The PELAGIOS Graph Explorer HTTP API exposes the data from the visualization in JSON format, with Place geometry encoded as GeoJSON. Since the Graph Explorer runs entirely on the API internally, all of the operations you can see in the visualization are also available through the API - from place search to finding intersections between datasets, to getting the source data references.
The design of the API has been somewhat ad hoc: the approach was to be pragmatic, quick, and build in the essentials needed for the visuals, no more, no less. If you are re-using it for other purposes which we haven't anticipated (different types of visualizations, analyses, etc.) you are likely to encounter situations where you lack certain features, would have expected things to be named or organized differently, or miss a bit of documentation here and there, I assume. But bear with us - it's an alpha version & this is exactly the kind of feedback that's valuable for us! We're excited about anyone trying out things with the API we haven't thought of!
Visit our Wiki to learn the basics and see some live examples.
P.S.: An online demo of the PELAGIOS Graph Explorer is available here. Screencasts explaining the basic usage are in this blogpost: The PELAGIOS Graph Explorer: A First Look