# Recovering Conductances of Resistor Networks in a Punctured Disk

## TL;DR

This paper extends the theory of reconstructing resistor network conductances from response matrices to networks on a punctured disk, introducing new local moves, medial graph properties, and broader recoverability conditions.

## Contribution

It generalizes existing results for circular planar networks to those on a punctured disk, including new local moves, medial graph analysis, and broader classes of recoverable networks.

## Key findings

- Networks on a punctured disk are recoverable from response matrices.
- Introduction of $z$-sequences for medial graphs.
- Necessary conditions for network recoverability.

## Abstract

The response matrix of a resistor network is the linear map from the potential at the boundary vertices to the net current at the boundary vertices. For circular planar resistor networks, Curtis, Ingerman, and Morrow have given a necessary and sufficient condition for recovering the conductance of each edge in the network uniquely from the response matrix using local moves and medial graphs. We generalize their results for resistor networks on a punctured disk. First we discuss additional local moves that occur in our setting, prove several results about medial graphs of resistor networks on a punctured disk, and define the notion of $z$-sequences for such graphs. We then define certain circular planar graphs that are electrically equivalent to standard graphs and turn them into networks on a punctured disk by adding a boundary vertex in the middle. We prove such networks are recoverable and are able to generalize this result to a much broader family of networks. A necessary condition for recoverability is also introduced.