Notion of "face" in the graph theory I am working out the Euler's Formula for Planar Graphs. For this the notion of "face" is introduced. In our script they just say: A plane graph seperates the plane into regions, called faces. Well, I can't start a lot with the definition and also my research on the web doesn't helps me to find a good definition of this notion of "face". Can someone, maybe with an example, try to illustrate for me this notion? Many thanks!
 A: This is a detail most graph theorists go over rather quickly, while it is actually quite hard to define properly.
Recall that a planar graph is a graph, which admits an embedding into the plane/sphere. This definition alone is quite sketchy, but can be made rigid by means of topology in saying that the topological realization of the graph, given by gluing copies of the unit interval $[0,1]$ to the vertices according to the incidence relation of the graph, admits a topological embedding into the plane/sphere. The gluing part is a properly defined construction, commonly called CW-complex.
Okay but what are faces then? A non-trivial but quite intuitive fact from topology is, that any embedding of a circle into a sphere separates the latter into two separate connected components. This is the Jordan Curve Theorem. Now we can use this inductively to show that an embedding of the topological realization of our (finite) graph separates the sphere into a finite amount of connected components. This connected components are what graph theorists call the faces of the embedding.
It is a remarkable fact, that the number of faces is independent of the embedding we chose and thus an invariant of the graph itself. This is the Euler Formula.
Addendum In case the topological realization thingy was too fast, consider the following more down-to-earth description. Consider a graph $G$ consisting of vertices $V$ and edges $E$. For any $v \in V$ pick a point $(v_1,v_2)\in\Bbb R^2$ (in such a way that no two distinct vertices hit the same point). Draw a straight line between the points $(v_1,v_2)$ and $(w_1,w_2)$ whenever $vw\in E$ is an edge. We call this an embedding of our graph $G$, if any two straight lines only intersect nontrivially in their endpoints and do not contain vertices other than the endpoints. If $G$ admits a cycle, this will be realized as a piecewise linear cycle in the plane. Vertices of $G$, which do not lie on this cycle either have to be embedded „inside“ or „outside“ the cycle. So the cycle and thus our graph separate the plane into (at least) two regions/faces.
