What class this graph connectivity problem belong to

We are given a Boolean circuit $ C(x_1, \dots, x_n,y_1, \dots, y_n)$ with $ 2n$ inputs and we have to determine if the graph $ G_C = (V_C, E_C)$ (specified below) is connected: $ $ V_C = \{0,1\}^n, \ E_C = \{((a_1, \dots, a_n), (b_1, \dots, b_n)) : C(a_1, \dots, a_n,b_1, \dots,b_n) = 1\}.$ $

To which class this problem belongs? I know that graph connectivity can be done so fast (as fast as to be in P) and using a small amount of space. Does this is different in this particular graph? Is this problem in P, EXP, PSPACE?