Correspondent Banking (CB) Network refers to a network of financial institutions providing cross-border payment services for customers through different channels such as SWIFT, Fedwire, etc. Through the CB network, banks and their customers can access financial services in different jurisdictions and provide cross-border payment services to their customers, supporting, among other things, international trade and financial inclusion. We employ the mathematical programming approach in conjunction with the graph theory to optimize a CB network. Optimizing the network requires decisions to be made to onboard, terminate or restrict the bank relationships to optimize the size and overall risk of the network. This study provides theoretical foundation to detect the components, the removal of which does not affect some key properties of the network such as connectivity and diameter. We find that the correspondent banking networks have a feature we call k-accessibility, which helps to drastically reduce the computational burden required for finding the above mentioned components.