The effective conductivities b/w boundary nodes and the Dirichlet-to-Neumann map of a network are invariant under the following star-mesh transform and Y-Δ move (a special case of the star-mesh transform in which the center is an interior node and has the degree 3), illustrated by the following drawings from Wikipedia:
Exercise (**). Let d be a diagonal entry of the Kirchhoff matrix K of a network G, corresponding to an interior node. Use the Schur complement formula for the Dirichlet-to-Neumann map to prove the invariance.
The series or parallel connection rules for replacing conductors follow from the invariance property of the Y-Δ move and can be viewed as its special cases, as also erasing an interior spike or a loop.