  Paolo Aluffi, inspired by an algebro-geometric problem, asked when the
Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff
polynomial of the same graph with one edge deleted.
  We give some results on which graph-edge pairs have this property. In
particular we show that multiple edges can be reduced to double edges, we
characterize which edges of wheel graphs satisfy the property, we consider a
stronger condition which guarantees the property for any parallel join, and we
find a class of series-parallel graphs with the property.
