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In mathematics, 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.[1] Hypertrees have also been called arboreal hypergraphs[2] or tree hypergraphs.[3]

A hypertree (blue vertices and yellow hyperedges) and its host tree (red)

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.[4] 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).[5]

By results of Duchet, Flament and Slater[6] 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.[7]
A hypergraph is a hypertree if and only if its dual hypergraph is alpha-acyclic in the sense of Fagin.[8]

It is possible to recognize hypertrees (as duals of alpha-acyclic hypergraphs) in linear time.[9] 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.[10]

Dually chordal graph, a graph whose maximal cliques form a hypertree

Notes

Brandstädt et al. (1998)
Berge (1989)
McKee & McMorris (1999)
Berge (1989); Voloshin (2002)
Berge (1989); Voloshin (2002)
See, e.g., Brandstädt, Le & Spinrad (1999); McKee & McMorris (1999)
Berge (1989)
Fagin (1983)
Tarjan & Yannakakis (1984).

Brandstädt, Leitert & Rautenbach (2012)

References

Berge, Claude (1989), Hypergraphs: Combinatorics of Finite Sets, North-Holland Mathematical Library, 45, Amsterdam: North Holland, ISBN 0-444-87489-5, MR 1013569.
Brandstädt, Andreas; Dragan, Feodor; Chepoi, Victor; Voloshin, Vitaly (1998), "Dually chordal graphs" (PDF), SIAM Journal on Discrete Mathematics, 11: 437–455, doi:10.1137/s0895480193253415, MR 1628114.
Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X, MR 1686154.
Brandstädt, Andreas; Leitert, Arne; Rautenbach, Dieter (2012), "Efficient dominating and edge dominating sets for graphs and hypergraphs", Algorithms and Computation: 23rd International Symposium, ISAAC 2012, Taipei, Taiwan, December 19-21, 2012, Proceedings, Lecture Notes in Computer Science, 7676, pp. 267–277, arXiv:1207.0953, doi:10.1007/978-3-642-35261-4_30, MR 3065639.
Fagin, Ronald (1983), "Degrees of acyclicity for hypergraphs and relational database schemes", Journal of the ACM, 30: 514–550, doi:10.1145/2402.322390, MR 0709831.
McKee, T.A.; McMorris, F.R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910.
Tarjan, Robert E.; Yannakakis, Mihalis (1984), "Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs" (PDF), SIAM Journal on Computing, 13 (3): 566–579, doi:10.1137/0213035, MR 0749707.
Voloshin, Vitaly (2002), Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, Fields Institute Monographs, 17, Providence, RI: American Mathematical Society, ISBN 0-8218-2812-6, MR 1912135.