  Genevois recently classified which graph braid groups on $\ge 3$ strands are
word hyperbolic. In the $3$-strand case, he asked whether all such word
hyperbolic groups are actually free; this reduced to checking two infinite
classes of graphs: sun and pulsar graphs. We prove that $3$-strand braid groups
of sun graphs are free. On the other hand, it was known to experts that
$3$-strand braid groups of most pulsar graphs contain surface subgroups. We
provide a simple proof of this and prove an additional structure theorem for
these groups.
