What happens if we assume that the vitiligo is generic disorder then can we check if it's a dominant, recesive and carrier etc. using karyotyping.?

Autoimmune: Vitiligo is usually considered an autoimmune problem. If you want to find out whether it is genetic, you will need to get a detailed history on family members with this problem. I am not aware of any genetic markers uncovered thus far. Nonetheless if there has been no history of such a disorder among your immediate family members within 3 generations, genetic study would be meaningless.