  We study the Teichm\"uller theory of Riemann surfaces with orbifold points of
order two using the fat graph technique. The previously developed technique of
quantization, classical and quantum mapping-class group transformations, and
Poisson and quantum algebras of geodesic functions is applicable to the
surfaces with orbifold points. We describe classical and quantum braid group
relations for particular sets of geodesic functions corresponding to $A_n$ and
$D_n$ algebras and describe their central elements for the Poisson and quantum
algebras.
