Is there a notion of "face" for bounded genus graphs?
For instance given fixed genus $k$ is there some $q$ for which there are $q$ sets of cycles $S_1, S_2,.., S_q$ each $G[S_i]$ is planar $S_i$ is a subset of faces of $G[S_i]$ each cycle of $G$ can be written as the symmetric difference of cycles in $\cup_{i=1}^q S_i $?
Not sure to understand fully the question, but there sure is a notion of face for any genus graph, as Euler's formulas can be generalised as $$ V-E+F=2-2g$$ Where $g$ is the genus of the graph's embedding.
You might want ot have a look at this post