Is it possible to robustly find all the graph minors within an arbitrary node graph where the pinch points are generally not single nodes? I have read some other posts on here about how to break up your graph into a Hamiltonian cycle and then from that find the graph minors but it seems to be such an algorithm would require that each "room" had "doorways" consisting of single nodes.
To explain a bit more a visual aid is necessary. Lets say the nodes below are an example of the typical node graph. What I am looking for is a way to automatically find the different colored regions of the graph (or graph minors)