In prior work, we established exact graphical conditions for the structural controllability of discrete-time rank-one bilinear systems. Controllability of these systems, with a single-input and rank-one input matrix, involves checking a greatest common divisor condition, which translates to finding a set of walks of coprime lengths in the network of connectivity implied by the state interconnections. Although we established this graphical condition, there was no approach to check for these coprime walks except through brute force matrix products. Here we present a graphical algorithm to guarantee the existence of coprime walks by constructing a cyclic partition of the state graph. This graphical approach provides both computational advantages and additional theoretical insight by identifying an equivalence between the cyclic partition and the existence of controllably invariant subspaces in the state-space.