In this paper, the idea of minimal inverse dominating set (IDS) in a graph G(V,E) is presented. A set I  (V − D) ofG (V,E) is anIDSof G , if I is the dominating set of the sub graph V − D  , D is a minimal dominating set ofthe G. A minimal IDS in a graph G is anIDSthat contains no IDSas a proper subset. A minimum cardinality among all the IDSis an ID number of G(V,E), and it is denoted by (G) ID  . Further theIDS and ID number of various special graphs like Bidiakis cube,Durer graph,Golomb graph and etc. have been discussed.