Cadences are complex structures that have been driving music from the beginning of contrapuntal polyphony until today. Detecting such structures is vital for numerous MIR tasks such as musicological analysis, key detection, or music segmentation. However, automatic cadence detection remains challenging mainly because it involves a combination of high-level musical elements like harmony, voice leading, and rhythm. In this work, we present a graph representation of symbolic scores as an intermediate means to solve the cadence detection task. We approach cadence detection as an imbalanced node classification problem using a Graph Convolutional Network. We obtain results that are roughly on par with the state of the art, and we present a model capable of making predictions at multiple levels of granularity, from individual notes to beats, thanks to the fine-grained, note-by-note representation. Moreover, our experiments suggest that graph convolution can learn non-local features that assist in cadence detection, freeing us from the need of having to devise specialized features that encode non-local context. We argue that this general approach to modeling musical scores and classification tasks has a number of potential advantages, beyond the specific recognition task presented here.