An attempt has been made to present an up to date survey of related works done on different graph classes. This monogram explores some new variations of domination problems on some special graphs. It is written in a way that it is self contained for independent study. The lemmas and theorems have been stated in logical way and illustration are given whenever necessary. The presentation is logical and simple. It is hoped that the readers will benefit immensely from this monogram. Also examples are given for algorithms.