This paper formalizes {\em cross-space} active learning on a graph convolutional network (GCN). The objective is to attain the most accurate hypothesis available in any of the instance spaces generated by the GCN. Subject to the objective, the challenge is to minimize the {\em label cost}, measured in the number of vertices whose labels are requested. Our study covers both {\em budget algorithms} which terminate after a designated number of label requests, and {\em verifiable algorithms} which terminate only after having found an accurate hypothesis. A new separation in label complexity between the two algorithm types is established. The separation is unique to GCNs.