In this thesis, I use a branch and cut implementation to solve the feedback vertex set problem and add new facet defining inequalities to the literature. Feedback in a system arises when repeated traversal over some cycle causes unwanted growth or decay in the data. This can cause problems in many systems such as excess noise at a concert or deadlock forming cyclical dependence on processes in a group. To eliminate the feedback, I implement a branch and cut algorithm to make the calculations, selecting inequalities based on results from the literature. I report improvements upon previous branch and cut work in a collection of random cases. Along with this implementation, I show that the clique inequalities, which are known to be valid for the feedback vertex problem, are in fact facet inducing for all maximal clique subgraphs of the original network.