We study the problem of actively learning the vertex labels of a graph, assuming the classes form geodesically convex subgraphs, which is related to linear separability in the Euclidean setting. The main result of this paper is a novel query-efficient active learning algorithm with label-independent upper bounds on the number of queries needed to learn all labels. For that, we use shortest path covers and provide a logarithmic approximation for the sub-problem of computing a shortest path cover

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... of minimum size. We extend the approach to arbitrarily labeled graphs using a convexity-based selection criterion. Finally, we discuss whether the convexity assumption holds on real-world data and give some first preliminary results on citation and image benchmark datasets.