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

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.