          ! "#  1                !" #$%          !" && ’&( ) )  &( *+  !  Let G = (V, E) be a simple graph. Let S be a maximum independent set of G. A subset T of S is called a forcing subset if T is contained in no other maximum independent subset in G. The independent forcing number of S denoted by f I (G, S) is the cardinality of a minimum forcing subset of S. The independent forcing number of G is the minimum of the independent forcing number of S, where S is a maximum independent subset in G. The independent forcing spectrum of G denoted by Spec I (G) is defined as the set Spec I (G) = {k : there exists a maximum independent set S of G such that f I (G, S) = k}. In this paper, a study of Spec I (G) is made..  Forcing domination number of a graph, Forcing spectrum of a graph and Forcing independent spectrum of a graph.   The forcing sets in a graph are a very interesting concept. In the management of an institution, the executive committee consists of senior members who have adequate rapport with other members of the institution. Some members of the executive committee may sit in other important committees also. Sometimes, restrictions are imposed on members that they can be part of exactly one committee. This precisely leads to the concept of forcing set. A subset of a minimum dominating set S is called a forcing subset with respect to S if this subset is contained in no other minimum dominating set of G.Many authors have studied this forcing concept with respect to several parameters like domination ,matching ,geodetic domination, chromatic partition, etc. This chapter studies the forcing concept with respect to maximum independence. A subset of a maximum independent set may be contained in other maximum independent sets also. For example, in C 5 ,every vertex is contained in at least two maximum independent sets. The natural curiosity is to study such subsets which are constrained to remain only in one maximum independent set which is forced to remain only in that set. We consider only finite ,simple and undirected graphs G=(V,E).[2]Gary Chartered, Gavlas and Robert C.Vandell introduced the concept of Forcing domination number of a graph.   [2] A subset T of a minimum dominating set S is a forcing Subset for S if S is the unique minimum dominating set containing T. S is called the forcing dominating set of T. The minimum cardinality among the forcing subsets of S is called the forcing domination number of S and is denoted by f(S, γ (G)). The minimum forcing domination number among the minimum dominating sets of G is denoted by f(G, γ). That is f(G, γ) = min f(S i , γ) , where S i ’ s are the minimum dominating set of G. Cleary, for any graph G, f(G, γ) ≤ γ (G).   [1] Let G be a simple graph. The forcing spectrum of G denoted by Spec γ (G) is defined as the set Spec γ (G) = {k: there exists a minimum dominating set S of G such that f(S, γ(G)) = k}. The above two concepts are extended in the context of maximum independent sets.   Let S be a maximum independent set of G. A subset T of S is called a forcing subset if T is contained in no other maximum independent subset in G. The independent forcing number of S denoted by f I (G, S) is the cardinality of a minimum forcing subset of S. The independent forcing number of G is the minimum of the independent forcing number of S, where S is a maximum independent subset in G. The independent forcing spectrum of G denoted by Spec I (G) is defined as the set Spec I (G) = {k : there exists a maximum independent set S of G such that f I (G, S) = k}.  !"#