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

Deletion and contraction in configuration spaces of graphs