A formula that states necessary but not sufficient conditions for an object to be a simple polyhedron. An object with V vertices, E edges, and F faces satisfies the formula
where χ is called the
Euler characteristic of the surface in which the object is embedded. For a plane connected graph, the formula takes the form
For a simple polyhedron in Euclidean 3-space, the formula has the form
A simple polyhedron is any polyhedron that can be continuously deformed into a sphere, assuming that its faces are treated like sheets of rubber. All faces are bounded by a single ring of edges: there are no holes in the faces; each edge joins exactly two faces and is terminated by a vertex at each end. At least three edges meet at each vertex.