Euler-Poincaré Formula:

The Euler-Poincaré formula explained the relationship of the number of vertices, the number of edges & the number of faces of a manifold. This has been generalized to include potholes & holes that penetrate the solid. To state the Euler-Poincaré formula, we required the following definitions :

• V :  it refer to the number of vertices
• E : it refer to the number of edges
• F : it refer to the number of faces
• G : it refer to the number of holes that penetrate the solid, usually referred to as genus in topology
• S : it refer to the number of shells. A shell is an internal void of a solid. A shell is bounded by a 2-manifold surface, which can have its own genus value. Note that the solid itself is counted as a shell. Therefore, the value for S is at least 1.
• L : it refer to the number of loops, all outer and inner loops of faces are counted.

After that, the Euler-Poincaré formula is the following :

V - E + F - (L - F) - 2 (S - G) = 0