I'm working on a project dealing with graphs, specifically planar graphs with connectivity >=2. Trying to randomly generate these graphs with a wide range of vertices, say between 4 and 50. For graphs with fewer than around 9 or 10 vertices it's pretty easy since there are relatively few graphs to choose from, I just have a database of adjacency lists and perform a random lookup. As the vertex count gets higher though that becomes less viable. I want to try and stitch planar graphs together to create a new graph that is still planar.
So let's say I want to generate a planar graph of k >=2 with 24 vertices. I'd like to stitch together two 9-vertex graphs and one 6-vertex and wind up with the 24 vertices. I'm just not sure how to actually implement this while guaranteeing that the resulting graph is still planar.
Unless there's an algorithm to just generate the graphs I'm looking for outright without stitching together smaller ones.