Hypertree



In the mathematical field of graph theory, a hypergraph $H$ is called a hypertree if it admits a host graph $T$ such that $T$ is a tree. In other words, $H$ is a hypertree if there exists a tree $T$ such that every hyperedge of $H$ is the set of vertices of a connected subtree of $T$. Hypertrees have also been called arboreal hypergraphs or tree hypergraphs.

Every tree $T$ is itself a hypertree: $T$ itself can be used as the host graph, and every edge of $T$ is a subtree of this host graph. Therefore, hypertrees may be seen as a generalization of the notion of a tree for hypergraphs. They include the connected Berge-acyclic hypergraphs, which have also been used as a (different) generalization of trees for hypergraphs.

Properties
Every hypertree has the Helly property (2-Helly property): if a subset $S$ of its hyperedges has the property that every two hyperedges in $S$ have a nonempty intersection, then $S$ itself has a nonempty intersection (a vertex that belongs to all hyperedges in $S$).

By results of Duchet, Flament and Slater hypertrees may be equivalently characterized in the following ways.
 * A hypergraph $H$ is a hypertree if and only if it has the Helly property and its line graph is a chordal graph.
 * A hypergraph $H$ is a hypertree if and only if its dual hypergraph $H*$ is conformal and the 2-section graph of $H*$ is chordal.
 * A hypergraph is a hypertree if and only if its dual hypergraph is alpha-acyclic in the sense of Fagin.

It is possible to recognize hypertrees (as duals of alpha-acyclic hypergraphs) in linear time. The exact cover problem (finding a set of non-overlapping hyperedges that covers all the vertices) is solvable in polynomial time for hypertrees but remains NP-complete for alpha-acyclic hypergraphs.