Polytree
In graph theory, polytree is a name for an oriented tree; that is, a polytree is a directed graph formed by giving a direction to each edge of a tree, forming an orientation of the tree. In other words, it is a directed graph with exactly one directed path between any two vertices; equivalently, it is a directed acyclic graph (DAG) for which there are no undirected cycles and for which the underlying undirected graph is connected.
The name "polytree" was coined by Rebane & Pearl (1987);[1] polytrees have also been referred to as singly connected networks[2] and oriented trees.[3][4]
Contents |
Related structures[edit]
Every directed tree (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 a directed tree. 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.[5]
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.
Enumeration[edit]
The number of distinct polytrees on n unlabeled nodes, for n = 1, 2, 3, ..., is
Sumner's conjecture[edit]
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.[6]
Applications[edit]
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.[1][2]
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.[7]
References[edit]
- ^ a b Rebane, George; Pearl, Judea (1987), "The recovery of causal poly-trees from statistical data", Proceedings of UAI, pp. 222–228.
- ^ a b Kim, J., J. H.; Pearl (1983), "A computational model for causal and diagnostic reasoning in inference engines", Proc. of the Eighth International Joint Conference on Artificial Intelligence, pp. 190–193.
- ^ Harary, Frank; Sumner, David (1980), "The dichromatic number of an oriented tree", Journal of Combinatorics, Information & System Sciences 5 (3): 184–187, MR 603363
- ^ Simion, Rodica (1991), "Trees with 1-factors and oriented trees", Discrete Mathematics 88 (1): 93–104, doi:10.1016/0012-365X(91)90061-6, MR 1099270.
- ^ Trotter, William T., Jr.; Moore, John I., Jr. (1977), "The dimension of planar posets", Journal of Combinatorial Theory, Series B 22 (1): 54–67, doi:10.1016/0095-8956(77)90048-X.
- ^ Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society. Third Series 102 (4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448.
- ^ Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions", Proc. 11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926.
