Topology Reconstruction of a Circular Planar Resistor Network

The reconstruction problem is an inverse problem and, in general, has no unique solution. We consider the problem of reconstructing all possible topologies with their edge resistance values of an unknown circular planar passive resistive network, whose response matrix is known apriori. The response matrix is used to deduce a set of all possible connections (such as 1-connection, 2-connection, $\ldots$ k-connection [6]) in the unknown circular planar network. Since number of such connections are large, a reduced set of connections is derived. For each connection in reduced connection set, we generate all possible path permutations along with there graph representations. The graph representations of all path permutations are used to generate several candidate planar graphs using union and edge deletion operations. A method is proposed wherein, the candidate planar graphs are posed as a set of non-linear multivariate polynomials. We then use the Gröbner basis to simultaneously reconstruct all possible topologies and the edge resistance values of an electrical network enclosed inside a black box. Numerical simulation establishes the effectiveness of the proposed strategy.