I am studying a problem concerned with the embedding of a graph $G=(V,E)$ in spaces that are not simply connected, e.g. $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$. My understanding is that an embedding is an identification of vertices in $V$ with distinct points in $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$ and edges in $E$ with simple paths (or even straight arcs) connecting two vertices. I have to admit I'm not an expert of the topic so feel free to correct me if I'm saying anything wrong.
An embedding would then be a tuple of a set of points $\{x_v\}_{v\in V}$ with $ x_v\in(\mathbb{C}\setminus\{0\})\times\mathbb{R}$ and $x_i \neq x_j$ for $ i\neq j$ and simple curves $\{f_e\}_{e\in E}$ with $f_e(0)=x_v$ and $f_e(1)=x_{v'}$ for $e=\{v,v'\}$. The graph, even before the embedding, carries non-trivial topological information. One can define its fundamental group and find the generators of the fundamental group. The graph, as embedded in $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$, is even richer of topological information. For example, a cycle of the graph can be contractible or not when thought of as a closed curve in $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$ (although it cannot be contractible when thought of as part of the graph because $x_i\neq x_j$). So one can for example define winding numbers for each of the cycles of the graph when thought of as closed curves in $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$, indicating how many times they wind around the hole of $\mathbb{C}\setminus\{0\}$.
My guess is, that for the specific example I presented, the entire topological information of the embedding is contained in a choice of generators of the fundamental group of the graph and the winding numbers associated to them.
My vague question is: is this a legit problem and is what I just presented correct? Is there any literature on the topic? I could find a lot of resources on the embedding of graphs on 2D surfaces associated with the problem of planarity or non-planarity, but these do not really address my (simpler) problem.