Polytree
In graph theory, a polytree is a directed graph with at most one undirected path between any two vertices. In other words, a polytree is a directed acyclic graph (DAG) for which there are no undirected cycles either. Equivalently, a polytree is a directed graph formed by giving a direction to each edge of a forest.
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]
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[edit] Related structures
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.
[edit] Enumeration
The number of distinct polytrees on n unlabeled nodes, for n = 1, 2, 3, ..., is
[edit] Sumner's conjecture
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 that tournaments with 3n − 3 vertices are universal in this sense.[6][7]
[edit] Applications
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.[8]
[edit] References
- ^ a b Rebane, George; Pearl, Judea (1987), "The recovery of causal poly-trees from statistical data", Proceedings of UAI, pp. 222–228, ftp://ftp.cs.ucla.edu/tech-report/198_-reports/870031.pdf.
- ^ 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, MR603363
- ^ Simion, Rodica (1991), "Trees with 1-factors and oriented trees", Discrete Mathematics 88 (1): 93–104, doi:10.1016/0012-365X(91)90061-6, MR1099270.
- ^ 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.
- ^ Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17.
- ^ El Sahili, A. (2004), "Trees in tournaments", Journal of Combinatorial Theory. Series B 92 (1): 183–187, doi:10.1016/j.jctb.2004.04.002, MR2078502
- ^ 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, http://portal.acm.org/citation.cfm?id=338659.
