A 1-forest (a maximal pseudoforest), formed by three 1-trees
In graph theory, a pseudoforest is an undirected graph[1] 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[2] attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.[3] Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – their number of edges is linearly bounded in terms of their number of vertices (in fact, they have at most as many edges as they have vertices) – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests. The name "pseudoforest" comes from Picard & Queyranne (1982).
^The kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph.
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...
directed graph in which the cycle lengths have no nontrivial common divisor Pseudoforest, a directed or undirected graph in which every connected component includes...
succeeds if and only if the cuckoo graph for this set of values is a pseudoforest, a graph with at most one cycle in each of its connected components....
is not invertible. Finite endofunctions are equivalent to directed pseudoforests. For sets of size n there are nn endofunctions on the set. Particular...
Linear complementarity problem Max-flow min-cut theorem of networks Pseudoforest Vehicle routing problem Dantzig's simplex algorithm Dantzig–Wolfe decomposition...
definition of a property is also called an invariant of the graph. pseudoforest A pseudoforest is an undirected graph in which each connected component has...
n-node directed pseudoforests. If there are Tn n-node trees, then there are n2Tn trees with two designated nodes. And a pseudoforest may be determined...
and graphs with arboricity k are exactly the (k,k)-sparse graphs. Pseudoforests are exactly the (1,0)-sparse graphs, and the Laman graphs arising in...
Based on Erdős' proof, one can infer that every linear thrackle is a pseudoforest. Every cycle of odd length may be arranged to form a linear thrackle...
reduce the rank by one. In other contexts (such as with the study of pseudoforests) it makes more sense to allow the deletion of a cut-edge, and to allow...
are forbidden minors, the family of graphs obtained is the family of pseudoforests. The full automorphism group of the diamond graph is a group of order...
families of graphs with bounded treewidth include the cactus graphs, pseudoforests, series–parallel graphs, outerplanar graphs, Halin graphs, and Apollonian...
find optimal colorings for certain classes of graphs, including trees, pseudoforests, and crown graphs. Markossian, Gasparian & Reed (1996) define a graph...
slope number. The pseudoarboricity of a graph is the minimum number of pseudoforests into which its edges can be partitioned. Equivalently, it is the maximum...
describe other important families of sparse graphs, including trees, pseudoforests, and graphs of bounded arboricity. Based on this characterization, it...
number 1, with a vertex ordering given by a breadth-first traversal. Pseudoforests and grid graphs also have queue number 1. Outerplanar graphs have queue...
are the edges of G and whose independent sets are the edge sets of pseudoforests of G, that is, the edge sets in which each connected component contains...
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