author would like to send thanks to Jordan Ellenberg, John Wiltshire-Gordon, Jennifer Wilson and Graham White for various useful conversations during the writing and conception of this work. The author would also like to send very special thanks to Steven Sam for his support during the initial stages of this work. Finally, the author would like to acknowledge the generous support of the NSF via the grant DMS-1502553.\

Preliminary notions
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Graphs and sink configuration spaces
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A **graph** $(G,V,E)$ is a triple of a compact connected CW-complex of dimension one $G$, along with the set of its 0-cells $V$ and its 1-cells $E$. An **isomorphism** of graphs $(G,V,E) \cong (G',V',E')$ is a cellular homeomorphism $\phi:G \rightarrow G'$. A **tree** is a graph $(G,V,E)$ for which $G$ is contractible. A **loop** in a graph $(G,V,E)$ is a 1-cell $e \in E$ whose endpoints are identified in $G$.\

Given a graph $(G,V,E)$, we will often slightly abuse nomenclature and refer to $G$ as being a graph. While this distinction is unimportant in many instances, it turns out to be quite relevant in the study of sink configuration spaces (see Definition \[sinkdef\]). In the coming sections we will find that given a 