Given a set of $n$ vertices and the fact that none of them is of degree greater than $2$, how many distinct such graphs are there?

This is OEIS A003292 Number of 4-line partitions of n decreasing across rows.

a(n) is the number of unlabeled graphs on n nodes whose connected components are a path or a cycle. - Geoffrey Critzer, Nov 28 2011

OEIS gives generating function and references which may contain additional information.

```
G.f.: Product (1 - x^k)^-{c(k)}; c(k) = 1, 1, 2, 2, 2, 2, ....
```