Euler formula, polyhedron with a hole

Question

I'd like to ask help for this question. I know that I need to have a little more concept on what kind of polyhedra that Euler's formula is valid.

Choose the false alternative:

a) F = 12
b) V = 20
c) E = 30
d) Euler's formula F + V - E = 2 applies
e) All above are true

Actually, by looking at this picture we can conclude that E is correct, since all of them seem to be true. However, shouldn't this formula be F + V - E = -2, since the genus is 2 ?

Answer 1

The number of faces is listed as 12. There are ten vertical faces and two "horizontal" faces (namely, top level and bottom level of the polyhedron). However, the latter pair are not admissible as faces because a face has to be a convex polygon for the formula to apply.

Answer 2

See: https://en.wikipedia.org/wiki/Polygon_with_holes#In_polyhedra

$\chi =V-E+F-\text{holes}=2(1-g)$

$V=20, E=30, F=12, H=4$ (2 faces each with 2 holes)

$\chi=20-30+12-4=-2=2(1-g) \implies g=2$ so it's $2$-holed polyhedron.