Polytree

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.

The term polytree was coined in 1987 by Rebane and Pearl.

Related structures

• An arborescence is 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. Every arborescence is a polytree, but not every polytree is an arborescence.
• A multitree is a directed acyclic graph in which the subgraph reachable from any node forms a tree. Every polytree is a multitree.
• 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 ${\displaystyle x}$, ${\displaystyle y_i}$, and ${\displaystyle z_i}$ (for ${\displaystyle i=0,1,2}$) such that, for each ${\displaystyle i}$, either ${\displaystyle x\le y_i\ge z_i}$ or ${\displaystyle x\ge y_i\le z_i}$, 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.