On 2D Inverse Problems/Y-Δ and star-mesh transforms

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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:

Star-mesh transform

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 \Lambda_G = K/C = (K/d)/(C/d) 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.