Let $G$ be a finite connected graph and $f:G\to S^2$ be an embedding of $G$ into $S^2$ (we are assuming that $G$ can be embedded in $S^2$). Think of the graph as a $1$-dimensional CW complex.
Is it true that each component of $S^2\setminus f(G)$ is homeomorphic to a disc?
For simplicity one may assume that the graph is simple and that each vertex has degree at least 2 (if this makes life easier).
I want to read planarity of graphs, but the graph theory texts I have access to do not treat the topological aspects rigorously. Can anyone suggest me a book where planarity is treated rigorously?
The embedding of graphs into all types of surfaces is discussed in extensive detail in Gross and Tucker's Topological Graph Theory. I read the first two or three chapters, but I don't remember if the answer to your specific question was addressed -- that being said, I would be very surprised if it wasn't, since the book covers embeddings in surfaces far more exotic than the sphere.