  We study the quasi-isometric rigidity of a large family of finitely generated
groups that split as graphs of groups with virtually free vertex groups and
two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic
relative to virtually abelian subgroups, and has JSJ decomposition over
two-ended subgroups containing only virtually free vertex groups that aren't
quadratically hanging. Our main result is that any group quasi-isometric to $G$
is abstractly commensurable to $G$. In particular, our result applies to
certain "generic" HNN extensions of a free group over cyclic subgroups.
