I am new to Algebraic Topology. So, I would like to have a detailed proof of this following problem as there are many similar and corollary problems in the book that I am reading.
Suppose we are given an embedding of a graph $G$ in the plane $\mathbb{R}^{2}$. Here $G$ could be a multigraph, i.e. multiple edges between two vertices as well as loop edges on a single vertex are allowed. And suppose the number of vertices is $V$, the number of edges is $E$ and the number of connected components of $G$ is $C$. I am required to show that the number of connected components of the complement $\mathbb{R}^{2}-G$ is $E - V + C + 1$.
The only result related to graphs that I have seen so far is the calculation of fundamental group using the Van Kampen theorem. But I am totally at loss regarding how to approach this problem.
Thanks in advance!
Hint: construct a clever choice of CW structure on $S^2$ so that your result follows from the fact that $\chi(S^2) = 2$ and that the Euler characteristic of a CW complex is given by the number of vertices less the number of 1-cells plus the number of 2-cells.
(A careful proof that this works uses the Jordan curve theorem, but you are surely expected to sweep this detail under the rug.)