  We prove a deletion-contraction formula for motivic Feynman rules given by
the classes of the affine graph hypersurface complement in the Grothendieck
ring of varieties. We derive explicit recursions and generating series for
these motivic Feynman rules under the operation of multiplying edges in a graph
and we compare it with similar formulae for the Tutte polynomial of graphs,
both being specializations of the same universal recursive relation. We obtain
similar recursions for graphs that are chains of polygons and for graphs
obtained by replacing an edge by a chain of triangles. We show that the
deletion-contraction relation can be lifted to the level of the category of
mixed motives in the form of a distinguished triangle, similarly to what
happens in categorifications of graph invariants.
