  We study the category whose objects are graphs of fixed genus and whose
morphisms are contractions. We show that the corresponding contravariant module
categories are Noetherian and we study two families of modules over these
categories. The first takes a graph to a graded piece of the homology of its
unordered configuration space and the second takes a graph to an intersection
homology group whose dimension is given by a Kazhdan-Lusztig coefficient; in
both cases we prove that the module is finitely generated. This allows us to
draw conclusions about torsion in the homology groups of graph configuration
spaces, and about the growth of Betti numbers of graph configuration spaces and
Kazhdan-Lusztig coefficients of graphical matroids. We also explore the
relationship between our category and outer space, which is used in the study
of outer automorphisms of free groups.
