Situation is the following. I have a genus $g$ surface $M_g$ and a graph $G$. There is an embedding $f: G \to M_g$ and, actually, $G$ triangulates $M_g$. Our graph has a cycle $C=(v_1,v_2,v_2)$ (graph $G$ is not directed, but we care about orientation of the cycles, i.e. $(v_1,v_2,v_3)\neq (v_3,v_2,v_1)$). This cycle is said to be "good" if $f(C)$ is the boundary of one component of $M_g\setminus f(G)$ and if the orientation on $f(C)$ induced by $f$ from the orientation on $C$ is compatible with the orientation of $M_g$.
I struggle to understand the second part of the definition of a "good" cycle. Ok, there is orientation on $f(C)$, but what will we consider compatible orientation on $M_g$? Obviously, homological/smooth definitions are useless in this context. Should I consider orientation on every face of $G$, since it basically is $1$-skeleton of the simplicial complex which is triangulation of $M_g$? Then we do have an orientation on $M_g$, but it still is not very clear when it is compatible with orientation on $f(C)$. Does anyone know what is meant here?
Thank you.