider the completeness argument. Say $$G$$ has a $$VC$$ with size at least $$\beta n = k$$ (a "yes" instance), then $$G'$$ has a $$DS$$ with a size at least $$k=\beta' n' = \beta' (n+n^2) \rightarrow \beta' = \frac {1}{n+1}\beta$$.

It doesn't make sense that $$\beta'$$ is dependent on $$n$$, because that loses the whole point of gap reductions. What am I missing?