I've seen the following definition in a paper on approximation preserving reductions.
Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} \pi_{2}$ if there exist two polynomial time computable functions $f, g$ that satisfy the following properties:
$1.f : I_{\pi_{1}}\rightarrow I_{\pi_{2}}$ such that $x_{1} \in I_{\pi_{1}}$, $f(x_{1})\in I_{\pi_{2}}$; in other words, given an instance $x_{1} \in \pi_{1}$, $f$ allows to build an instance $x_{2}= f(x_{1}) \in \pi_{2}$;
2.$g : I_{\pi_{1}} \times \mathrm{Sol}_{\pi_{2}} \rightarrow \mathrm{Sol}_{\pi_{1}}$ such that, $\forall (x_{1},y_{2})\in (I_{\pi_{1}} \times \mathrm{Sol}_{\pi_{2}}(f(x_{1})))$ , $g(x_{1},y_{2}) \in \mathrm{Sol}_{\pi_{1}}(x_{1})$; in other words, starting from a solution $y_{2}$ of the instance $x_{2}$, $g$ determines a solution $y_{1} = g(x_{1}, y_{2})$ of the initial instance $x_{1}$.
What if for example i had the following scenario
I have two instances of the maximum independent set problem $\pi_{1}$ and $\pi_{2}$ for different classes of graphs call the graphs $G$ and $G'$ respectively . In addition suppose given an instance $x_{1} \in \pi_{1}$ i can build an instance $f(x_{1})$ such that i can find an independent set of size $k$ in $G$ if and only if i can find an independent set of size $F(k)$ in $G'$ (for some function $F$), this at least to me seems fine in order to describe the function $f$ above.
But what about $g$? What if i just said let $g$ map every feasible solution $y_{2}$ of the problem $\pi_{2}$ to a single vertex of a graph in $G$, technically a single vertex would be a feasible solution to the MIS problem in $\pi_{1}$ right? But this seems a bit like cheating.
Could someone please clarify the flaw in my argument, if there is one?
