Good evening,
I am new at the MSE, I signed up just now, so I greet you all; please bare with the newcomer.
I have a graph theory problem, which has come up in an entirely different context, a problem in Mathematical Physics. As I am not familiar with graph theory, I decided to post this question, being fully aware that, in my ignorance, I am unable to assess this problem anywhere between trivial and impossible... Here goes:
Consider a set of vertices on the plane; specifically, $N_A$ vertices of valence 4, and $N_B$ vertices of valence 1. How many planar, connected graphs can one construct with the given vertices, up to isomorphism?
Many thanks!