Presentations of graph braid groups

Let be a graph. The (unlabeled) configuration space of points on is the space of -element subsets of . The -strand braid group of , denoted , is the fundamental group of . This paper applies the methods of discrete Morse theory to the spaces . We describe how to compute presentations for , where is an arbitrary natural number and is an arbitrary finite connected graph. Particular attention is paid to the case , and many examples are given.