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