"faces" of a non-planar graph Good afternoon,
I have a question concerning concepts in graph theory. Graph theory is a field quite strange to my knowledge, so my question is maybe stupid.
For a planar graph, we can define its faces as follows : we delete all its edges and its vertices from the plane. Then the remaining part of the plane is a collection of pieces (connected components). Each such piece is called a face.
So how to define faces of a non-planar graph? I would like to know if we can define it from the intuitive figure of a graph.
Thanks in advance,
Duc Anh
EDIT : I would like to explain more where my question comes from. In this slide http://www.aimath.org/~hogben/Goins.pdf, the author writes $K_{3,3}$ has $6$ vertices, $9$ edges and $3$ faces, so I wonder how these faces are defined? Or we only can define them after embedding the graph into some surface of genus $g$?
 A: Maybe this is something you want: Any graph $G$ can be embedded in a compact surface $S_g$ of genus $g$, for some $g$. The minimum of such $g$ is called the genus of $G$. Then faces of a non-planar graph may be defined as the regions of the embedding of $G$ in the $S_g$. 
A: There is no reasonable definition of faces for an arbitrary graph. Even for a planar graph the definition of faces assumes that in addition to the (abstract) graph a concrete embedding in the plane is given. For instance take a graph constructed starting form two vertices $A,B$ by linking them with simple paths of lengths $1,3,2,3$ (introducing the necessary $5$ intermediate vertices). This graph is certainly planar, but does it contain a triangular face? That depends on how those paths are embedded in the plane (the order given suggests an embedding that does not give rise to a triangular face, but it there does exist an embedding of this graph with a triangular face. For more general graphs you could arrange that any cycle in the graph defines a face for some embedding in a sufficiently complicated surface.
A: In general, the answer to your question is no---from an abstract representation $G=(V,E)$ of a graph, you cannot immediately get any information about its faces in an embedding into a surface. All is not lost, though: You might want to look at rotations. In simple terms, if for each vertex $v$ of $G$ you define an order in which all of the edges adjacent to $v$ would be visited in a rotation about $v$, these orders will determine an embedding of $G$ in some orientable surface: selecting different orders for rotations will give you different embeddings.
A: For a planar graph, we can define its faces as follows: we delete all its edges and its vertices from the plane. Then the remaining part of the plane is a collection of pieces (connected components). Each such piece is called a face.
It is similar to planar.
A face of the embedded graph is an arcwise connected component of the surface minus the graph.
For example, the torus is up and down connected, left and right connected.
So $K_{3,3}$ has 3 faces on the torus.
