  We provide an explicit construction that allows one to easily decompose a
graph braid group as a graph of groups. This allows us to compute the braid
groups of a wide range of graphs, as well as providing two general criteria for
a graph braid group to split as a non-trivial free product, answering two
questions of Genevois. We also use this to distinguish certain right-angled
Artin groups and graph braid groups. Additionally, we provide an explicit
example of a graph braid group that is relatively hyperbolic, but is not
hyperbolic relative to braid groups of proper subgraphs. This answers another
question of Genevois in the negative.
