This paper deals with the multi-consensus problem in discrete-time directed networks with second-order agent dynamics. Necessary and sufficient conditions to ensure multi-equilibria consensus are obtained by utilising some concepts related to graph structures. It is shown that while the choice of control parameters directly affects the stability of the consensus problem, the equilibria states do not depend on them. Furthermore, the initial values of the states of the secondary layer subgraphs have no effect on the consensus equilibria of any agent in the network. A systematic method for choosing the control parameters is proposed and it is proved that this choice ensures multi-consensus of the network. Finally, illustrative examples are provided to better highlight the theoretical results.