  In this paper we study the inverse Dirichlet-to-Neumann problem for certain
cylindrical electrical networks. We define and study a birational
transformation acting on cylindrical electrical networks called the electrical
$R$-matrix. We use this transformation to formulate a general conjectural
solution to this inverse problem on the cylinder. This conjecture extends work
of Curtis, Ingerman, and Morrow, and of de Verdi\`ere, Gitler, and Vertigan for
circular planar electrical networks. We show that our conjectural solution
holds for certain "purely cylindrical" networks. Here we apply the grove
combinatorics introduced by Kenyon and Wilson.
