# Counting number of graphs of size $n$ with a given graph as a minor

I'm interested in either a formula or algorithm for figuring out the number of graphs of size $n$ which have a given graph as a minor. Are there any results regarding this?

• I'm sure you've thought of the obvious algorithm with horrific time complexity. I'm sure one could do better than that, but I would bet not much better. – Matt Samuel Feb 16 '16 at 2:20
• Yes. I'd be disappointed if the only algorithm was "enumerate all the graphs of size n", as it doesn't give any insight into the problem. – Davis Yoshida Feb 16 '16 at 2:27
• Mathoverflow might be a better place to ask this. I think this qualifies as research level combinatorics. – Matt Samuel Feb 16 '16 at 2:28
• I'm not really a research level mathematician, so I don't usually post there. – Davis Yoshida Feb 16 '16 at 2:31