Author : Sloane, N. J. A
Book or Source : Sequence A134964 in "The On-Line Encyclopedia of Integer Sequences."
page : ...
A pseudoforest is an undirected graph in which every connected component contains at most one graph cycle. A pseudotree is therefore a connected pseudoforest and a forest (i.e., not-necessarily-connected acyclic graph) is a trivial pseudoforest.
Some care is needed when encountering pseudoforests as some authors use the term to mean "a pseudoforest that is not a forest."
The numbers of pseudoforests on 1, 2, 3, ... vertices are 1, 2, 4, 9, 19, 46, 108, 273 ... (OEIS A134964), the first few of which are illustrated above.
Sloane, N. J. A. Sequence A134964 in "The On-Line Encyclopedia of Integer Sequences."