Genus and embedding

Let $G$ be a graph of genus $g$.

Let $S_g$ be a surface of genus $g$ (equivalent to the Torus with $g$ handles).

The question is: If we embed $G$ on $S_g$, are the faces that we obtain $2-$cells (homeomorphic to disks)?

I believe the answer is yes (intuitively). But, is there an argument that is more mathematic to express this? If there is some sort of page or paper that talks about this, that would be very helpful.

Note: I can see that if we try to embed a 3-cycle( genus 0) on a Torus(genus 1 $\neq 0$ ), in a way that the cycle goes 'around' the handle of the Torus, then the faces are not $2-$cells.