Permalink Submitted by Anonymous on February 29, 2012

Using the fact that every vertex 'u' is connected to d(u) (degree of 'u') faces of the polyhedron, we try and see what happens to |V|,|F| and |E| when a vertex is removed and a new polyhedron is formed with |V'| , |F'| and |E'| (# of vertices, faces and edges).

|V'|= |V|-1 (trivial)

|F'|=|F|-d(u)+1
all the d(u) faces vertex 'u' is connected to will merge into a single face

|E'|=|E|-d(u) (trivial)

observe that |V|+|F|-|E|=|V'|+|F'|-|E'|

repeat this process on the given polyhedron until only a cycle is left.
And we can see that |V|+|F|-|E|=2
(the face formed by merging the faces to be removed and the face already present is the same, i.e our cycle has two faces)

## An alternate proof

Using the fact that every vertex 'u' is connected to d(u) (degree of 'u') faces of the polyhedron, we try and see what happens to |V|,|F| and |E| when a vertex is removed and a new polyhedron is formed with |V'| , |F'| and |E'| (# of vertices, faces and edges).

|V'|= |V|-1 (trivial)

|F'|=|F|-d(u)+1

all the d(u) faces vertex 'u' is connected to will merge into a single face

|E'|=|E|-d(u) (trivial)

observe that |V|+|F|-|E|=|V'|+|F'|-|E'|

repeat this process on the given polyhedron until only a cycle is left.

And we can see that |V|+|F|-|E|=2

(the face formed by merging the faces to be removed and the face already present is the same, i.e our cycle has two faces)