The rules for subdividing a surface using vertices V, edges E and faces F are:
1. each vertex is an endpoint of at least one edge;
2. each endpoint of an edge is a vertex;
3. the vertices and edges form a connected graph;
4. no two edges have any points in common;
5. if the surface has a boundary, the boundary consists only of vertices and edges;
6. the space obtained from the surface by removing the vertices and edges is a union of disjoint pieces, called faces, each of which is homeomorphic to an open disc.
With these rules, we can give each sueface a number,
\[\chi =V-E+F\]
called the Euler characteristic. No matter hgow we subdivide the surface, the Euler characteristic for that surface does not change. This can be shown in the following table.
| V | E | F | Remarks |
| 0 | 0 | 0 | No Change |
| 1 | 0 | 0 | Not possible by 1 |
| 0 | 1 | 0 | Not possible by 2 |
| 0 | 0 | 1 | Not possible by 5 |
| 1 | 1 | 0 | Only possible, if edge is not a loop. Either subdivision would be disconnected with an extra face |
| 1 | 0 | 1 | Not possible by 1 |
| 0 | 1 | 1 | Not possible since an extra face reqquires extra edges. |
| 1 | 1 | 1 | Would mean a disconnected subdivision. |