In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.
A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic.
A polytree is an example of an oriented graph.
Every arborescence (a directed rooted tree, i.e. a directed acyclic graph in which there exists a single source node that has a unique path to every other node) is a polytree, but not every polytree is an arborescence. Every polytree is a multitree, a directed acyclic graph in which the subgraph reachable from any node forms a tree.
The reachability relationship among the nodes of a polytree forms a partial order that has order dimension at most three. If the order dimension is three, there must exist a subset of seven elements x, yi, and zi (for i = 0, 1, 2) such that, for each i, either x ≤ yi ≥ zi, or x ≥ yi ≤ zi, with these six inequalities defining the polytree structure on these seven elements.
A fence or zigzag poset is a special case of a polytree in which the underlying tree is a path and the edges have orientations that alternate along the path. The reachability ordering in a polytree has also been called a generalized fence.
The number of distinct polytrees on n unlabeled nodes, for n = 1, 2, 3, ..., is
Sumner's conjecture, named after David Sumner, states that tournaments are universal graphs for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of n.
Polytrees have been used as a graphical model for probabilistic reasoning. If a Bayesian network has the structure of a polytree, then belief propagation may be used to perform inference efficiently on it.
The contour tree of a real-valued function on a vector space is a polytree that describes the level sets of the function. The nodes of the contour tree are the level sets that pass through a critical point of the function and the edges describe contiguous sets of level sets without a critical point. The orientation of an edge is determined by the comparison between the function values on the corresponding two level sets.
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