TITLE: Number of graphs on a given set of vertices with maximum degree of $2$
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: 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, ....