Cutpoint decoupling and first passage times for random walks on graphs

One approach to the computations for Markov chains, due to Meyer, is to break a problem down into corresponding computations for several related chains involving a smaller number of states. In this spirit, we focus on the mean first passage matrix associated with a random walk on a connected graph, and consider the problem of transforming the computation of that matrix into smaller tasks. We show that this is possible when there is a cutpoint in the graph and provide an explicit formula for the mean first passage matrix when this is the case.