Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.