Multitree
Type of graph in mathematics
In combinatorics and order theory, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two vertices, or equivalently in which the subgraph reachable from any vertex induces an undirected tree, or a partially ordered set (poset) that does not have four items a, b, c, and d forming a diamond suborder with a ≤ b ≤ d and a ≤ c ≤ d but with b and c incomparable to each other (also called a diamond-free poset).
In computational complexity theory, multitrees have also been called strongly unambiguous graphs or mangroves; they can be used to model nondeterministic algorithms in which there is at most one computational path connecting any two states.
Multitrees may be used to represent multiple overlapping taxonomies over the same ground set. If a family tree may contain multiple marriages from one family to another, but does not contain marriages between any two blood relatives, then it forms a multitree.
01Equivalence between DAG and poset definitions
In a directed acyclic graph, if there is at most one directed path between any two vertices, or equivalently if the subgraph reachable from any vertex induces an undirected tree, then its reachability relation is a diamond-free partial order. Conversely, in a diamond-free partial order, the transitive reduction identifies a directed acyclic graph in which the subgraph reachable from any vertex induces an undirected tree.
02Diamond-free families
A diamond-free family of sets is a family F of sets whose inclusion ordering forms a diamond-free poset. If D(n) denotes the largest possible diamond-free family of subsets of an n-element set, then it is known that
,
and it is conjectured that the limit is 2.
Sources and credits
This article is adapted from the Wikipedia article “Multitree”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Butterfly multitree.svg by David Eppstein, CC0
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