refined enough to determine anything particularly significant about the homology groups of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$. Towards this end, we will apply techniques from discrete Morse theory.\

\[sinkdmt\] There exists a discrete gradient vector field $X$ on the space $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ for which the number of critical $i$-cells of $X$ is bounded by $p_i(n)(i+1)^n$, where $p_i$ is a polynomial of degree $i$.\

For the reader unfamiliar with discrete Morse theory, Section \[dmt\] provides a self-contained introduction to the subject. Put briefly, discrete Morse theory is a means by which one can determine which cells of a CW complex are “critical” to the homotopy type of the space. Similar to classical Morse theory, one defines a map, called in this context a **discrete vector field**, which divides the cells of the space to three categories: redundant, collapsible, and critical. It is then shown that the space will deformation retract onto a complex with $m_i$ $i$-cells, where $m_i$ is the number of critical $i$-cells. Discrete Morse theory was originally developed by Foreman [@Fo1], and has since seen tremendous application in a variety of ways throughout combinatorial topology. Farley and Sabalka, for instance, placed a discrete Morse structure on the so-called unordered configuration spaces of graphs in 