I have a dude with the following problem. Suppose you have an 2-cell embedding of some simple graph $G$ on a orientable type surface $S_g$ (for example a plane graph), and you desire to find a set $A \subset V(G)$ with minimum cardinality, such that every face boundary have at least one vertex from $A$.
Does anyone knows if exist related studies or parameters of this type? I revised some cover problems like vertex cover or face cover problems, unfortunately without success.
Thanks in advance for any suggestion or commentary