single cycle within them are called 1-trees or unicyclic graphs. That is, a 1-tree is a connected graph containing exactly one cycle. A pseudoforest with a single connected component (usually called a pseudotree, although some authors define a pseudotree to be a 1-tree) is either a tree or a 1-tree; in general a pseudoforest may have multiple connected components as long as all of them are trees or 1-trees. If one removes from a 1-tree one of the edges in its cycle, the result is a tree. Reversing this process, if one augments a tree by connecting any two of its vertices by a new edge, the result is a 1-tree; the path in the tree connecting the two endpoints of the added edge, together with the added edge itself, form the 1-tree's unique cycle. If one augments a 1-tree by adding an edge that connects one of its vertices to a newly added vertex, the result is again a 1-tree, with one more vertex; an alternative method for constructing 1-trees is to start with a single cycle and then repeat this augmentation operation any number of times. The edges of any 1-tree can be partitioned in a unique way into two subgraphs, one of which is a cycle and the other of which is a forest, such that each tree of the forest contains exactly one vertex of the cycle. Certain more specific types of pseudoforests have also been studied. A 1-forest, sometimes called a maximal pseudoforest, is a pseudoforest to which no more edges can be added without causing some component of the graph to contain multiple cycles. If a pseudoforest contains a tree as one of its components, it cannot be a 1-forest, for one can add either an edge connecting two vertices within that tree, forming a single cycle, or an edge connecting that tree to some other component. Thus, the 1-forests are exactly the pseudoforests in which every component is a 1-tree. The spanning pseudoforests of an undirected graph G are the pseudoforest subgraphs of G that have all the vertices of G. Such a pseudoforest need not have any edges, since for example the subgraph that has all the vertices of G and no edges is a pseudoforest (whose components are trees consisting of a single vertex). The maximal pseudoforests of G are the pseudoforest subgraphs of G that are not contained within any