This paper investigates the minimum-time optimal control problem in a Frechet graph space G_T associated with a continuous linear transformation T : F × I → X, where F is a Frechet space, I = [t₀, t₁] is the time interval, and X is the state space. The study is motivated by the need to extend classical time-optimal control theory from Banach and finite-dimensional settings to more general Frechet spaces. A continuous linear dynamical system Σ = {G_T,U^T_G, I, X, Y, ϕ, η} is formulated on the Frechet graph space G_Tand the unit ball U^T_G is employed as a net to characterise the structure of G_T. A key contribution of this work is to determine controls in Frechet graph space through a unit ball framework U^T_G for the analysis of admissible controls and attainable states, aiming to reach a prescribed target state from an initial state while minimizing a time cost function J_t. Under suitable assumptions including reflexivity and rotundity of the graph space, sufficient conditions are established for the existence and uniqueness of minimum-time optimal controls. Furthermore, necessary and sufficient conditions are derived for both normal and proper systems. In the normal case, the optimal control is shown to be unique and is characterised by a sign condition involving the outward normal vector η to a supporting hyperplane M of the attainable set A^T_G(t) ⊆ X whereas in the proper case optimal controls exist but need not be unique. The proposed graph space framework provides a novel approach to characterize optimal trajectories and derive optimal control laws in Frechet graph space. Finally, illustrative examples are presented to demonstrate and validate the theoretical results.