Since the use of multi-agent systems has significantly increased over the past few decades, it is crucial to study the methods used in this area and take implementation-related practical issues into consideration. For instance, in the formulation of the consensus problem, the communication between the agents is depicted using a graph, where each edge represents a communication link between the agents. In practical terms, there will be significant operational challenges and costs associated with implementing each of these communication channels in various forms. Therefore, in this study, the graph reduction problem has been presented as an optimization problem, accompanied by six distinct cost functions. The objective is to reduce the size of a graph without compromising its fundamental characteristics. Hence, the cost functions provided are formulated to minimize the number of edges in the resulting graph while ensuring that the consensus error remains bounded. Ultimately, simulations were performed to demonstrate the effectiveness of the proposed approach. The outcomes derived from the specified set of cost functions were then compared to several conventional topologies in multi-agent systems.