The problem of classifying graphs with respect to connectivity via partial observations of nodes is posed as a composite hypothesis testing problem with controlled sensing. An observation at a node is a subset of edges incident to the node on the complete graph drawn according to a probability model, which are modeled as conditionally independent given their neighborhoods. Connectivity is measured through average node degree and is classified with respect to a threshold. A simple approximation of the controlled sensing test is derived and simulated on Erdös-Rènyi Model A graphs to characterize error probabilities as a function of expected stopping times. It is shown that the proposed test achieves favorable tradeoffs between the classification error and the number of measurements and further outperforms existing approaches, especially at low target error rates. Furthermore, the proposed test achieves asymptotically optimal error performance, as the error rate goes to zero.