We systematically study the query complexity of learning geodesically convex halfspaces on vertex-labelled graphs. Geodesic convexity is a natural generalisation of Euclidean convexity and allows the definition of convex sets and halfspaces on graphs. We prove upper bounds on the query complexity linear in the treewidth and the minimum hull set size but only logarithmic in the diameter. We show tight lower bounds corresponding to well-established separation axioms. Additionally, we identify the Radon number as a central parameter bounding the query complexity and the VC dimension. While previous bounds typically depend on the cut size of the labelling, all parameters in our bounds can be computed from the unlabelled graph. We empirically compare our proposed approach with other active learning algorithms and provide evidence that ground-truth communities in real-world graphs are often convex.