I would like to construct graphs under the following conditions:
- No loops
- Maximum of one edge between any nodes
- Connected
- No intersection between the edges may occur on a plane.
Now, a similar problem minus the condition of non-intersection on the plane results in the Catalan numbers, A000108. However, I wish to find the sequence that expresses $a_n$ as the number of unique graphs as above with n nodes. So far I have $1,1,2,5,13$ [sic]. For these I am sure, but it is daunting to try and draw out all possible graphs of $6$ without me missing any, and I have not yet worked out an algorithm to generate them.
Also, I am sorry if I am not using correct terminology, I have searched for the notation to describe these nodes but I am feeling slightly overwhelmed by Wikipedia's glossary. If someone could express my "rules" with the correct jargon that would make me very happy.
Finally, I have a suspicion that A001519 is the answer I am searching for. If it is indeed the correct solution, then why is it referring to trees when I allow cycles, is it merely coincidence?
You appear to be trying to count connected simple planar graphs, but if so, you’re missing one on four nodes and seven on five nodes: this is OEIS A003094, and the six of them on four nodes are pictured at the first link.