There are several ways in which we can characterize classes of graphs. One such way of classifying graphs is by their brachwidth. In working to characterize the class of graphs with brachwidth at most four beta4 we have found a set of reductions that reduces members of beta4 to the zero graph. We have also computed several planar members of the obstruction set O_beta4 for graphs with branchwidth at most four. This thesis will summarize previous results on branchwidth and reveal the previously mentioned new results.