In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.
The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan attribute the naming of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – they have very few edges relative to their number of vertices – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests.
Read more about Pseudoforest: Definitions and Structure, Directed Pseudoforests, Number of Edges, Enumeration, Graphs of Functions, Bicircular Matroid, Forbidden Minors, Algorithms