  We employ the mathematical programming approach in conjunction with the graph
theory to study the structure of correspondent banking networks. 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. We prove a number
of fundamental theorems related to k-accessible directed graphs, which should
be also applicable beyond the particular problem of financial networks. The
theoretical findings are verified through the data from a large international
bank.
