Abrams' stabilization theorem for no-k-equal configuration spaces on graphs

For a graph $G$, let Conf$(G,n)$ denote the classical configuration space of $n$ labelled points in $G$. Abrams introduced a cubical complex, denoted here by DConf$(G,n)$, sitting inside Conf$(G,n)$ as a strong deformation retract provided $G$ is suitably subdivided. Using discrete Morse Theory techniques, we extend Abrams' result to the realm of configurations having no $k$-fold collisions.