The middle graph of a graph G denoted by M(G) is a graph whose vertex set is V(G)UE(G) and two vertices are adjacent if they are adjacent edges of G or one is a vertex and other is a edge incident with it. The Line graph of G written L(G) is the simple graph whose vertices are the edges of G with ef Є E(L(G)) when e and f have a common end vertex in G. A set S of vertices of graph M(G) if S is an independent dominating set of M(G) if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent middle domination number of G, denoted by iM(G) is the minimum cardinality of an independent dominating set of M(G).A dominating set D is a connected dominating set if D is connected. The connected domination number, denoted by Ƴc, is the minimum number of vertices in a connected dominating set. In this paper many bounds on iL(G), iM(G), ƳM(G) were obtained in terms of element of G, but not in terms of elements of L(G) or M(G).