Side note: From Wikipedia:

> The surfaces of nonconvex polyhedra can have various Euler characteristics:

It's strange because the examples use weird faces, but in most (all?) of them it is possible to split the weird faces into a few poligonal faces and get V-E+F=2. For example https://en.wikipedia.org/wiki/Small_stellated_dodecahedron use faces that are stars that intersect other faces and the intersection is not an edge. Replacing each star with 5 triangles the Euler characteristic is 2.