  The aim of this article is to provide space level maps between configuration
spaces of graphs that are predicted by algebraic manipulations of cellular
chains. More explicitly, we consider edge contraction and half-edge deletion,
and identify the homotopy cofibers in terms of configuration spaces of simpler
graphs. The construction's main benefit lies in making the operations
functorial - in particular, graph minors give rise to compatible maps at the
level of fundamental groups as well as generalized (co)homology theories.
  As applications we provide a long exact sequence for half-edge deletion in
any generalized cohomology theory, compatible with cohomology operations such
as the Steenrod and Adams operations, allowing for inductive calculations in
this general context. We also show that the generalized homology of unordered
configuration spaces is finitely generated as a representation of the opposite
graph minor category.
