I'd like to know if Inclusion/Exclusion could be useful to count all the possible graphs that can be drawn with n vertices and no vertex having 2 or more lines attached, and how to apply it.
Would it be too naive to expect a nice-looking formula out of this?. Maybe... we need Polya counting?, maybe another theorem to solve it?, something involving complements to simplify the question?. I'd appreciate to see your thoughts on this, guys.
EDIT: I'm considering graphs with general types of symmetries (not just $S_n$).
If no vertex has $2$ or more lines attached, then every vertex has at most one line. Then every vertex is either isolated, having no edges, or it is in a pair. If the vertices are unlabeled then this comes down to writing $n$ as a sum of $1$'s and $2$'s, so there are $\lfloor\tfrac n2\rfloor+1$ such graphs on $n$ vertices.