Let $M$ be orientable 3-manifold admitting a Heegaard splitting $V\cup_{S}W$.

Let $X$ be a carrier graph of $M$ such that rank($X$)=rank($\pi_{1} M$).

Note: A connected graph is called a carrier graph of $M$ if there is a map $f: X\rightarrow M$ such that $f: \pi_{1} X\rightarrow \pi_{1}M$ is surjective. And we call $f$ a carrier map of $X$.

Thank Agol for comments. I have editted my questions again.

Now I want to know

If we fix the carrier graph$X$, is it possible that there is a carrier map $f$ of $X$, $f(X)\subset S$? Can we ask $f(X)$ to be embedded into $S$?

Or more weakerly, Is there a pair of $(X,f)$ such that $X$ is a carrier graph of $M$ with rank(X)=rank($\pi_{1} M$) and $f(X)$ can be embedded into $S$?