  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.
