In cooperative localization, the aim is to compute the locations in Euclidean space of a set of nodes performing pairwise distance measurements. In cases of lack of measurements, several nodes might have multiple feasible solutions meeting the distance constraints. In this paper, we are interested in identifying the nodes that have a unique solution. By employing a semidefmite programming (SDP) formulation of the problem, it is possible to identify only a portion of the uniquely solvable nodes. To improve the identification of these nodes, we develop an iterative algorithm based on SDP. At each iteration, the objective function of the SDP problem is modified in order to identify additional uniquely solvable nodes. We apply this algorithm to study the statistical occurrence of uniquely solvable nodes in uniformly generated networks, and compare the results with the simple SDP. We also investigate the errors in the computed locations for both methods and a variant of the SDP method augmented by bounding constraints on unobserved distances.