 | ||
In combinatorics and order-theoretic mathematics, a multitree may describe either of two equivalent structures: a directed acyclic graph in which the set of nodes reachable from any node form a tree, or a partially ordered set 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).
Contents
Equivalence between directed acyclic graph and poset definitions
If G is a directed acyclic graph ("DAG") in which the nodes reachable from each vertex form a tree (or equivalently, if G is a directed graph in which there is at most one directed path between any two nodes, in either direction) then the reachability relation in G forms a diamond-free partial order. Conversely, if P is a diamond-free partial order, its transitive reduction forms a DAG in which the successors of any node form a tree.
Diamond-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.
Applications
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. In the context of 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.
Related structures
A polytree, a directed acyclic graph formed by assigning an orientation to each edge of an undirected tree, may be viewed as a special case of a multitree.
The set of all nodes connected to any node u in a multitree forms an arborescence.
The word "multitree" has also been used to refer to a series-parallel partial order, or to other structures formed by combining multiple trees.